WORST_CASE(?,O(n^1))
proof of input_u6ue9ictZA.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(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), 5 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), 7171 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 4269 ms]
(20) CpxRNTS
(21) FinalProof [FINISHED, 0 ms]
(22) BOUNDS(1, n^1)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f4(S(x''), S(x'), x3, x4, S(x)) -> f2(S(x''), x', x3, x4, x)
   f4(S(x'), 0, x3, x4, S(x)) -> f3(x', 0, x3, x4, x)
   f2(x1, x2, S(x''), S(x'), S(x)) -> f5(x1, x2, S(x''), x', x)
   f2(x1, x2, S(x'), 0, S(x)) -> f3(x1, x2, x', 0, x)
   f4(S(x'), S(x), x3, x4, 0) -> 0
   f4(S(x), 0, x3, x4, 0) -> 0
   f2(x1, x2, S(x'), S(x), 0) -> 0
   f2(x1, x2, S(x), 0, 0) -> 0
   f0(x1, S(x'), x3, S(x), x5) -> f1(x', S(x'), x, S(x), S(x))
   f0(x1, S(x), x3, 0, x5) -> 0
   f6(x1, x2, x3, x4, S(x)) -> f0(x1, x2, x4, x3, x)
   f5(x1, x2, x3, x4, S(x)) -> f6(x2, x1, x3, x4, x)
   f3(x1, x2, x3, x4, S(x)) -> f4(x1, x2, x4, x3, x)
   f1(x1, x2, x3, x4, S(x)) -> f2(x2, x1, x3, x4, x)
   f6(x1, x2, x3, x4, 0) -> 0
   f5(x1, x2, x3, x4, 0) -> 0
   f4(0, x2, x3, x4, x5) -> 0
   f3(x1, x2, x3, x4, 0) -> 0
   f2(x1, x2, 0, x4, x5) -> 0
   f1(x1, x2, x3, x4, 0) -> 0
   f0(x1, 0, x3, x4, x5) -> 0

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^1).


The TRS R consists of the following rules:

   f4(S(x''), S(x'), x3, x4, S(x)) -> f2(S(x''), x', x3, x4, x) [1]
   f4(S(x'), 0, x3, x4, S(x)) -> f3(x', 0, x3, x4, x) [1]
   f2(x1, x2, S(x''), S(x'), S(x)) -> f5(x1, x2, S(x''), x', x) [1]
   f2(x1, x2, S(x'), 0, S(x)) -> f3(x1, x2, x', 0, x) [1]
   f4(S(x'), S(x), x3, x4, 0) -> 0 [1]
   f4(S(x), 0, x3, x4, 0) -> 0 [1]
   f2(x1, x2, S(x'), S(x), 0) -> 0 [1]
   f2(x1, x2, S(x), 0, 0) -> 0 [1]
   f0(x1, S(x'), x3, S(x), x5) -> f1(x', S(x'), x, S(x), S(x)) [1]
   f0(x1, S(x), x3, 0, x5) -> 0 [1]
   f6(x1, x2, x3, x4, S(x)) -> f0(x1, x2, x4, x3, x) [1]
   f5(x1, x2, x3, x4, S(x)) -> f6(x2, x1, x3, x4, x) [1]
   f3(x1, x2, x3, x4, S(x)) -> f4(x1, x2, x4, x3, x) [1]
   f1(x1, x2, x3, x4, S(x)) -> f2(x2, x1, x3, x4, x) [1]
   f6(x1, x2, x3, x4, 0) -> 0 [1]
   f5(x1, x2, x3, x4, 0) -> 0 [1]
   f4(0, x2, x3, x4, x5) -> 0 [1]
   f3(x1, x2, x3, x4, 0) -> 0 [1]
   f2(x1, x2, 0, x4, x5) -> 0 [1]
   f1(x1, x2, x3, x4, 0) -> 0 [1]
   f0(x1, 0, x3, x4, x5) -> 0 [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:

