/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^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 34 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 16 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 206 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) 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^2).


The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
S tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, IF_3, G_2

Compound Symbols:   c1, c2, c3_1, c4, c5, c6_1, c7_4


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   IF(false, s(z0), s(z1)) -> c5
   F(1) -> c2
   F(0) -> c1
   IF(true, s(z0), s(z1)) -> c4

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_4


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_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(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = 0
   POL(G(x_1, x_2)) = [2]x_2
   POL(c(x_1)) = [3] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [3] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0)) -> c3(F(z0))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = [2] + x_1
   POL(G(x_1, x_2)) = [2]x_2^2 + x_1*x_2
   POL(c(x_1)) = [2] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [2] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:none
K tuples:
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
   F(s(z0)) -> c3(F(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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

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

(14)
BOUNDS(1, 1)

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

(16)
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(0') -> true
   f(1') -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c

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

(19) 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

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(20)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c


Generator Equations:
gen_0':1':s:c3_0(0) <=> 0'
gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
f(gen_0':1':s:c3_0(0)) ->_R^Omega(1)
true

Induction Step:
f(gen_0':1':s:c3_0(+(n5_0, 1))) ->_R^Omega(1)
f(gen_0':1':s:c3_0(n5_0)) ->_IH
true

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

(22)
Complex Obligation (BEST)

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

(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c


Generator Equations:
gen_0':1':s:c3_0(0) <=> 0'
gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))


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

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

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
f(0') -> true
f(1') -> false
f(s(x)) -> f(x)
if(true, s(x), s(y)) -> s(x)
if(false, s(x), s(y)) -> s(y)
g(x, c(y)) -> c(g(x, y))
g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

Types:
f :: 0':1':s:c -> true:false
0' :: 0':1':s:c
true :: true:false
1' :: 0':1':s:c
false :: true:false
s :: 0':1':s:c -> 0':1':s:c
if :: true:false -> 0':1':s:c -> 0':1':s:c -> 0':1':s:c
g :: 0':1':s:c -> 0':1':s:c -> 0':1':s:c
c :: 0':1':s:c -> 0':1':s:c
hole_true:false1_0 :: true:false
hole_0':1':s:c2_0 :: 0':1':s:c
gen_0':1':s:c3_0 :: Nat -> 0':1':s:c


Lemmas:
f(gen_0':1':s:c3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':1':s:c3_0(0) <=> 0'
gen_0':1':s:c3_0(+(x, 1)) <=> s(gen_0':1':s:c3_0(x))


The following defined symbols remain to be analysed:
g