KoAT2 Proof Output

Initial Problem

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

Preprocessing

Eliminate variables [X₃] that do not contribute to the problem

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₀ for location l1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₀ for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₁₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₃
t₁₁: l1(X₀, X₁, X₂, X₃) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃
t₁₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃
t₁₃: l2(X₀, X₁, X₂, X₃) → l1(X₃, 2⋅X₃, 3⋅X₃, X₃-1) :|: 1 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃
t₁₄: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃

MPRF for transition t₁₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• l1: [1+X₃]
• l2: [X₃]

MPRF for transition t₁₃: l2(X₀, X₁, X₂, X₃) → l1(X₃, 2⋅X₃, 3⋅X₃, X₃-1) :|: 1 ≤ X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• l1: [1+X₃]
• l2: [1+X₃]

TWN: t₁₁: l1→l1

cycle: [t₁₁: l1→l1]
original loop: (1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃,(X₀,X₃) -> (X₀-1,X₃))
transformed loop: (1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃,(X₀,X₃) -> (X₀-1,X₃))
loop: (1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃,(X₀,X₃) -> (X₀-1,X₃))
order: [X₃; X₀]
closed-form:

X₃: X₃
X₀: X₀ + [[n != 0]]⋅-1⋅n^1

Termination: true
Formula:

0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₀ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 2 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 2 ≤ X₀ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₀ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀+X₃ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀+X₃ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀ ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ X₃
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 0 ≤ X₀+X₃ ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 ≤ 0 ∧ 0 ≤ X₃

Stabilization-Threshold for: 0 ≤ X₀+X₃

alphas_abs: 1+X₀+X₃
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₃+4 {O(n)}

Stabilization-Threshold for: 0 ≤ X₀

alphas_abs: 1+X₀
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}

Stabilization-Threshold for: 1 ≤ X₀

alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}

original loop: (1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃,(X₀,X₃) -> (X₀-1,X₃))
transformed loop: (1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃,(X₀,X₃) -> (X₀-1,X₃))
loop: (1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃,(X₀,X₃) -> (X₀-1,X₃))
order: [X₃; X₀]
closed-form:

X₃: X₃
X₀: X₀ + [[n != 0]]⋅-1⋅n^1

Termination: true
Formula:

Stabilization-Threshold for: 0 ≤ X₀+X₃

alphas_abs: 1+X₀+X₃
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₃+4 {O(n)}

Stabilization-Threshold for: 0 ≤ X₀

alphas_abs: 1+X₀
M: 0
N: 1
Bound: 2⋅X₀+4 {O(n)}

Stabilization-Threshold for: 1 ≤ X₀

alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}

TWN - Lifting for [11: l1->l1] of 2⋅X₃+6⋅X₀+12 {O(n)}

relevant size-bounds w.r.t. t₁₀: l0→l1:
X₀: X₀ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₀: 1 {O(1)}
Results in: 2⋅X₃+6⋅X₀+12 {O(n)}

TWN - Lifting for [11: l1->l1] of 2⋅X₃+6⋅X₀+12 {O(n)}

relevant size-bounds w.r.t. t₁₃: l2→l1:
X₀: 2⋅X₃ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₃: X₃+1 {O(n)}
Results in: 14⋅X₃⋅X₃+26⋅X₃+12 {O(n^2)}

MPRF for transition t₁₄: l2(X₀, X₁, X₂, X₃) → l2(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

14⋅X₂⋅X₃+176⋅X₃⋅X₃+23⋅X₁⋅X₃+14⋅X₂+176⋅X₃+23⋅X₁ {O(n^2)}

MPRF:

• l1: [0]
• l2: [X₁]

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₀ for location l2

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₀ for location l1_v1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location l2_v1

Found invariant 0 ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₀ for location l1

All Bounds

Timebounds

Overall timebound:14⋅X₂⋅X₃+190⋅X₃⋅X₃+23⋅X₁⋅X₃+14⋅X₂+206⋅X₃+23⋅X₁+6⋅X₀+27 {O(n^2)}
t₁₀: 1 {O(1)}
t₁₁: 14⋅X₃⋅X₃+28⋅X₃+6⋅X₀+24 {O(n^2)}
t₁₂: X₃+1 {O(n)}
t₁₃: X₃+1 {O(n)}
t₁₄: 14⋅X₂⋅X₃+176⋅X₃⋅X₃+23⋅X₁⋅X₃+14⋅X₂+176⋅X₃+23⋅X₁ {O(n^2)}

Costbounds

Overall costbound: 14⋅X₂⋅X₃+190⋅X₃⋅X₃+23⋅X₁⋅X₃+14⋅X₂+206⋅X₃+23⋅X₁+6⋅X₀+27 {O(n^2)}
t₁₀: 1 {O(1)}
t₁₁: 14⋅X₃⋅X₃+28⋅X₃+6⋅X₀+24 {O(n^2)}
t₁₂: X₃+1 {O(n)}
t₁₃: X₃+1 {O(n)}
t₁₄: 14⋅X₂⋅X₃+176⋅X₃⋅X₃+23⋅X₁⋅X₃+14⋅X₂+176⋅X₃+23⋅X₁ {O(n^2)}

Sizebounds

t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₁, X₀: 2⋅X₃+X₀ {O(n)}
t₁₁, X₁: 14⋅X₂+172⋅X₃+22⋅X₁ {O(n)}
t₁₁, X₂: 24⋅X₂+296⋅X₃+38⋅X₁ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: 2⋅X₀+4⋅X₃ {O(n)}
t₁₂, X₁: 14⋅X₂+176⋅X₃+23⋅X₁ {O(n)}
t₁₂, X₂: 25⋅X₂+302⋅X₃+38⋅X₁ {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₃ {O(n)}
t₁₃, X₁: 4⋅X₃ {O(n)}
t₁₃, X₂: 6⋅X₃ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: 2⋅X₀+4⋅X₃ {O(n)}
t₁₄, X₁: 14⋅X₂+176⋅X₃+23⋅X₁ {O(n)}
t₁₄, X₂: 25⋅X₂+302⋅X₃+38⋅X₁ {O(n)}
t₁₄, X₃: X₃ {O(n)}