   f4(S(x''), S(x'), x3, x4, S(x)) -> f2(S(x''), x', x3, x4, x) [1]
   f4(S(x'), 0, x3, x4, S(x)) -> f3(x', 0, x3, x4, x) [1]
   f2(x1, x2, S(x''), S(x'), S(x)) -> f5(x1, x2, S(x''), x', x) [1]
   f2(x1, x2, S(x'), 0, S(x)) -> f3(x1, x2, x', 0, x) [1]
   f4(S(x'), S(x), x3, x4, 0) -> 0 [1]
   f4(S(x), 0, x3, x4, 0) -> 0 [1]
   f2(x1, x2, S(x'), S(x), 0) -> 0 [1]
   f2(x1, x2, S(x), 0, 0) -> 0 [1]
   f0(x1, S(x'), x3, S(x), x5) -> f1(x', S(x'), x, S(x), S(x)) [1]
   f0(x1, S(x), x3, 0, x5) -> 0 [1]
   f6(x1, x2, x3, x4, S(x)) -> f0(x1, x2, x4, x3, x) [1]
   f5(x1, x2, x3, x4, S(x)) -> f6(x2, x1, x3, x4, x) [1]
   f3(x1, x2, x3, x4, S(x)) -> f4(x1, x2, x4, x3, x) [1]
   f1(x1, x2, x3, x4, S(x)) -> f2(x2, x1, x3, x4, x) [1]
   f6(x1, x2, x3, x4, 0) -> 0 [1]
   f5(x1, x2, x3, x4, 0) -> 0 [1]
   f4(0, x2, x3, x4, x5) -> 0 [1]
   f3(x1, x2, x3, x4, 0) -> 0 [1]
   f2(x1, x2, 0, x4, x5) -> 0 [1]
   f1(x1, x2, x3, x4, 0) -> 0 [1]
   f0(x1, 0, x3, x4, x5) -> 0 [1]

The TRS has the following type information:
f4 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
S :: S:0 -> S:0
f2 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
0 :: S:0
f3 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f5 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f0 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f1 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f6 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0

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:

   f4_5
   f2_5
   f0_5
   f6_5
   f5_5
   f3_5
   f1_5

(c) The following functions are completely defined:
none

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:

   f4(S(x''), S(x'), x3, x4, S(x)) -> f2(S(x''), x', x3, x4, x) [1]
   f4(S(x'), 0, x3, x4, S(x)) -> f3(x', 0, x3, x4, x) [1]
   f2(x1, x2, S(x''), S(x'), S(x)) -> f5(x1, x2, S(x''), x', x) [1]
   f2(x1, x2, S(x'), 0, S(x)) -> f3(x1, x2, x', 0, x) [1]
   f4(S(x'), S(x), x3, x4, 0) -> 0 [1]
   f4(S(x), 0, x3, x4, 0) -> 0 [1]
   f2(x1, x2, S(x'), S(x), 0) -> 0 [1]
   f2(x1, x2, S(x), 0, 0) -> 0 [1]
   f0(x1, S(x'), x3, S(x), x5) -> f1(x', S(x'), x, S(x), S(x)) [1]
   f0(x1, S(x), x3, 0, x5) -> 0 [1]
   f6(x1, x2, x3, x4, S(x)) -> f0(x1, x2, x4, x3, x) [1]
   f5(x1, x2, x3, x4, S(x)) -> f6(x2, x1, x3, x4, x) [1]
   f3(x1, x2, x3, x4, S(x)) -> f4(x1, x2, x4, x3, x) [1]
   f1(x1, x2, x3, x4, S(x)) -> f2(x2, x1, x3, x4, x) [1]
   f6(x1, x2, x3, x4, 0) -> 0 [1]
   f5(x1, x2, x3, x4, 0) -> 0 [1]
   f4(0, x2, x3, x4, x5) -> 0 [1]
   f3(x1, x2, x3, x4, 0) -> 0 [1]
   f2(x1, x2, 0, x4, x5) -> 0 [1]
   f1(x1, x2, x3, x4, 0) -> 0 [1]
   f0(x1, 0, x3, x4, x5) -> 0 [1]

The TRS has the following type information:
f4 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
S :: S:0 -> S:0
f2 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
0 :: S:0
f3 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f5 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f0 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f1 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f6 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0

