WORST_CASE(?,O(n^1))
proof of input_TEmEnZ0Chq.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTypedWeightedTrs
(7) CompletionProof [UPPER BOUND(ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
(12) CpxRNTS
(13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 404 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 9 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 910 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 490 ms]
(28) CpxRNTS
(29) FinalProof [FINISHED, 0 ms]
(30) BOUNDS(1, n^1)


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(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:

   del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
   f(true, x, y, z) -> del(.(y, z))
   f(false, x, y, z) -> .(x, del(.(y, z)))
   =(nil, nil) -> true
   =(.(x, y), nil) -> false
   =(nil, .(y, z)) -> false
   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(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:

   del(.(x, .(y, z))) -> f(=(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   =(nil, nil) -> true [1]
   =(.(x, y), nil) -> false [1]
   =(nil, .(y, z)) -> false [1]
   =(.(x, y), .(u, v)) -> and(=(x, u), =(y, v)) [1]

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

(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

= => eq

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

(4)
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:

   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]

The TRS has the following type information:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
eq :: .:nil:u:v -> .:nil:u:v -> true:false:and
true :: true:false:and
false :: true:false:and
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and -> true:false:and -> true:false:and

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

(7) 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:

   del_1
   f_4

(c) The following functions are completely defined:

   eq_2

Due to the following rules being added:

   eq(v0, v1) -> null_eq [0]

And the following fresh constants:    null_eq

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

(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:

   del(.(x, .(y, z))) -> f(eq(x, y), x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
   eq(v0, v1) -> null_eq [0]

The TRS has the following type information:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
eq :: .:nil:u:v -> .:nil:u:v -> true:false:and:null_eq
true :: true:false:and:null_eq
false :: true:false:and:null_eq
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and:null_eq -> true:false:and:null_eq -> true:false:and:null_eq
null_eq :: true:false:and:null_eq

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

(10)
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:

   del(.(nil, .(nil, z))) -> f(true, nil, nil, z) [2]
   del(.(.(x', y'), .(nil, z))) -> f(false, .(x', y'), nil, z) [2]
   del(.(nil, .(.(y'', z'), z))) -> f(false, nil, .(y'', z'), z) [2]
   del(.(.(x'', y1), .(.(u, v), z))) -> f(and(eq(x'', u), eq(y1, v)), .(x'', y1), .(u, v), z) [2]
   del(.(x, .(y, z))) -> f(null_eq, x, y, z) [1]
   f(true, x, y, z) -> del(.(y, z)) [1]
   f(false, x, y, z) -> .(x, del(.(y, z))) [1]
   eq(nil, nil) -> true [1]
   eq(.(x, y), nil) -> false [1]
   eq(nil, .(y, z)) -> false [1]
   eq(.(x, y), .(u, v)) -> and(eq(x, u), eq(y, v)) [1]
   eq(v0, v1) -> null_eq [0]

The TRS has the following type information:
del :: .:nil:u:v -> .:nil:u:v
. :: .:nil:u:v -> .:nil:u:v -> .:nil:u:v
f :: true:false:and:null_eq -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v -> .:nil:u:v
eq :: .:nil:u:v -> .:nil:u:v -> true:false:and:null_eq
true :: true:false:and:null_eq
false :: true:false:and:null_eq
nil :: .:nil:u:v
u :: .:nil:u:v
v :: .:nil:u:v
and :: true:false:and:null_eq -> true:false:and:null_eq -> true:false:and:null_eq
null_eq :: true:false:and:null_eq

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

   true => 1
   false => 0
   nil => 0
   u => 1
   v => 2
   null_eq => 0

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z) :|: z >= 0, z'' = 1 + 0 + (1 + 0 + z)
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + y + z) :|: z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = 1, z1 = x
   f(z'', z1, z2, z3) -{ 1 }-> 1 + x + del(1 + y + z) :|: z'' = 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z1 = x


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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0


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

   { eq }
   { del, f }

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {eq}, {del,f}

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {eq}, {del,f}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {eq}, {del,f}
Previous analysis results are:
  eq: runtime: ?, size: O(1) [1]

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(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 2 }-> f(1 + eq(x'', 1) + eq(y1, 2), 1 + x'' + y1, 1 + 1 + 2, z) :|: y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 1 }-> 1 + eq(x, 1) + eq(y, 2) :|: z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {del,f}
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]

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

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

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {del,f}
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z1

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: {del,f}
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]
  del: runtime: ?, size: O(1) [0]
  f: runtime: ?, size: O(n^1) [1 + z1]

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(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 34*z''

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 35 + 34*z2 + 34*z3

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

   del(z'') -{ 2 }-> f(1, 0, 0, z'' - 2) :|: z'' - 2 >= 0
   del(z'') -{ 1 }-> f(0, x, y, z) :|: z >= 0, x >= 0, y >= 0, z'' = 1 + x + (1 + y + z)
   del(z'') -{ 2 }-> f(0, 0, 1 + y'' + z', z) :|: z >= 0, z'' = 1 + 0 + (1 + (1 + y'' + z') + z), z' >= 0, y'' >= 0
   del(z'') -{ 2 }-> f(0, 1 + x' + y', 0, z) :|: z >= 0, x' >= 0, y' >= 0, z'' = 1 + (1 + x' + y') + (1 + 0 + z)
   del(z'') -{ 8 }-> f(1 + s + s', 1 + x'' + y1, 1 + 1 + 2, z) :|: s >= 0, s <= 1, s' >= 0, s' <= 1, y1 >= 0, z >= 0, x'' >= 0, z'' = 1 + (1 + x'' + y1) + (1 + (1 + 1 + 2) + z)
   eq(z'', z1) -{ 1 }-> 1 :|: z'' = 0, z1 = 0
   eq(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' = 1 + x + y, x >= 0, y >= 0
   eq(z'', z1) -{ 1 }-> 0 :|: z'' = 0, z >= 0, z1 = 1 + y + z, y >= 0
   eq(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   eq(z'', z1) -{ 7 }-> 1 + s'' + s1 :|: s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 1, z'' = 1 + x + y, x >= 0, y >= 0, z1 = 1 + 1 + 2
   f(z'', z1, z2, z3) -{ 1 }-> del(1 + z2 + z3) :|: z3 >= 0, z1 >= 0, z2 >= 0, z'' = 1
   f(z'', z1, z2, z3) -{ 1 }-> 1 + z1 + del(1 + z2 + z3) :|: z'' = 0, z3 >= 0, z1 >= 0, z2 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  eq: runtime: O(1) [3], size: O(1) [1]
  del: runtime: O(n^1) [34*z''], size: O(1) [0]
  f: runtime: O(n^1) [35 + 34*z2 + 34*z3], size: O(n^1) [1 + z1]

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(29) FinalProof (FINISHED)
Computed overall runtime complexity
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(30)
BOUNDS(1, n^1)