Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: U, V
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, U, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃+1, X₄) :|: 1 ≤ X₃ ∧ X₃ ≤ 3 ∧ V ≤ 0 ∧ 0 ≤ V
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄-1)
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄)
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀

Preprocessing

Found invariant 1 ≤ X₄ for location l2

Found invariant 1 ≤ X₄ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: U, V
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, U, X₄)
t₁: l1(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃+1, X₄) :|: 1 ≤ X₃ ∧ X₃ ≤ 3 ∧ V ≤ 0 ∧ 0 ≤ V
t₃: l1(X₀, X₁, X₂, X₃, X₄) → l2(X₀, X₁, X₂, X₃, X₄) :|: 0 < X₄ ∧ V ≤ 1 ∧ 1 ≤ V
t₂: l2(X₀, X₁, X₂, X₃, X₄) → l1(X₀, X₁, X₂, X₃, X₄-1) :|: 1 ≤ X₄
t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄
t₅: l3(X₀, X₁, X₂, X₃, X₄) → l3(5⋅X₀+(X₂)², 2⋅X₁, X₂, X₃, X₄) :|: X₀ < (X₁)² ∧ 0 < X₀ ∧ 1 ≤ X₄

Analysing control-flow refined program

Found invariant 0 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ 4+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___2

Found invariant 1 ≤ X₄ ∧ 3 ≤ X₃+X₄ ∧ X₃ ≤ 3+X₄ ∧ X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l2___3

Found invariant X₃ ≤ 4 ∧ 2 ≤ X₃ for location n_l1___5

Found invariant 1 ≤ X₄ for location l3

Found invariant 0 ≤ X₄ for location n_l1___1

Found invariant 1 ≤ X₄ for location n_l2___4

CFR did not improve the program. Rolling back

Found invariant 1 ≤ X₄ for location l2

Found invariant 1 ≤ X₄ for location l3

Time-Bound by TWN-Loops:

TWN-Loops: t₅ 16⋅X₂⋅X₂+32⋅X₁⋅X₁+16 {O(n^2)}

TWN-Loops:

entry: t₄: l2(X₀, X₁, X₂, X₃, X₄) → l3(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₄
results in twn-loop: twn:Inv: [1 ≤ X₄] , (X₀,X₁,X₂,X₃,X₄) -> (5⋅X₀+(X₂)²,2⋅X₁,X₂,X₃,X₄) :|: X₀ < (X₁)² ∧ 0 < X₀
order: [X₂; X₀; X₁; X₄]
closed-form:
X₂: X₂
X₀: X₀ * 5^n + [[n != 0]] * 1/4⋅(X₂)² * 5^n + [[n != 0]] * -1/4⋅(X₂)²
X₁: X₁ * 2^n
X₄: X₄

Termination: true
Formula:

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

Stabilization-Threshold for: 0 < X₀
alphas_abs: (X₂)²
M: 0
N: 1
Bound: 2⋅X₂⋅X₂+2 {O(n^2)}
Stabilization-Threshold for: X₀ < (X₁)²
alphas_abs: 4⋅(X₁)²+(X₂)²
M: 11
N: 1
Bound: 2⋅X₂⋅X₂+8⋅X₁⋅X₁+12 {O(n^2)}

relevant size-bounds w.r.t. t₄:
X₁: 2⋅X₁ {O(n)}
X₂: 2⋅X₂ {O(n)}
Runtime-bound of t₄: 1 {O(1)}
Results in: 16⋅X₂⋅X₂+32⋅X₁⋅X₁+16 {O(n^2)}

16⋅X₂⋅X₂+32⋅X₁⋅X₁+16 {O(n^2)}

Analysing control-flow refined program

Eliminate variables {X₁,X₂} that do not contribute to the problem

Found invariant 1 ≤ X₄ for location l2

Found invariant 1 ≤ X₄ for location l3

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: inf {Infinity}
t₃: inf {Infinity}
t₂: inf {Infinity}
t₄: 1 {O(1)}
t₅: 16⋅X₂⋅X₂+32⋅X₁⋅X₁+16 {O(n^2)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: inf {Infinity}
t₃: inf {Infinity}
t₂: inf {Infinity}
t₄: 1 {O(1)}
t₅: 16⋅X₂⋅X₂+32⋅X₁⋅X₁+16 {O(n^2)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: 2⋅X₀ {O(n)}
t₁, X₁: 2⋅X₁ {O(n)}
t₁, X₂: 2⋅X₂ {O(n)}
t₁, X₃: 4 {O(1)}
t₁, X₄: 2⋅X₄ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₂ {O(n)}
t₂, X₄: 2⋅X₄ {O(n)}
t₄, X₀: 2⋅X₀ {O(n)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₁: 2⋅2^(16⋅X₂⋅X₂+32⋅X₁⋅X₁+16)⋅X₁ {O(EXP)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}