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:

   f4(S(x''), S(x'), x3, x4, S(x)) -> f2(S(x''), x', x3, x4, x) [1]
   f4(S(x'), 0, x3, x4, S(x)) -> f3(x', 0, x3, x4, x) [1]
   f2(x1, x2, S(x''), S(x'), S(x)) -> f5(x1, x2, S(x''), x', x) [1]
   f2(x1, x2, S(x'), 0, S(x)) -> f3(x1, x2, x', 0, x) [1]
   f4(S(x'), S(x), x3, x4, 0) -> 0 [1]
   f4(S(x), 0, x3, x4, 0) -> 0 [1]
   f2(x1, x2, S(x'), S(x), 0) -> 0 [1]
   f2(x1, x2, S(x), 0, 0) -> 0 [1]
   f0(x1, S(x'), x3, S(x), x5) -> f1(x', S(x'), x, S(x), S(x)) [1]
   f0(x1, S(x), x3, 0, x5) -> 0 [1]
   f6(x1, x2, x3, x4, S(x)) -> f0(x1, x2, x4, x3, x) [1]
   f5(x1, x2, x3, x4, S(x)) -> f6(x2, x1, x3, x4, x) [1]
   f3(x1, x2, x3, x4, S(x)) -> f4(x1, x2, x4, x3, x) [1]
   f1(x1, x2, x3, x4, S(x)) -> f2(x2, x1, x3, x4, x) [1]
   f6(x1, x2, x3, x4, 0) -> 0 [1]
   f5(x1, x2, x3, x4, 0) -> 0 [1]
   f4(0, x2, x3, x4, x5) -> 0 [1]
   f3(x1, x2, x3, x4, 0) -> 0 [1]
   f2(x1, x2, 0, x4, x5) -> 0 [1]
   f1(x1, x2, x3, x4, 0) -> 0 [1]
   f0(x1, 0, x3, x4, x5) -> 0 [1]

The TRS has the following type information:
f4 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
S :: S:0 -> S:0
f2 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
0 :: S:0
f3 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f5 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f0 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f1 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0
f6 :: S:0 -> S:0 -> S:0 -> S:0 -> S:0 -> S:0

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:

   0 => 0

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

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

   f0(z, z', z'', z1, z2) -{ 1 }-> f1(x', 1 + x', x, 1 + x, 1 + x) :|: x1 >= 0, x5 >= 0, z' = 1 + x', x' >= 0, x >= 0, z1 = 1 + x, z = x1, z'' = x3, z2 = x5, x3 >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = 1 + x, z1 = 0, x1 >= 0, x5 >= 0, x >= 0, z = x1, z'' = x3, z2 = x5, x3 >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = x4, x1 >= 0, x4 >= 0, x5 >= 0, z = x1, z'' = x3, z2 = x5, x3 >= 0, z' = 0
   f1(z, z', z'', z1, z2) -{ 1 }-> f2(x2, x1, x3, x4, x) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 1 + x, x >= 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f1(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f5(x1, x2, 1 + x'', x', x) :|: z' = x2, x1 >= 0, z2 = 1 + x, x' >= 0, x >= 0, z = x1, z'' = 1 + x'', z1 = 1 + x', x'' >= 0, x2 >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f3(x1, x2, x', 0, x) :|: z' = x2, z1 = 0, x1 >= 0, z2 = 1 + x, z'' = 1 + x', x' >= 0, x >= 0, z = x1, x2 >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, x1 >= 0, z'' = 1 + x', z2 = 0, x' >= 0, x >= 0, z1 = 1 + x, z = x1, x2 >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, z1 = 0, x1 >= 0, z2 = 0, x >= 0, z = x1, z'' = 1 + x, x2 >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z' = x2, z1 = x4, x1 >= 0, x4 >= 0, x5 >= 0, z = x1, z2 = x5, x2 >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> f4(x1, x2, x4, x3, x) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 1 + x, x >= 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f3(x', 0, x3, x4, x) :|: z = 1 + x', z1 = x4, x4 >= 0, z2 = 1 + x, x' >= 0, x >= 0, z'' = x3, x3 >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f2(1 + x'', x', x3, x4, x) :|: z = 1 + x'', z1 = x4, x4 >= 0, z' = 1 + x', z2 = 1 + x, x' >= 0, x >= 0, z'' = x3, x'' >= 0, x3 >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z = 1 + x', z' = 1 + x, z1 = x4, x4 >= 0, z2 = 0, x' >= 0, x >= 0, z'' = x3, x3 >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = x4, x4 >= 0, z2 = 0, x >= 0, z = 1 + x, z'' = x3, x3 >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, z1 = x4, x4 >= 0, x5 >= 0, z'' = x3, z = 0, z2 = x5, x2 >= 0, x3 >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> f6(x2, x1, x3, x4, x) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 1 + x, x >= 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> f0(x1, x2, x4, x3, x) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 1 + x, x >= 0, z = x1, z'' = x3, x2 >= 0, x3 >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, z2 = 0, z = x1, z'' = x3, x2 >= 0, x3 >= 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:

