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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 2 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 176 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), 239 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)


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

(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(C(x, y), y') -> add0(mul0(y, y'), y')
   mul0(Z, y) -> Z
   add0(C(x, y), y') -> add0(y, C(S, y'))
   add0(Z, y) -> y
   second(C(x, y)) -> y
   isZero(C(x, y)) -> False
   isZero(Z) -> True
   goal(xs, ys) -> mul0(xs, ys)

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Z, z0) -> Z
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   add0(Z, z0) -> z0
   second(C(z0, z1)) -> z1
   isZero(C(z0, z1)) -> False
   isZero(Z) -> True
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Z, z0) -> c1
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
   ADD0(Z, z0) -> c3
   SECOND(C(z0, z1)) -> c4
   ISZERO(C(z0, z1)) -> c5
   ISZERO(Z) -> c6
   GOAL(z0, z1) -> c7(MUL0(z0, z1))
S tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Z, z0) -> c1
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
   ADD0(Z, z0) -> c3
   SECOND(C(z0, z1)) -> c4
   ISZERO(C(z0, z1)) -> c5
   ISZERO(Z) -> c6
   GOAL(z0, z1) -> c7(MUL0(z0, z1))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, second_1, isZero_1, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2, SECOND_1, ISZERO_1, GOAL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4, c5, c6, c7_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c7(MUL0(z0, z1))
Removed 5 trailing nodes:
   ISZERO(C(z0, z1)) -> c5
   SECOND(C(z0, z1)) -> c4
   ADD0(Z, z0) -> c3
   MUL0(Z, z0) -> c1
   ISZERO(Z) -> c6

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Z, z0) -> Z
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   add0(Z, z0) -> z0
   second(C(z0, z1)) -> z1
   isZero(C(z0, z1)) -> False
   isZero(Z) -> True
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
S tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, second_1, isZero_1, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Z, z0) -> Z
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   add0(Z, z0) -> z0
Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
S tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(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


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

(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(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = 0
   POL(C(x_1, x_2)) = [1] + x_1 + x_2
   POL(MUL0(x_1, x_2)) = x_1
   POL(S) = [1]
   POL(Z) = [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(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Z, z0) -> Z
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   add0(Z, z0) -> z0
Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
S tuples:
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
K tuples:
   MUL0(C(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


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

(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(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
We considered the (Usable) Rules:
   mul0(Z, z0) -> Z
   add0(Z, z0) -> z0
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   mul0(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
And the Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = x_1
   POL(C(x_1, x_2)) = [1] + x_2
   POL(MUL0(x_1, x_2)) = x_1^2*x_2 + x_1*x_2^2
   POL(S) = 0
   POL(Z) = 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(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Z, z0) -> Z
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   add0(Z, z0) -> z0
Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
S tuples:none
K tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
Defined Rule Symbols:   mul0_2, add0_2

Defined Pair Symbols:   MUL0_2, ADD0_2

Compound Symbols:   c_2, c2_1


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

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

(12)
BOUNDS(1, 1)

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

(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(C(x, y), y') -> add0(mul0(y, y'), y')
   mul0(Z, y) -> Z
   add0(C(x, y), y') -> add0(y, C(S, y'))
   add0(Z, y) -> y
   second(C(x, y)) -> y
   isZero(C(x, y)) -> False
   isZero(Z) -> True
   goal(xs, ys) -> mul0(xs, ys)

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

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
C/0

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

(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(C(y), y') -> add0(mul0(y, y'), y')
   mul0(Z, y) -> Z
   add0(C(y), y') -> add0(y, C(y'))
   add0(Z, y) -> y
   second(C(y)) -> y
   isZero(C(y)) -> False
   isZero(Z) -> True
   goal(xs, ys) -> mul0(xs, ys)

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

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

(18)
Obligation:
Innermost TRS:
Rules:
mul0(C(y), y') -> add0(mul0(y, y'), y')
mul0(Z, y) -> Z
add0(C(y), y') -> add0(y, C(y'))
add0(Z, y) -> y
second(C(y)) -> y
isZero(C(y)) -> False
isZero(Z) -> True
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat -> C:Z

