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

(0) Obligation:

The Runtime Complexity (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))

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed 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) transformation)

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

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

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) transformation)

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
f

(c) The following functions are completely defined:

eq

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

(9) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

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

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

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

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ eq } { del, f }

(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}

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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}
Previous analysis results are:

eq: runtime: ?, size: O(1) [1]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(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: {del,f}
Previous analysis results are:

eq: runtime: O(1) [3], size: O(1) [1]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(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'') -{ 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]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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(n¹) with polynomial bound: 1 + z1

(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] del: runtime: ?, size: O(1) [0] f: runtime: ?, size: O(n1) [1 + z1]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: del
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 34·z''

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 35 + 34·z2 + 34·z3

(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:
Previous analysis results are:

eq: runtime: O(1) [3], size: O(1) [1] del: runtime: O(n1) [34·z''], size: O(1) [0] f: runtime: O(n1) [35 + 34·z2 + 34·z3], size: O(n1) [1 + z1]

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)