   f0(z, z', z'', z1, z2) -{ 1 }-> f1(z' - 1, 1 + (z' - 1), z1 - 1, 1 + (z1 - 1), 1 + (z1 - 1)) :|: z >= 0, z2 >= 0, z' - 1 >= 0, z1 - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 >= 0, z' - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 >= 0, z'' >= 0, z' = 0
   f1(z, z', z'', z1, z2) -{ 1 }-> f2(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f1(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f5(z, z', 1 + (z'' - 1), z1 - 1, z2 - 1) :|: z >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f3(z, z', z'' - 1, 0, z2 - 1) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z2 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z2 = 0, z'' - 1 >= 0, z1 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 = 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 >= 0, z2 >= 0, z' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> f4(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f3(z - 1, 0, z'', z1, z2 - 1) :|: z1 >= 0, z - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f2(1 + (z - 1), z' - 1, z'', z1, z2 - 1) :|: z1 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z' - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 >= 0, z = 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> f6(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> f0(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0


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

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

   { f1, f0, f4, f2, f6, f3, f5 }

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

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

   f0(z, z', z'', z1, z2) -{ 1 }-> f1(z' - 1, 1 + (z' - 1), z1 - 1, 1 + (z1 - 1), 1 + (z1 - 1)) :|: z >= 0, z2 >= 0, z' - 1 >= 0, z1 - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 >= 0, z' - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 >= 0, z'' >= 0, z' = 0
   f1(z, z', z'', z1, z2) -{ 1 }-> f2(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f1(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f5(z, z', 1 + (z'' - 1), z1 - 1, z2 - 1) :|: z >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f3(z, z', z'' - 1, 0, z2 - 1) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z2 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z2 = 0, z'' - 1 >= 0, z1 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 = 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 >= 0, z2 >= 0, z' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> f4(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f3(z - 1, 0, z'', z1, z2 - 1) :|: z1 >= 0, z - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f2(1 + (z - 1), z' - 1, z'', z1, z2 - 1) :|: z1 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z' - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 >= 0, z = 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> f6(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> f0(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {f1,f0,f4,f2,f6,f3,f5}

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

(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:

   f0(z, z', z'', z1, z2) -{ 1 }-> f1(z' - 1, 1 + (z' - 1), z1 - 1, 1 + (z1 - 1), 1 + (z1 - 1)) :|: z >= 0, z2 >= 0, z' - 1 >= 0, z1 - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 >= 0, z' - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 >= 0, z'' >= 0, z' = 0
   f1(z, z', z'', z1, z2) -{ 1 }-> f2(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f1(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f5(z, z', 1 + (z'' - 1), z1 - 1, z2 - 1) :|: z >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f3(z, z', z'' - 1, 0, z2 - 1) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z2 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z2 = 0, z'' - 1 >= 0, z1 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 = 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 >= 0, z2 >= 0, z' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> f4(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f3(z - 1, 0, z'', z1, z2 - 1) :|: z1 >= 0, z - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f2(1 + (z - 1), z' - 1, z'', z1, z2 - 1) :|: z1 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z' - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 >= 0, z = 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> f6(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> f0(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {f1,f0,f4,f2,f6,f3,f5}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: f0
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: f4
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: f2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: f6
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: f3
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: f5
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   f0(z, z', z'', z1, z2) -{ 1 }-> f1(z' - 1, 1 + (z' - 1), z1 - 1, 1 + (z1 - 1), 1 + (z1 - 1)) :|: z >= 0, z2 >= 0, z' - 1 >= 0, z1 - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 >= 0, z' - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 >= 0, z'' >= 0, z' = 0
   f1(z, z', z'', z1, z2) -{ 1 }-> f2(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f1(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f5(z, z', 1 + (z'' - 1), z1 - 1, z2 - 1) :|: z >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f3(z, z', z'' - 1, 0, z2 - 1) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z2 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z2 = 0, z'' - 1 >= 0, z1 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 = 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 >= 0, z2 >= 0, z' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> f4(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f3(z - 1, 0, z'', z1, z2 - 1) :|: z1 >= 0, z - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f2(1 + (z - 1), z' - 1, z'', z1, z2 - 1) :|: z1 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z' - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 >= 0, z = 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> f6(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> f0(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {f1,f0,f4,f2,f6,f3,f5}
Previous analysis results are:
  f1: runtime: ?, size: O(1) [0]
  f0: runtime: ?, size: O(1) [0]
  f4: runtime: ?, size: O(1) [0]
  f2: runtime: ?, size: O(1) [0]
  f6: runtime: ?, size: O(1) [0]
  f3: runtime: ?, size: O(1) [0]
  f5: runtime: ?, size: O(1) [0]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: f1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 48 + 36*z' + 45*z''

