KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars: Y
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(X₆, X₆, 2, X₂, X₆, X₆, X₆-Y) :|: 1 ≤ Y ∧ Y ≤ X₆
t₅: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₀) :|: X₆ ≤ 0
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(3⋅X₀+2⋅X₁, -5⋅X₀-3⋅X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄, X₆) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1+X₃ ≤ 0
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(3⋅X₀+2⋅X₁, -5⋅X₀-3⋅X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄, X₆) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1 ≤ X₃
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆-1) :|: 1 ≤ X₆

Preprocessing

Found invariant 1+X₆ ≤ X₄ ∧ 1 ≤ X₄+X₆ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ X₂ ≤ 1+X₄ ∧ X₂ ≤ 2 ∧ 2 ≤ X₂ for location l2

Found invariant X₆ ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars: Y
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(X₆, X₆, 2, X₂, X₆, X₆, X₆-Y) :|: 1 ≤ Y ∧ Y ≤ X₆
t₅: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₀) :|: X₆ ≤ 0
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₂ ≤ 2 ∧ X₂ ≤ 1+X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄+X₆ ∧ 1+X₆ ≤ X₄ ∧ 2 ≤ X₂ ∧ 3 ≤ X₂+X₄
t₃: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(3⋅X₀+2⋅X₁, -5⋅X₀-3⋅X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄, X₆) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1+X₃ ≤ 0 ∧ X₂ ≤ 2 ∧ X₂ ≤ 1+X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄+X₆ ∧ 1+X₆ ≤ X₄ ∧ 2 ≤ X₂ ∧ 3 ≤ X₂+X₄
t₄: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(3⋅X₀+2⋅X₁, -5⋅X₀-3⋅X₁, X₂, -2⋅X₃, (X₂)²+X₄+(X₄)², (X₂)²+2⋅(X₄)²+3⋅X₅-4⋅X₄, X₆) :|: 1+(X₃)²+X₅ ≤ (X₂)⁵+2⋅X₄ ∧ 1 ≤ X₃ ∧ X₂ ≤ 2 ∧ X₂ ≤ 1+X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄+X₆ ∧ 1+X₆ ≤ X₄ ∧ 2 ≤ X₂ ∧ 3 ≤ X₂+X₄
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆-1) :|: 1 ≤ X₆ ∧ X₆ ≤ X₀

MPRF for transition t₁: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l2(X₆, X₆, 2, X₂, X₆, X₆, X₆-Y) :|: 1 ≤ Y ∧ Y ≤ X₆ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• l1: [X₆]
• l2: [X₆]

MPRF for transition t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₂ ≤ 2 ∧ X₂ ≤ 1+X₄ ∧ 1 ≤ X₄ ∧ 1 ≤ X₄+X₆ ∧ 1+X₆ ≤ X₄ ∧ 2 ≤ X₂ ∧ 3 ≤ X₂+X₄ of depth 1:

new bound:

X₆ {O(n)}

MPRF:

• l1: [0]
• l2: [1]

Found invariant 1+X₆ ≤ X₅ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ 1 ≤ X₅+X₆ ∧ 1 ≤ X₄+X₆ ∧ 1 ≤ X₁+X₆ ∧ 1 ≤ X₀+X₆ ∧ X₅ ≤ X₄ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1 ≤ X₅ ∧ 2 ≤ X₄+X₅ ∧ X₄ ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ X₂ ≤ 1+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ X₂ ≤ 1+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₂ ≤ 2 ∧ X₂ ≤ 1+X₁ ∧ X₂ ≤ 1+X₀ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location l2

Found invariant 6+X₆ ≤ X₄ ∧ 6 ≤ X₄+X₆ ∧ 6 ≤ X₄ ∧ 8 ≤ X₃+X₄ ∧ 8 ≤ X₂+X₄ ∧ 4+X₂ ≤ X₄ ∧ 2 ≤ X₃ ∧ 4 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 2 ∧ 2 ≤ X₂ for location l2_v2

Found invariant 6+X₆ ≤ X₄ ∧ 6 ≤ X₄+X₆ ∧ 6 ≤ X₄ ∧ 8+X₃ ≤ X₄ ∧ 8 ≤ X₂+X₄ ∧ 4+X₂ ≤ X₄ ∧ 2+X₃ ≤ 0 ∧ 4+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 0 ∧ X₂ ≤ 2 ∧ 2 ≤ X₂ for location l2_v1

Found invariant X₆ ≤ X₀ for location l3

Found invariant 1+X₆ ≤ X₄ ∧ 1 ≤ X₄+X₆ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ X₂ ≤ 1+X₄ ∧ X₂ ≤ 2 ∧ 2 ≤ X₂ for location l2

Found invariant X₆ ≤ X₀ ∧ X₀ ≤ X₆ for location l3

Found invariant 1+X₆ ≤ X₀ ∧ 0 ≤ X₆ ∧ 1 ≤ X₀+X₆ ∧ 1 ≤ X₀ for location l3_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: X₆ {O(n)}
t₂: X₆ {O(n)}
t₃: inf {Infinity}
t₄: inf {Infinity}
t₅: 1 {O(1)}
t₆: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: X₆ {O(n)}
t₂: X₆ {O(n)}
t₃: inf {Infinity}
t₄: inf {Infinity}
t₅: 1 {O(1)}
t₆: inf {Infinity}

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)}
t₀, X₅: X₅ {O(n)}
t₀, X₆: X₆ {O(n)}
t₁, X₀: 2⋅X₆ {O(n)}
t₁, X₁: 2⋅X₆ {O(n)}
t₁, X₂: 2 {O(1)}
t₁, X₃: X₂+2 {O(n)}
t₁, X₄: 2⋅X₆ {O(n)}
t₁, X₅: 2⋅X₆ {O(n)}
t₁, X₆: X₆ {O(n)}
t₂, X₂: 2 {O(1)}
t₂, X₆: X₆ {O(n)}
t₃, X₂: 2 {O(1)}
t₃, X₆: X₆ {O(n)}
t₄, X₂: 2 {O(1)}
t₄, X₆: X₆ {O(n)}
t₅, X₂: X₂+2 {O(n)}
t₆, X₂: X₂+2 {O(n)}