Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₁)² ≤ X₀
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(5⋅X₀+(X₄)², 2⋅X₁, X₂, X₃, X₄)
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Found invariant X₂ ≤ X₀ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₀ for location l4

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄)
t₂: l1(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀ ∧ X₂ ≤ X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: (X₁)² ≤ X₀ ∧ X₂ ≤ X₀
t₄: l1(X₀, X₁, X₂, X₃, X₄) → l4(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₂ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l1(5⋅X₀+(X₄)², 2⋅X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀ ∧ 1 ≤ X₀
t₆: l4(X₀, X₁, X₂, X₃, X₄) → l5(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₀

Found invariant X₂ ≤ X₀ for location l5

Found invariant X₂ ≤ X₀ for location l1

Found invariant X₂ ≤ X₀ for location l4

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₀ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₂ 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}

TWN-Loops:

entry: t₁: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₂, X₃, X₂, X₃, X₄)
results in twn-loop: twn:Inv: [X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₀] , (X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₄)²,2⋅X₁,X₂,X₃,X₄) :|: X₀ < (X₁)² ∧ 0 < X₀
order: [X₄; X₀; X₁; X₂]
closed-form:
X₄: X₄
X₀: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₄)² * 5^n + [[n != 0]] * -1/4⋅(X₄)²
X₁: X₁ * 2^n
X₂: X₂

Termination: true
Formula:

4⋅X₀+(X₄)² < 0 ∧ 0 < 4⋅X₀+(X₄)²
∨ 0 < 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² < 0 ∧ (X₄)²+4⋅X₂ < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0
∨ 0 < 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ (X₄)²+4⋅X₂ ≤ 0 ∧ 0 ≤ (X₄)²+4⋅X₂
∨ 0 < 4⋅(X₁)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 0 < 4⋅X₀+(X₄)²
∨ 0 < 4⋅X₀+(X₄)² ∧ 0 < 4⋅(X₁)² ∧ (X₄)²+4⋅X₂ < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0
∨ 0 < 4⋅X₀+(X₄)² ∧ 0 < 4⋅(X₁)² ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ (X₄)²+4⋅X₂ ≤ 0 ∧ 0 ≤ (X₄)²+4⋅X₂
∨ 0 < (X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0 ∧ 0 < 4⋅X₀+(X₄)²
∨ 0 < 4⋅X₀+(X₄)² ∧ 0 < (X₄)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0 ∧ (X₄)²+4⋅X₂ < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0
∨ 0 < 4⋅X₀+(X₄)² ∧ 0 < (X₄)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ (X₄)²+4⋅X₂ ≤ 0 ∧ 0 ≤ (X₄)²+4⋅X₂
∨ (X₄)² < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ 4⋅X₀+(X₄)² < 0 ∧ 0 < 4⋅X₀+(X₄)²
∨ (X₄)² < 0 ∧ 4⋅X₀+(X₄)² < 0 ∧ (X₄)²+4⋅X₂ < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0
∨ (X₄)² < 0 ∧ 4⋅X₀+(X₄)² < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ (X₄)²+4⋅X₂ ≤ 0 ∧ 0 ≤ (X₄)²+4⋅X₂
∨ (X₄)² < 0 ∧ 0 < 4⋅(X₁)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 0 < 4⋅X₀+(X₄)²
∨ (X₄)² < 0 ∧ 0 < 4⋅(X₁)² ∧ (X₄)²+4⋅X₂ < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0
∨ (X₄)² < 0 ∧ 0 < 4⋅(X₁)² ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ (X₄)²+4⋅X₂ ≤ 0 ∧ 0 ≤ (X₄)²+4⋅X₂
∨ (X₄)² < 0 ∧ 0 < (X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0 ∧ 0 < 4⋅X₀+(X₄)²
∨ (X₄)² < 0 ∧ 0 < (X₄)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0 ∧ (X₄)²+4⋅X₂ < 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0
∨ (X₄)² < 0 ∧ 0 < (X₄)² ∧ 0 ≤ 4⋅(X₁)² ∧ 4⋅(X₁)² ≤ 0 ∧ 0 ≤ 4⋅X₀+(X₄)² ∧ 4⋅X₀+(X₄)² ≤ 0 ∧ (X₄)²+4⋅X₂ ≤ 0 ∧ 0 ≤ (X₄)²+4⋅X₂

Stabilization-Threshold for: 0 < X₀
alphas_abs: (X₄)²
M: 0
N: 1
Bound: 2⋅X₄⋅X₄+2 {O(n^2)}
Stabilization-Threshold for: X₀ < (X₁)²
alphas_abs: 4⋅(X₁)²+(X₄)²
M: 11
N: 1
Bound: 2⋅X₄⋅X₄+8⋅X₁⋅X₁+12 {O(n^2)}

relevant size-bounds w.r.t. t₁:
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}

4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}

relevant size-bounds w.r.t. t₁:
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}

4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}

Analysing control-flow refined program

Eliminate variables {X₄} that do not contribute to the problem

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location n_l3___3

Found invariant 1 ≤ X₂ for location n_l1___2

Found invariant X₂ ≤ X₀ for location l5

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₂ ≤ X₀ for location l4

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location n_l3___1

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:16⋅X₃⋅X₃+8⋅X₄⋅X₄+37 {O(n^2)}
t₀: 1 {O(1)}
t₂: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₃⋅X₃+8⋅X₄⋅X₄+37 {O(n^2)}
t₀: 1 {O(1)}
t₂: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₂, X₁: 2^(4⋅X₄⋅X₄+8⋅X₃⋅X₃+16)⋅X₃ {O(EXP)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₁: 2^(4⋅X₄⋅X₄+8⋅X₃⋅X₃+16)⋅X₃+X₃ {O(EXP)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₁: 2^(4⋅X₄⋅X₄+8⋅X₃⋅X₃+16)⋅X₃+X₃ {O(EXP)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₅, X₁: 2^(4⋅X₄⋅X₄+8⋅X₃⋅X₃+16)⋅X₃ {O(EXP)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₆, X₁: 2⋅2^(4⋅X₄⋅X₄+8⋅X₃⋅X₃+16)⋅X₃+2⋅X₃ {O(EXP)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₆, X₄: 4⋅X₄ {O(n)}