WORST_CASE(Omega(n^3),O(n^3))
proof of input_rnhkVEf5hU.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 245 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 82 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 189 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 492 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 579 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^3)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 16 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 325 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 145 ms]
            (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 113 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 2157 ms]
            (68) proven lower bound
            (69) LowerBoundPropagationProof [FINISHED, 0 ms]
            (70) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   mul0(C(x, y), y') -> add0(mul0(y, y'), y') [1]
   mul0(Z, y) -> Z [1]
   add0(C(x, y), y') -> add0(y, C(S, y')) [1]
   add0(Z, y) -> y [1]
   second(C(x, y)) -> y [1]
   isZero(C(x, y)) -> False [1]
   isZero(Z) -> True [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   mul0(C(x, y), y') -> add0(mul0(y, y'), y') [1]
   mul0(Z, y) -> Z [1]
   add0(C(x, y), y') -> add0(y, C(S, y')) [1]
   add0(Z, y) -> y [1]
   second(C(x, y)) -> y [1]
   isZero(C(x, y)) -> False [1]
   isZero(Z) -> True [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

The TRS has the following type information:
mul0 :: C:Z -> C:Z -> C:Z
C :: S -> C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
S :: S
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   second_1
   isZero_1
   goal_2

(c) The following functions are completely defined:

   mul0_2
   add0_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   mul0(C(x, y), y') -> add0(mul0(y, y'), y') [1]
   mul0(Z, y) -> Z [1]
   add0(C(x, y), y') -> add0(y, C(S, y')) [1]
   add0(Z, y) -> y [1]
   second(C(x, y)) -> y [1]
   isZero(C(x, y)) -> False [1]
   isZero(Z) -> True [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

The TRS has the following type information:
mul0 :: C:Z -> C:Z -> C:Z
C :: S -> C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
S :: S
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z

Rewrite Strategy: INNERMOST
----------------------------------------

(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   mul0(C(x, C(x', y'')), y') -> add0(add0(mul0(y'', y'), y'), y') [2]
   mul0(C(x, Z), y') -> add0(Z, y') [2]
   mul0(Z, y) -> Z [1]
   add0(C(x, y), y') -> add0(y, C(S, y')) [1]
   add0(Z, y) -> y [1]
   second(C(x, y)) -> y [1]
   isZero(C(x, y)) -> False [1]
   isZero(Z) -> True [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

The TRS has the following type information:
mul0 :: C:Z -> C:Z -> C:Z
C :: S -> C:Z -> C:Z
add0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
S :: S
second :: C:Z -> C:Z
isZero :: C:Z -> False:True
False :: False:True
True :: False:True
goal :: C:Z -> C:Z -> C:Z

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Z => 0
   S => 0
   False => 0
   True => 1

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   add0(z, z') -{ 1 }-> add0(y, 1 + 0 + y') :|: z' = y', z = 1 + x + y, x >= 0, y >= 0, y' >= 0
   goal(z, z') -{ 1 }-> mul0(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', y'), y'), y') :|: z' = y', z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, y' >= 0
   mul0(z, z') -{ 2 }-> add0(0, y') :|: z' = y', x >= 0, y' >= 0, z = 1 + x + 0
   mul0(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0


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

(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(y, 1 + 0 + z') :|: z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0


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

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { add0 }
   { isZero }
   { second }
   { mul0 }
   { goal }

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(y, 1 + 0 + z') :|: z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {add0}, {isZero}, {second}, {mul0}, {goal}

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

(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(y, 1 + 0 + z') :|: z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {add0}, {isZero}, {second}, {mul0}, {goal}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(y, 1 + 0 + z') :|: z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {add0}, {isZero}, {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: ?, size: O(n^1) [z + z']

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(y, 1 + 0 + z') :|: z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {isZero}, {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {isZero}, {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: isZero
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {isZero}, {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isZero
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]

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

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: second
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {second}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: second
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]

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

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: mul0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z*z' + z'

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]
  mul0: runtime: ?, size: O(n^2) [2*z*z' + z']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mul0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4 + 4*z + 3*z*z' + 4*z^2*z'

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(y'', z'), z'), z') :|: z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']

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

(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 8 + s'' + s1 + 4*y'' + 3*y''*z' + 4*y''^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * y'') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z*z' + z'

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 8 + s'' + s1 + 4*y'' + 3*y''*z' + 4*y''^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * y'') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']
  goal: runtime: ?, size: O(n^2) [2*z*z' + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 5 + 4*z + 3*z*z' + 4*z^2*z'

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + y }-> s' :|: s' >= 0, s' <= y + (1 + 0 + z'), z = 1 + x + y, x >= 0, y >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0
   isZero(z) -{ 1 }-> 1 :|: z = 0
   isZero(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 8 + s'' + s1 + 4*y'' + 3*y''*z' + 4*y''^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * y'') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z = 1 + x + (1 + x' + y''), x >= 0, x' >= 0, y'' >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   second(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  isZero: runtime: O(1) [1], size: O(1) [1]
  second: runtime: O(1) [1], size: O(n^1) [z]
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']
  goal: runtime: O(n^3) [5 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']

