KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 1 ≤ X₃
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 1 ≤ X₂
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 1 ≤ X₀

Preprocessing

Found invariant X₃ ≤ 0 for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 1 ≤ X₃
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 1 ≤ X₂ ∧ X₃ ≤ 0
t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
t₄: l3(X₀, X₁, X₂, X₃) → l3(X₀+X₁, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂+X₃, X₃-1) :|: 1 ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• l1: [X₃]

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l3(X₂, X₂, X₂, X₃) :|: 1 ≤ X₂ ∧ X₃ ≤ 0 of depth 1:

new bound:

2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}

MPRF:

• l2: [X₂]
• l3: [X₂-1]

MPRF for transition t₅: l3(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0 of depth 1:

new bound:

2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}

MPRF:

• l2: [X₂]
• l3: [X₂]

TWN: t₄: l3→l3

cycle: [t₄: l3→l3]
original loop: (1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0,(X₀,X₁,X₂,X₃) -> (X₀+X₁,X₁-1,X₂,X₃))
transformed loop: (1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0,(X₀,X₁,X₂,X₃) -> (X₀+X₁,X₁-1,X₂,X₃))
loop: (1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0,(X₀,X₁,X₂,X₃) -> (X₀+X₁,X₁-1,X₂,X₃))
order: [X₁; X₃; X₂; X₀]
closed-form:

X₁: X₁ + [[n != 0]]⋅-1⋅n^1
X₃: X₃
X₂: X₂
X₀: X₀ + [[n != 0]]⋅X₁⋅n^1 + [[n != 0, n != 1]]⋅-1/2⋅n^2 + [[n != 0, n != 1]]⋅1/2⋅n^1

Termination: true
Formula:

0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 2⋅X₀ ∧ 1 ≤ X₀ ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 2⋅X₀ ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3 ≤ 2⋅X₀ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3 ≤ 2⋅X₀ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 2⋅X₀ ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3+2⋅X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 2⋅X₀ ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 3+2⋅X₁ ∧ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3+2⋅X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3 ≤ 2⋅X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 0 ≤ 1+2⋅X₁ ∧ 1 ≤ 0 ∧ 1+2⋅X₁ ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 3 ≤ 2⋅X₀ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0
∨ 1 ≤ 0 ∧ 1 ≤ X₂ ∧ 1+X₃ ≤ X₂ ∧ X₁ ≤ X₂ ∧ X₃ ≤ 0

Stabilization-Threshold for: X₁ ≤ X₀

alphas_abs: 3+2⋅X₀+2⋅X₁
M: 0
N: 2
Bound: 4⋅X₀+4⋅X₁+9 {O(n)}

Stabilization-Threshold for: 1 ≤ X₀

alphas_abs: 1+2⋅X₀+2⋅X₁
M: 0
N: 2
Bound: 4⋅X₀+4⋅X₁+5 {O(n)}

TWN - Lifting for [4: l3->l3] of 8⋅X₀+8⋅X₁+16 {O(n)}

relevant size-bounds w.r.t. t₃: l2→l3:
X₀: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
X₁: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
Runtime-bound of t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
Results in: 128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+160⋅X₃⋅X₃+256⋅X₂⋅X₃+32⋅X₂+32⋅X₃ {O(n^4)}

Found invariant X₃ ≤ 0 for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ X₂ ≤ 1+X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location l3_v2

Found invariant X₃ ≤ 0 ∧ X₃ ≤ X₂ ∧ 2+X₁+X₃ ≤ 0 ∧ X₀+X₃ ≤ 0 ∧ 0 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 2+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location l2_v1

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l3_v1

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ 2+X₁ ≤ X₀ for location l3_v3

All Bounds

Timebounds

Overall timebound:128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+164⋅X₃⋅X₃+256⋅X₂⋅X₃+36⋅X₂+37⋅X₃+2 {O(n^4)}
t₀: 1 {O(1)}
t₁: X₃ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄: 128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+160⋅X₃⋅X₃+256⋅X₂⋅X₃+32⋅X₂+32⋅X₃ {O(n^4)}
t₅: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}

Costbounds

Overall costbound: 128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+164⋅X₃⋅X₃+256⋅X₂⋅X₃+36⋅X₂+37⋅X₃+2 {O(n^4)}
t₀: 1 {O(1)}
t₁: X₃ {O(n)}
t₂: 1 {O(1)}
t₃: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄: 128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+160⋅X₃⋅X₃+256⋅X₂⋅X₃+32⋅X₂+32⋅X₃ {O(n^4)}
t₅: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₃⋅X₃+2⋅X₃+X₂ {O(n^2)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₂, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₂, X₃: 2⋅X₃ {O(n)}
t₃, X₀: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
t₃, X₁: 4⋅X₃⋅X₃+4⋅X₂+4⋅X₃ {O(n^2)}
t₃, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 16384⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+107520⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+196608⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+98304⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+196608⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃+224256⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₂⋅X₂⋅X₂⋅X₃⋅X₃+93184⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+120832⋅X₂⋅X₃⋅X₃⋅X₃+125952⋅X₂⋅X₂⋅X₃⋅X₃+16384⋅X₂⋅X₂⋅X₂⋅X₂+45440⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₂⋅X₂⋅X₂⋅X₃+12032⋅X₃⋅X₃⋅X₃+27648⋅X₂⋅X₂⋅X₃+30464⋅X₂⋅X₃⋅X₃+9216⋅X₂⋅X₂⋅X₂+1408⋅X₂⋅X₂+1452⋅X₃⋅X₃+2816⋅X₂⋅X₃+44⋅X₂+44⋅X₃ {O(n^8)}
t₄, X₁: 128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+164⋅X₃⋅X₃+256⋅X₂⋅X₃+36⋅X₂+36⋅X₃ {O(n^4)}
t₄, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: 16384⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+107520⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+196608⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+98304⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+196608⋅X₂⋅X₂⋅X₃⋅X₃⋅X₃+224256⋅X₂⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₂⋅X₂⋅X₂⋅X₃⋅X₃+93184⋅X₃⋅X₃⋅X₃⋅X₃⋅X₃+120832⋅X₂⋅X₃⋅X₃⋅X₃+125952⋅X₂⋅X₂⋅X₃⋅X₃+16384⋅X₂⋅X₂⋅X₂⋅X₂+45440⋅X₃⋅X₃⋅X₃⋅X₃+65536⋅X₂⋅X₂⋅X₂⋅X₃+12032⋅X₃⋅X₃⋅X₃+27648⋅X₂⋅X₂⋅X₃+30464⋅X₂⋅X₃⋅X₃+9216⋅X₂⋅X₂⋅X₂+1408⋅X₂⋅X₂+1452⋅X₃⋅X₃+2816⋅X₂⋅X₃+44⋅X₂+44⋅X₃ {O(n^8)}
t₅, X₁: 128⋅X₃⋅X₃⋅X₃⋅X₃+256⋅X₂⋅X₃⋅X₃+256⋅X₃⋅X₃⋅X₃+128⋅X₂⋅X₂+164⋅X₃⋅X₃+256⋅X₂⋅X₃+36⋅X₂+36⋅X₃ {O(n^4)}
t₅, X₂: 2⋅X₃⋅X₃+2⋅X₂+2⋅X₃ {O(n^2)}
t₅, X₃: 2⋅X₃ {O(n)}