/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms]
    (8) CdtProblem
    (9) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (16) CdtProblem
    (17) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 3 ms]
    (20) CdtProblem
    (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) BOUNDS(1, 1)
(23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 0 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 312 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (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, 0) -> s(0)
   f(s(x), s(y)) -> s(f(x, y))
   g(0, x) -> g(f(x, x), x)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c, c1_1, c2_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(z0, 0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c1_1, c2_2


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

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c1_1, c2_2


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
We considered the (Usable) Rules:
   f(s(z0), s(z1)) -> s(f(z0, z1))
   f(z0, 0) -> s(0)
And the Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(F(x_1, x_2)) = 0
   POL(G(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2)) = 0
   POL(s(x_1)) = 0

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
K tuples:
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c1_1, c2_2


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

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0)) by 
   G(0, 0) -> c2(G(s(0), 0), F(0, 0))
   G(0, s(z0)) -> c2(G(s(f(z0, z0)), s(z0)), F(s(z0), s(z0)))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, 0) -> c2(G(s(0), 0), F(0, 0))
   G(0, s(z0)) -> c2(G(s(f(z0, z0)), s(z0)), F(s(z0), s(z0)))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
K tuples:
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c1_1, c2_2


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

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(0, 0) -> c2(G(s(0), 0), F(0, 0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, s(z0)) -> c2(G(s(f(z0, z0)), s(z0)), F(s(z0), s(z0)))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c1_1, c2_2


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

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, s(z0)) -> c2(F(s(z0), s(z0)))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c1_1, c2_1


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

(15) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   G(0, s(z0)) -> c2(F(s(z0), s(z0)))

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1


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

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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
S tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), s(z1)) -> c1(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
S tuples:none
K tuples:
   F(s(z0), s(z1)) -> c1(F(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1


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

(21) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(22)
BOUNDS(1, 1)

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

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

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(25) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(26)
Obligation:
Innermost TRS:
Rules:
f(x, 0') -> s(0')
f(s(x), s(y)) -> s(f(x, y))
g(0', x) -> g(f(x, x), x)

Types:
f :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
g :: 0':s -> 0':s -> g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat -> 0':s

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

(27) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, g

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

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

(28)
Obligation:
Innermost TRS:
Rules:
f(x, 0') -> s(0')
f(s(x), s(y)) -> s(f(x, y))
g(0', x) -> g(f(x, x), x)

Types:
f :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
g :: 0':s -> 0':s -> g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

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

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(1, n5_0)), rt in Omega(1 + n5_0)

Induction Base:
f(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
s(0')

Induction Step:
f(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
s(f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) ->_IH
s(gen_0':s3_0(+(1, c6_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(30)
Complex Obligation (BEST)

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

(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
f :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
g :: 0':s -> 0':s -> g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


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

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

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

(32) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(33)
BOUNDS(n^1, INF)

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

(34)
Obligation:
Innermost TRS:
Rules:
f(x, 0') -> s(0')
f(s(x), s(y)) -> s(f(x, y))
g(0', x) -> g(f(x, x), x)

Types:
f :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
g :: 0':s -> 0':s -> g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(1, n5_0)), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
g