/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 3 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (10) CdtProblem
    (11) CdtKnowledgeProof [FINISHED, 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 876 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

S is empty.
Rewrite Strategy: FULL
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(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   g(0, f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0)
   g(s(x), y) -> g(f(x, y), 0)
   g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   G_2

Compound Symbols:   c, c1_1, c2_1, c3_2


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(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   G(z0, s(z1)) -> c1(G(f(z0, z1), 0))
Removed 1 trailing nodes:
   G(0, f(z0, z0)) -> c

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))
Tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
S tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:   g_2

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1, c3_2


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g(0, f(z0, z0)) -> z0
   g(z0, s(z1)) -> g(f(z0, z1), 0)
   g(s(z0), z1) -> g(f(z0, z1), 0)
   g(f(z0, z1), 0) -> f(g(z0, 0), g(z1, 0))

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
S tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1, c3_2


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
We considered the (Usable) Rules:none
And the Tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(G(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
S tuples:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
K tuples:
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1, c3_2


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(11) CdtKnowledgeProof (FINISHED)
The following tuples could be moved from S to K by knowledge propagation:
   G(s(z0), z1) -> c2(G(f(z0, z1), 0))
   G(f(z0, z1), 0) -> c3(G(z0, 0), G(z1, 0))
Now S is empty
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(12)
BOUNDS(1, 1)

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(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   g(0', f(x, x)) -> x
   g(x, s(y)) -> g(f(x, y), 0')
   g(s(x), y) -> g(f(x, y), 0')
   g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))

S is empty.
Rewrite Strategy: FULL
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(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(16)
Obligation:
TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))

Types:
g :: 0':f:s -> 0':f:s -> 0':f:s
0' :: 0':f:s
f :: 0':f:s -> 0':f:s -> 0':f:s
s :: 0':f:s -> 0':f:s
hole_0':f:s1_0 :: 0':f:s
gen_0':f:s2_0 :: Nat -> 0':f:s

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(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g
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(18)
Obligation:
TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))

Types:
g :: 0':f:s -> 0':f:s -> 0':f:s
0' :: 0':f:s
f :: 0':f:s -> 0':f:s -> 0':f:s
s :: 0':f:s -> 0':f:s
hole_0':f:s1_0 :: 0':f:s
gen_0':f:s2_0 :: Nat -> 0':f:s


Generator Equations:
gen_0':f:s2_0(0) <=> 0'
gen_0':f:s2_0(+(x, 1)) <=> f(0', gen_0':f:s2_0(x))


The following defined symbols remain to be analysed:
g
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':f:s2_0(+(1, n4_0)), gen_0':f:s2_0(0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
g(gen_0':f:s2_0(+(1, 0)), gen_0':f:s2_0(0))

Induction Step:
g(gen_0':f:s2_0(+(1, +(n4_0, 1))), gen_0':f:s2_0(0)) ->_R^Omega(1)
f(g(0', 0'), g(gen_0':f:s2_0(+(1, n4_0)), 0')) ->_IH
f(g(0', 0'), *3_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
g(0', f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0')
g(s(x), y) -> g(f(x, y), 0')
g(f(x, y), 0') -> f(g(x, 0'), g(y, 0'))

Types:
g :: 0':f:s -> 0':f:s -> 0':f:s
0' :: 0':f:s
f :: 0':f:s -> 0':f:s -> 0':f:s
s :: 0':f:s -> 0':f:s
hole_0':f:s1_0 :: 0':f:s
gen_0':f:s2_0 :: Nat -> 0':f:s


Generator Equations:
gen_0':f:s2_0(0) <=> 0'
gen_0':f:s2_0(+(x, 1)) <=> f(0', gen_0':f:s2_0(x))


The following defined symbols remain to be analysed:
g
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(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(22)
BOUNDS(n^1, INF)