KoAT2 Proof Output

Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅) → l2(X₀, X₁, X₂, X₃, X₄, X₅)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l3(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ < X₄
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₀, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₅, X₁, X₂, X₃, X₄, X₅)
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅) → l1(X₀+1, X₁, X₂, X₃, X₄, X₅)
t₅: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l5(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ < X₃
t₆: l4(X₀, X₁, X₂, X₃, X₄, X₅) → l6(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₃ ≤ X₁
t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅) → l4(X₀, X₁+1, X₂, X₃, X₄, X₅)
t₈: l6(X₀, X₁, X₂, X₃, X₄, X₅) → l7(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

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

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l6

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁ for location l7

Found invariant 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁ for location l5

Found invariant X₅ ≤ X₀ for location l1

Found invariant X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ for location l4

Found invariant 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₃, X₄, X₅
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₈: l0(X₀, X₁, X₃, X₄, X₅) → l2(X₀, X₁, X₃, X₄, X₅)
t₁₉: l1(X₀, X₁, X₃, X₄, X₅) → l3(X₀, X₁, X₃, X₄, X₅) :|: X₀ < X₄ ∧ X₅ ≤ X₀
t₂₀: l1(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₀, X₃, X₄, X₅) :|: X₄ ≤ X₀ ∧ X₅ ≤ X₀
t₂₁: l2(X₀, X₁, X₃, X₄, X₅) → l1(X₅, X₁, X₃, X₄, X₅)
t₂₂: l3(X₀, X₁, X₃, X₄, X₅) → l1(X₀+1, X₁, X₃, X₄, X₅) :|: 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄
t₂₃: l4(X₀, X₁, X₃, X₄, X₅) → l5(X₀, X₁, X₃, X₄, X₅) :|: X₁ < X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁
t₂₄: l4(X₀, X₁, X₃, X₄, X₅) → l6(X₀, X₁, X₃, X₄, X₅) :|: X₃ ≤ X₁ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁
t₂₅: l5(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₁+1, X₃, X₄, X₅) :|: 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁
t₂₆: l6(X₀, X₁, X₃, X₄, X₅) → l7(X₀, X₁, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₁

MPRF for transition t₁₉: l1(X₀, X₁, X₃, X₄, X₅) → l3(X₀, X₁, X₃, X₄, X₅) :|: X₀ < X₄ ∧ X₅ ≤ X₀ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

l3 [X₄-X₀-1 ]
l1 [X₄-X₀ ]

MPRF for transition t₂₂: l3(X₀, X₁, X₃, X₄, X₅) → l1(X₀+1, X₁, X₃, X₄, X₅) :|: 1+X₅ ≤ X₄ ∧ X₅ ≤ X₀ ∧ 1+X₀ ≤ X₄ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

l3 [X₄-X₀ ]
l1 [X₄-X₀ ]

MPRF for transition t₂₃: l4(X₀, X₁, X₃, X₄, X₅) → l5(X₀, X₁, X₃, X₄, X₅) :|: X₁ < X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₃+3⋅X₅+X₄ {O(n)}

MPRF:

l5 [X₃-X₁-1 ]
l4 [X₃-X₁ ]

MPRF for transition t₂₅: l5(X₀, X₁, X₃, X₄, X₅) → l4(X₀, X₁+1, X₃, X₄, X₅) :|: 1+X₅ ≤ X₃ ∧ X₅ ≤ X₁ ∧ X₅ ≤ X₀ ∧ 1+X₄ ≤ X₃ ∧ X₄ ≤ X₁ ∧ X₄ ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₃+3⋅X₅+X₄ {O(n)}

MPRF:

l5 [X₃-X₁ ]
l4 [X₃-X₁ ]

All Bounds

Timebounds

Overall timebound:4⋅X₃+4⋅X₄+8⋅X₅+5 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: X₄+X₅ {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₄+X₅ {O(n)}
t₂₃: 2⋅X₃+3⋅X₅+X₄ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 2⋅X₃+3⋅X₅+X₄ {O(n)}
t₂₆: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+4⋅X₄+8⋅X₅+5 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: X₄+X₅ {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: X₄+X₅ {O(n)}
t₂₃: 2⋅X₃+3⋅X₅+X₄ {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 2⋅X₃+3⋅X₅+X₄ {O(n)}
t₂₆: 1 {O(1)}

Sizebounds

t₁₈, X₀: X₀ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₉, X₀: 2⋅X₅+X₄ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}
t₂₀, X₀: 3⋅X₅+X₄ {O(n)}
t₂₀, X₁: 3⋅X₅+X₄ {O(n)}
t₂₀, X₃: 2⋅X₃ {O(n)}
t₂₀, X₄: 2⋅X₄ {O(n)}
t₂₀, X₅: 2⋅X₅ {O(n)}
t₂₁, X₀: X₅ {O(n)}
t₂₁, X₁: X₁ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₂, X₀: 2⋅X₅+X₄ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: 3⋅X₅+X₄ {O(n)}
t₂₃, X₁: 2⋅X₃+2⋅X₄+6⋅X₅ {O(n)}
t₂₃, X₃: 2⋅X₃ {O(n)}
t₂₃, X₄: 2⋅X₄ {O(n)}
t₂₃, X₅: 2⋅X₅ {O(n)}
t₂₄, X₀: 2⋅X₄+6⋅X₅ {O(n)}
t₂₄, X₁: 2⋅X₃+3⋅X₄+9⋅X₅ {O(n)}
t₂₄, X₃: 4⋅X₃ {O(n)}
t₂₄, X₄: 4⋅X₄ {O(n)}
t₂₄, X₅: 4⋅X₅ {O(n)}
t₂₅, X₀: 3⋅X₅+X₄ {O(n)}
t₂₅, X₁: 2⋅X₃+2⋅X₄+6⋅X₅ {O(n)}
t₂₅, X₃: 2⋅X₃ {O(n)}
t₂₅, X₄: 2⋅X₄ {O(n)}
t₂₅, X₅: 2⋅X₅ {O(n)}
t₂₆, X₀: 2⋅X₄+6⋅X₅ {O(n)}
t₂₆, X₁: 2⋅X₃+3⋅X₄+9⋅X₅ {O(n)}
t₂₆, X₃: 4⋅X₃ {O(n)}
t₂₆, X₄: 4⋅X₄ {O(n)}
t₂₆, X₅: 4⋅X₅ {O(n)}