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


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WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 61 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 210 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(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), 256 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) BOUNDS(n^3, INF)


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y)
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y))
   mul0(Nil, y) -> Nil
   add0(Nil, y) -> y
   goal(xs, ys) -> mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2, GOAL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
Removed 2 trailing nodes:
   MUL0(Nil, z0) -> c1
   ADD0(Nil, z0) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> mul0(z0, z1)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(MUL0(x_1, x_2)) = x_1
   POL(Nil) = [1]
   POL(S) = [1]
   POL(add0(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(mul0(x_1, x_2)) = [1] + x_1 + x_2

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
K tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
We considered the (Usable) Rules:
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
And the Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = x_1
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(MUL0(x_1, x_2)) = x_1^2*x_2 + x_1*x_2^2
   POL(Nil) = 0
   POL(S) = 0
   POL(add0(x_1, x_2)) = x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(mul0(x_1, x_2)) = x_2 + x_1*x_2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
S tuples:none
K tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set 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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y)
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y))
   mul0(Nil, y) -> Nil
   add0(Nil, y) -> y
   goal(xs, ys) -> mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST
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(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
   add0(Cons(xs), y) -> add0(xs, Cons(y))
   mul0(Nil, y) -> Nil
   add0(Nil, y) -> y
   goal(xs, ys) -> mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST
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(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(18)
Obligation:
Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil

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(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mul0, add0

They will be analysed ascendingly in the following order:
add0 < mul0

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(20)
Obligation:
Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
add0, mul0

They will be analysed ascendingly in the following order:
add0 < mul0

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(+(n4_1, b)), rt in Omega(1 + n4_1)

Induction Base:
add0(gen_Cons:Nil2_1(0), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
gen_Cons:Nil2_1(b)

Induction Step:
add0(gen_Cons:Nil2_1(+(n4_1, 1)), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
add0(gen_Cons:Nil2_1(n4_1), Cons(gen_Cons:Nil2_1(b))) ->_IH
gen_Cons:Nil2_1(+(+(b, 1), c5_1))

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

(22)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
add0, mul0

They will be analysed ascendingly in the following order:
add0 < mul0

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(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(25)
BOUNDS(n^1, INF)

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(26)
Obligation:
Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(+(n4_1, b)), rt in Omega(1 + n4_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
mul0
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mul0(gen_Cons:Nil2_1(n457_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(*(n457_1, b)), rt in Omega(1 + b*n457_1^2 + n457_1)

Induction Base:
mul0(gen_Cons:Nil2_1(0), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
Nil

Induction Step:
mul0(gen_Cons:Nil2_1(+(n457_1, 1)), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
add0(mul0(gen_Cons:Nil2_1(n457_1), gen_Cons:Nil2_1(b)), gen_Cons:Nil2_1(b)) ->_IH
add0(gen_Cons:Nil2_1(*(c458_1, b)), gen_Cons:Nil2_1(b)) ->_L^Omega(1 + b*n457_1)
gen_Cons:Nil2_1(+(*(n457_1, b), b))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(28)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(+(n4_1, b)), rt in Omega(1 + n4_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
mul0
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(29) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(30)
BOUNDS(n^3, INF)