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

(45) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(46)
BOUNDS(1, n^3)

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

(47) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(48)
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


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

(49) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(50)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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)

Rewrite Strategy: INNERMOST
----------------------------------------

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

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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)

Rewrite Strategy: INNERMOST
----------------------------------------

(53) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(54)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MUL0, ADD0, mul0, add0

They will be analysed ascendingly in the following order:
ADD0 < MUL0
mul0 < MUL0
add0 < mul0

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

(56)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3


Generator Equations:
gen_c:c19_8(0) <=> c1
gen_c:c19_8(+(x, 1)) <=> c(c3, gen_c:c19_8(x))
gen_C:Z10_8(0) <=> Z
gen_C:Z10_8(+(x, 1)) <=> C(S, gen_C:Z10_8(x))
gen_c2:c311_8(0) <=> c3
gen_c2:c311_8(+(x, 1)) <=> c2(gen_c2:c311_8(x))


The following defined symbols remain to be analysed:
ADD0, MUL0, mul0, add0

They will be analysed ascendingly in the following order:
ADD0 < MUL0
mul0 < MUL0
add0 < mul0

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ADD0(gen_C:Z10_8(n13_8), gen_C:Z10_8(b)) -> gen_c2:c311_8(n13_8), rt in Omega(1 + n13_8)

Induction Base:
ADD0(gen_C:Z10_8(0), gen_C:Z10_8(b)) ->_R^Omega(1)
c3

Induction Step:
ADD0(gen_C:Z10_8(+(n13_8, 1)), gen_C:Z10_8(b)) ->_R^Omega(1)
c2(ADD0(gen_C:Z10_8(n13_8), C(S, gen_C:Z10_8(b)))) ->_IH
c2(gen_c2:c311_8(c14_8))

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

(58)
Complex Obligation (BEST)

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

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3


Generator Equations:
gen_c:c19_8(0) <=> c1
gen_c:c19_8(+(x, 1)) <=> c(c3, gen_c:c19_8(x))
gen_C:Z10_8(0) <=> Z
gen_C:Z10_8(+(x, 1)) <=> C(S, gen_C:Z10_8(x))
gen_c2:c311_8(0) <=> c3
gen_c2:c311_8(+(x, 1)) <=> c2(gen_c2:c311_8(x))


The following defined symbols remain to be analysed:
ADD0, MUL0, mul0, add0

They will be analysed ascendingly in the following order:
ADD0 < MUL0
mul0 < MUL0
add0 < mul0

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

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3


Lemmas:
ADD0(gen_C:Z10_8(n13_8), gen_C:Z10_8(b)) -> gen_c2:c311_8(n13_8), rt in Omega(1 + n13_8)


Generator Equations:
gen_c:c19_8(0) <=> c1
gen_c:c19_8(+(x, 1)) <=> c(c3, gen_c:c19_8(x))
gen_C:Z10_8(0) <=> Z
gen_C:Z10_8(+(x, 1)) <=> C(S, gen_C:Z10_8(x))
gen_c2:c311_8(0) <=> c3
gen_c2:c311_8(+(x, 1)) <=> c2(gen_c2:c311_8(x))


The following defined symbols remain to be analysed:
add0, MUL0, mul0

They will be analysed ascendingly in the following order:
mul0 < MUL0
add0 < mul0

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_C:Z10_8(n564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(+(n564_8, b)), rt in Omega(0)

Induction Base:
add0(gen_C:Z10_8(0), gen_C:Z10_8(b)) ->_R^Omega(0)
gen_C:Z10_8(b)

Induction Step:
add0(gen_C:Z10_8(+(n564_8, 1)), gen_C:Z10_8(b)) ->_R^Omega(0)
add0(gen_C:Z10_8(n564_8), C(S, gen_C:Z10_8(b))) ->_IH
gen_C:Z10_8(+(+(b, 1), c565_8))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(64)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3


Lemmas:
ADD0(gen_C:Z10_8(n13_8), gen_C:Z10_8(b)) -> gen_c2:c311_8(n13_8), rt in Omega(1 + n13_8)
add0(gen_C:Z10_8(n564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(+(n564_8, b)), rt in Omega(0)


Generator Equations:
gen_c:c19_8(0) <=> c1
gen_c:c19_8(+(x, 1)) <=> c(c3, gen_c:c19_8(x))
gen_C:Z10_8(0) <=> Z
gen_C:Z10_8(+(x, 1)) <=> C(S, gen_C:Z10_8(x))
gen_c2:c311_8(0) <=> c3
gen_c2:c311_8(+(x, 1)) <=> c2(gen_c2:c311_8(x))


The following defined symbols remain to be analysed:
mul0, MUL0

They will be analysed ascendingly in the following order:
mul0 < MUL0

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mul0(gen_C:Z10_8(n1564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(*(n1564_8, b)), rt in Omega(0)

