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

TWN: t₂: l1→l3

cycle: [t₂: l1→l3; t₅: l3→l1]
loop: (X₀ < (X₁)² ∧ 0 < X₀,(X₀,X₁,X₄) -> (5⋅X₀+(X₄)²,2⋅X₁,X₄)
order: [X₄; X₀; X₁]
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

Termination: true
Formula:

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

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)}

TWN - Lifting for t₂: l1→l3 of 4⋅X₄⋅X₄+8⋅X₁⋅X₁+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)}

TWN: t₅: l3→l1

TWN - Lifting for t₅: l3→l1 of 4⋅X₄⋅X₄+8⋅X₁⋅X₁+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)}

Chain transitions t₅: l3→l1 and t₄: l1→l4 to t₄₅: l3→l4

Chain transitions t₁: l2→l1 and t₄: l1→l4 to t₄₆: l2→l4

Chain transitions t₁: l2→l1 and t₃: l1→l4 to t₄₇: l2→l4

Chain transitions t₅: l3→l1 and t₃: l1→l4 to t₄₈: l3→l4

Chain transitions t₁: l2→l1 and t₂: l1→l3 to t₄₉: l2→l3

Chain transitions t₅: l3→l1 and t₂: l1→l3 to t₅₀: l3→l3

Analysing control-flow refined program

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 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₀ for location l3

TWN: t₅₀: l3→l3

cycle: [t₅₀: l3→l3]
loop: (5⋅X₀+(X₄)² < 4⋅(X₁)² ∧ 0 < 5⋅X₀+(X₄)²,(X₀,X₁,X₄) -> (5⋅X₀+(X₄)²,2⋅X₁,X₄)
order: [X₄; X₀; X₁]
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

Termination: true
Formula:

0 < 20⋅X₀+5⋅(X₄)² ∧ 20⋅X₀+5⋅(X₄)² < 0
∨ 0 < 20⋅X₀+5⋅(X₄)² ∧ 0 < 16⋅(X₁)² ∧ 20⋅X₀+5⋅(X₄)² ≤ 0 ∧ 0 ≤ 20⋅X₀+5⋅(X₄)²
∨ 0 < 20⋅X₀+5⋅(X₄)² ∧ 0 < (X₄)² ∧ 20⋅X₀+5⋅(X₄)² ≤ 0 ∧ 0 ≤ 20⋅X₀+5⋅(X₄)² ∧ 0 ≤ 16⋅(X₁)² ∧ 16⋅(X₁)² ≤ 0
∨ (X₄)² < 0 ∧ 0 ≤ 20⋅X₀+5⋅(X₄)² ∧ 20⋅X₀+5⋅(X₄)² ≤ 0 ∧ 20⋅X₀+5⋅(X₄)² < 0
∨ (X₄)² < 0 ∧ 0 < 16⋅(X₁)² ∧ 20⋅X₀+5⋅(X₄)² ≤ 0 ∧ 0 ≤ 20⋅X₀+5⋅(X₄)²
∨ (X₄)² < 0 ∧ 0 < (X₄)² ∧ 20⋅X₀+5⋅(X₄)² ≤ 0 ∧ 0 ≤ 20⋅X₀+5⋅(X₄)² ∧ 0 ≤ 16⋅(X₁)² ∧ 16⋅(X₁)² ≤ 0

Stabilization-Threshold for: 0 < 5⋅X₀+(X₄)²
alphas_abs: (X₄)²
M: 0
N: 1
Bound: 2⋅X₄⋅X₄+2 {O(n^2)}
Stabilization-Threshold for: 5⋅X₀+(X₄)² < 4⋅(X₁)²
alphas_abs: 16⋅(X₁)²+(X₄)²
M: 11
N: 1
Bound: 2⋅X₄⋅X₄+32⋅X₁⋅X₁+12 {O(n^2)}

TWN - Lifting for t₅₀: l3→l3 of 32⋅X₁⋅X₁+4⋅X₄⋅X₄+16 {O(n^2)}

relevant size-bounds w.r.t. t₄₉:
X₁: X₃ {O(n)}
X₄: X₄ {O(n)}
Runtime-bound of t₄₉: 1 {O(1)}
Results in: 32⋅X₃⋅X₃+4⋅X₄⋅X₄+16 {O(n^2)}

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

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

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

CFR did not improve the program. Rolling back

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)}
t₁: 1 {O(1)}
t₂: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}
t₃: 1 {O(1)}
t₄: 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)}
t₁: 1 {O(1)}
t₂: 4⋅X₄⋅X₄+8⋅X₃⋅X₃+16 {O(n^2)}
t₃: 1 {O(1)}
t₄: 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)}
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)}
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)}
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)}