Computed RUNTIME bound using CoFloCo for: f0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 36*z' + 45*z1

Computed RUNTIME bound using CoFloCo for: f4
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 9 + 36*z + 45*z'' + 2*z2

Computed RUNTIME bound using CoFloCo for: f2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 7 + 36*z + 45*z'' + 2*z2

Computed RUNTIME bound using CoFloCo for: f6
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 36*z' + 45*z''

Computed RUNTIME bound using CoFloCo for: f3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 8 + 36*z + 45*z1 + 2*z2

Computed RUNTIME bound using CoFloCo for: f5
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6 + 36*z + 45*z''

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

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

   f0(z, z', z'', z1, z2) -{ 1 }-> f1(z' - 1, 1 + (z' - 1), z1 - 1, 1 + (z1 - 1), 1 + (z1 - 1)) :|: z >= 0, z2 >= 0, z' - 1 >= 0, z1 - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 >= 0, z' - 1 >= 0, z'' >= 0
   f0(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 >= 0, z'' >= 0, z' = 0
   f1(z, z', z'', z1, z2) -{ 1 }-> f2(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f1(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f5(z, z', 1 + (z'' - 1), z1 - 1, z2 - 1) :|: z >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> f3(z, z', z'' - 1, 0, z2 - 1) :|: z1 = 0, z >= 0, z'' - 1 >= 0, z2 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z2 = 0, z'' - 1 >= 0, z1 - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = 0, z >= 0, z2 = 0, z'' - 1 >= 0, z' >= 0
   f2(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 >= 0, z2 >= 0, z' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> f4(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f3(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f3(z - 1, 0, z'', z1, z2 - 1) :|: z1 >= 0, z - 1 >= 0, z2 - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> f2(1 + (z - 1), z' - 1, z'', z1, z2 - 1) :|: z1 >= 0, z' - 1 >= 0, z2 - 1 >= 0, z - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z' - 1 >= 0, z'' >= 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 = 0, z - 1 >= 0, z'' >= 0, z' = 0
   f4(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z1 >= 0, z2 >= 0, z = 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> f6(z', z, z'', z1, z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f5(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> f0(z, z', z1, z'', z2 - 1) :|: z >= 0, z1 >= 0, z2 - 1 >= 0, z' >= 0, z'' >= 0
   f6(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z2 = 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  f1: runtime: O(n^1) [48 + 36*z' + 45*z''], size: O(1) [0]
  f0: runtime: O(n^1) [4 + 36*z' + 45*z1], size: O(1) [0]
  f4: runtime: O(n^1) [9 + 36*z + 45*z'' + 2*z2], size: O(1) [0]
  f2: runtime: O(n^1) [7 + 36*z + 45*z'' + 2*z2], size: O(1) [0]
  f6: runtime: O(n^1) [5 + 36*z' + 45*z''], size: O(1) [0]
  f3: runtime: O(n^1) [8 + 36*z + 45*z1 + 2*z2], size: O(1) [0]
  f5: runtime: O(n^1) [6 + 36*z + 45*z''], size: O(1) [0]

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

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

(22)
BOUNDS(1, n^1)