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

(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

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

(20)
Obligation:
Innermost TRS:
Rules:
mul0(C(y), y') -> add0(mul0(y, y'), y')
mul0(Z, y) -> Z
add0(C(y), y') -> add0(y, C(y'))
add0(Z, y) -> y
second(C(y)) -> y
isZero(C(y)) -> False
isZero(Z) -> True
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat -> C:Z


Generator Equations:
gen_C:Z3_1(0) <=> Z
gen_C:Z3_1(+(x, 1)) <=> C(gen_C:Z3_1(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) -> gen_C:Z3_1(+(n5_1, b)), rt in Omega(1 + n5_1)

Induction Base:
add0(gen_C:Z3_1(0), gen_C:Z3_1(b)) ->_R^Omega(1)
gen_C:Z3_1(b)

Induction Step:
add0(gen_C:Z3_1(+(n5_1, 1)), gen_C:Z3_1(b)) ->_R^Omega(1)
add0(gen_C:Z3_1(n5_1), C(gen_C:Z3_1(b))) ->_IH
gen_C:Z3_1(+(+(b, 1), c6_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)

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

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

Innermost TRS:
Rules:
mul0(C(y), y') -> add0(mul0(y, y'), y')
mul0(Z, y) -> Z
add0(C(y), y') -> add0(y, C(y'))
add0(Z, y) -> y
second(C(y)) -> y
isZero(C(y)) -> False
isZero(Z) -> True
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat -> C:Z


Generator Equations:
gen_C:Z3_1(0) <=> Z
gen_C:Z3_1(+(x, 1)) <=> C(gen_C:Z3_1(x))


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

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

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

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

(25)
BOUNDS(n^1, INF)

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

(26)
Obligation:
Innermost TRS:
Rules:
mul0(C(y), y') -> add0(mul0(y, y'), y')
mul0(Z, y) -> Z
add0(C(y), y') -> add0(y, C(y'))
add0(Z, y) -> y
second(C(y)) -> y
isZero(C(y)) -> False
isZero(Z) -> True
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat -> C:Z


Lemmas:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) -> gen_C:Z3_1(+(n5_1, b)), rt in Omega(1 + n5_1)


Generator Equations:
gen_C:Z3_1(0) <=> Z
gen_C:Z3_1(+(x, 1)) <=> C(gen_C:Z3_1(x))


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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mul0(gen_C:Z3_1(n523_1), gen_C:Z3_1(b)) -> gen_C:Z3_1(*(n523_1, b)), rt in Omega(1 + b*n523_1^2 + n523_1)

Induction Base:
mul0(gen_C:Z3_1(0), gen_C:Z3_1(b)) ->_R^Omega(1)
Z

Induction Step:
mul0(gen_C:Z3_1(+(n523_1, 1)), gen_C:Z3_1(b)) ->_R^Omega(1)
add0(mul0(gen_C:Z3_1(n523_1), gen_C:Z3_1(b)), gen_C:Z3_1(b)) ->_IH
add0(gen_C:Z3_1(*(c524_1, b)), gen_C:Z3_1(b)) ->_L^Omega(1 + b*n523_1)
gen_C:Z3_1(+(*(n523_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(C(y), y') -> add0(mul0(y, y'), y')
mul0(Z, y) -> Z
add0(C(y), y') -> add0(y, C(y'))
add0(Z, y) -> y
second(C(y)) -> y
isZero(C(y)) -> False
isZero(Z) -> True
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z
hole_C:Z1_1 :: C:Z
hole_False:True2_1 :: False:True
gen_C:Z3_1 :: Nat -> C:Z


Lemmas:
add0(gen_C:Z3_1(n5_1), gen_C:Z3_1(b)) -> gen_C:Z3_1(+(n5_1, b)), rt in Omega(1 + n5_1)


Generator Equations:
gen_C:Z3_1(0) <=> Z
gen_C:Z3_1(+(x, 1)) <=> C(gen_C:Z3_1(x))


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

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

(30)
BOUNDS(n^3, INF)