Induction Base:
mul0(gen_C:Z10_8(0), gen_C:Z10_8(b)) ->_R^Omega(0)
Z

Induction Step:
mul0(gen_C:Z10_8(+(n1564_8, 1)), gen_C:Z10_8(b)) ->_R^Omega(0)
add0(mul0(gen_C:Z10_8(n1564_8), gen_C:Z10_8(b)), gen_C:Z10_8(b)) ->_IH
add0(gen_C:Z10_8(*(c1565_8, b)), gen_C:Z10_8(b)) ->_L^Omega(0)
gen_C:Z10_8(+(*(n1564_8, b), b))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(66)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3


Lemmas:
ADD0(gen_C:Z10_8(n13_8), gen_C:Z10_8(b)) -> gen_c2:c311_8(n13_8), rt in Omega(1 + n13_8)
add0(gen_C:Z10_8(n564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(+(n564_8, b)), rt in Omega(0)
mul0(gen_C:Z10_8(n1564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(*(n1564_8, b)), rt in Omega(0)


Generator Equations:
gen_c:c19_8(0) <=> c1
gen_c:c19_8(+(x, 1)) <=> c(c3, gen_c:c19_8(x))
gen_C:Z10_8(0) <=> Z
gen_C:Z10_8(+(x, 1)) <=> C(S, gen_C:Z10_8(x))
gen_c2:c311_8(0) <=> c3
gen_c2:c311_8(+(x, 1)) <=> c2(gen_c2:c311_8(x))


The following defined symbols remain to be analysed:
MUL0
----------------------------------------

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MUL0(gen_C:Z10_8(+(1, n2692_8)), gen_C:Z10_8(b)) -> *12_8, rt in Omega(b*n2692_8 + b*n2692_8^2 + n2692_8)

Induction Base:
MUL0(gen_C:Z10_8(+(1, 0)), gen_C:Z10_8(b))

Induction Step:
MUL0(gen_C:Z10_8(+(1, +(n2692_8, 1))), gen_C:Z10_8(b)) ->_R^Omega(1)
c(ADD0(mul0(gen_C:Z10_8(+(1, n2692_8)), gen_C:Z10_8(b)), gen_C:Z10_8(b)), MUL0(gen_C:Z10_8(+(1, n2692_8)), gen_C:Z10_8(b))) ->_L^Omega(0)
c(ADD0(gen_C:Z10_8(*(+(1, n2692_8), b)), gen_C:Z10_8(b)), MUL0(gen_C:Z10_8(+(1, n2692_8)), gen_C:Z10_8(b))) ->_L^Omega(1 + b + b*n2692_8)
c(gen_c2:c311_8(+(b, *(n2692_8, b))), MUL0(gen_C:Z10_8(+(1, n2692_8)), gen_C:Z10_8(b))) ->_IH
c(gen_c2:c311_8(+(b, *(n2692_8, b))), *12_8)

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

(68)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
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))
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)

Types:
MUL0 :: C:Z -> C:Z -> c:c1
C :: S -> C:Z -> C:Z
c :: c2:c3 -> c:c1 -> c:c1
ADD0 :: C:Z -> C:Z -> c2:c3
mul0 :: C:Z -> C:Z -> C:Z
Z :: C:Z
c1 :: c:c1
c2 :: c2:c3 -> c2:c3
S :: S
c3 :: c2:c3
SECOND :: C:Z -> c4
c4 :: c4
ISZERO :: C:Z -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: C:Z -> C:Z -> c7
c7 :: c:c1 -> c7
add0 :: C:Z -> C: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:c11_8 :: c:c1
hole_C:Z2_8 :: C:Z
hole_S3_8 :: S
hole_c2:c34_8 :: c2:c3
hole_c45_8 :: c4
hole_c5:c66_8 :: c5:c6
hole_c77_8 :: c7
hole_False:True8_8 :: False:True
gen_c:c19_8 :: Nat -> c:c1
gen_C:Z10_8 :: Nat -> C:Z
gen_c2:c311_8 :: Nat -> c2:c3


Lemmas:
ADD0(gen_C:Z10_8(n13_8), gen_C:Z10_8(b)) -> gen_c2:c311_8(n13_8), rt in Omega(1 + n13_8)
add0(gen_C:Z10_8(n564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(+(n564_8, b)), rt in Omega(0)
mul0(gen_C:Z10_8(n1564_8), gen_C:Z10_8(b)) -> gen_C:Z10_8(*(n1564_8, b)), rt in Omega(0)


Generator Equations:
gen_c:c19_8(0) <=> c1
gen_c:c19_8(+(x, 1)) <=> c(c3, gen_c:c19_8(x))
gen_C:Z10_8(0) <=> Z
gen_C:Z10_8(+(x, 1)) <=> C(S, gen_C:Z10_8(x))
gen_c2:c311_8(0) <=> c3
gen_c2:c311_8(+(x, 1)) <=> c2(gen_c2:c311_8(x))


The following defined symbols remain to be analysed:
MUL0
----------------------------------------

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

(70)
BOUNDS(n^3, INF)