Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₂: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 0 < X₀
t₃: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l4(X₀, X₁, X₂, X₁, X₂, X₅, X₆, X₇, X₈) :|: X₀ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l1(X₅, X₆, X₇, X₃, X₄, X₅, X₆, X₇, X₈)
t₄: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₅: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 0 < X₃+X₄
t₆: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: X₃+X₄ ≤ 0
t₇: l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l4(X₀, X₁, X₂, X₃-1, X₄-1, X₅, X₆, X₇, X₈)
t₈: l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → l7(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
Preprocessing
Eliminate variables {X₈} that do not contribute to the problem
Found invariant X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location l6
Found invariant X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location l7
Found invariant X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₁+X₄ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ 0 for location l5
Found invariant X₀ ≤ X₅ for location l1
Found invariant X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 for location l4
Found invariant 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₀ for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₇: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₁₈: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 0 < X₀ ∧ X₀ ≤ X₅
t₁₉: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l4(X₀, X₁, X₂, X₁, X₂, X₅, X₆, X₇) :|: X₀ ≤ 0 ∧ X₀ ≤ X₅
t₂₀: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l1(X₅, X₆, X₇, X₃, X₄, X₅, X₆, X₇)
t₂₁: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₀
t₂₂: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 0 < X₃+X₄ ∧ X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0
t₂₃: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃+X₄ ≤ 0 ∧ X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0
t₂₄: l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l4(X₀, X₁, X₂, X₃-1, X₄-1, X₅, X₆, X₇) :|: X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₁+X₄ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ 0
t₂₅: l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l7(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₀ ≤ X₅ ∧ X₃+X₄ ≤ 0 ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0
Solv. Size Bound: t₁₈: l1→l3 for X₁
cycle: [t₁₈: l1→l3; t₂₁: l3→l1]
loop: (0 < X₀,(X₁,X₂) -> (3⋅X₁+2⋅X₂,-5⋅X₁-3⋅X₂)
overappr. closed-form: 2⋅X₂+6⋅X₁ {O(n)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₁₈: l1→l3 and X₁: 2⋅X₇+6⋅X₆ {O(n)}
Solv. Size Bound: t₁₈: l1→l3 for X₂
cycle: [t₁₈: l1→l3; t₂₁: l3→l1]
loop: (0 < X₀,(X₁,X₂) -> (3⋅X₁+2⋅X₂,-5⋅X₁-3⋅X₂)
overappr. closed-form: 6⋅X₁+6⋅X₂ {O(n)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₁₈: l1→l3 and X₂: 6⋅X₆+6⋅X₇ {O(n)}
Solv. Size Bound: t₂₁: l3→l1 for X₁
cycle: [t₂₁: l3→l1; t₁₈: l1→l3]
loop: (1 < X₀,(X₁,X₂) -> (3⋅X₁+2⋅X₂,-5⋅X₁-3⋅X₂)
overappr. closed-form: 2⋅X₂+6⋅X₁ {O(n)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₂₁: l3→l1 and X₁: 2⋅X₇+6⋅X₆ {O(n)}
Solv. Size Bound: t₂₁: l3→l1 for X₂
cycle: [t₂₁: l3→l1; t₁₈: l1→l3]
loop: (1 < X₀,(X₁,X₂) -> (3⋅X₁+2⋅X₂,-5⋅X₁-3⋅X₂)
overappr. closed-form: 6⋅X₁+6⋅X₂ {O(n)}
runtime bound: X₀+1 {O(n)}
Solv. Size Bound - Lifting for t₂₁: l3→l1 and X₂: 6⋅X₆+6⋅X₇ {O(n)}
MPRF for transition t₁₈: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 0 < X₀ ∧ X₀ ≤ X₅ of depth 1:
new bound:
X₅ {O(n)}
MPRF for transition t₂₁: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l1(X₀-1, 3⋅X₁+2⋅X₂, -5⋅X₁-3⋅X₂, X₃, X₄, X₅, X₆, X₇) :|: 1 ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₅ {O(n)}
MPRF for transition t₂₂: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 0 < X₃+X₄ ∧ X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₀ ≤ 0 of depth 1:
new bound:
13⋅X₆+9⋅X₇ {O(n)}
MPRF for transition t₂₄: l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → l4(X₀, X₁, X₂, X₃-1, X₄-1, X₅, X₆, X₇) :|: X₀ ≤ X₅ ∧ X₄ ≤ X₂ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₁+X₄ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ 0 of depth 1:
new bound:
13⋅X₆+9⋅X₇ {O(n)}
All Bounds
Timebounds
Overall timebound:18⋅X₇+2⋅X₅+26⋅X₆+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₅ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₅ {O(n)}
t₂₂: 13⋅X₆+9⋅X₇ {O(n)}
t₂₃: 1 {O(1)}
t₂₄: 13⋅X₆+9⋅X₇ {O(n)}
t₂₅: 1 {O(1)}
Costbounds
Overall costbound: 18⋅X₇+2⋅X₅+26⋅X₆+5 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₅ {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₅ {O(n)}
t₂₂: 13⋅X₆+9⋅X₇ {O(n)}
t₂₃: 1 {O(1)}
t₂₄: 13⋅X₆+9⋅X₇ {O(n)}
t₂₅: 1 {O(1)}
Sizebounds
t₁₇, X₀: X₀ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₇, X₅: X₅ {O(n)}
t₁₇, X₆: X₆ {O(n)}
t₁₇, X₇: X₇ {O(n)}
t₁₈, X₀: X₅ {O(n)}
t₁₈, X₁: 2⋅X₇+6⋅X₆ {O(n)}
t₁₈, X₂: 6⋅X₆+6⋅X₇ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₈, X₄: X₄ {O(n)}
t₁₈, X₅: X₅ {O(n)}
t₁₈, X₆: X₆ {O(n)}
t₁₈, X₇: X₇ {O(n)}
t₁₉, X₀: 2⋅X₅ {O(n)}
t₁₉, X₁: 2⋅X₇+7⋅X₆ {O(n)}
t₁₉, X₂: 6⋅X₆+7⋅X₇ {O(n)}
t₁₉, X₃: 2⋅X₇+7⋅X₆ {O(n)}
t₁₉, X₄: 6⋅X₆+7⋅X₇ {O(n)}
t₁₉, X₅: 2⋅X₅ {O(n)}
t₁₉, X₆: 2⋅X₆ {O(n)}
t₁₉, X₇: 2⋅X₇ {O(n)}
t₂₀, X₀: X₅ {O(n)}
t₂₀, X₁: X₆ {O(n)}
t₂₀, X₂: X₇ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₀, X₅: X₅ {O(n)}
t₂₀, X₆: X₆ {O(n)}
t₂₀, X₇: X₇ {O(n)}
t₂₁, X₀: X₅ {O(n)}
t₂₁, X₁: 2⋅X₇+6⋅X₆ {O(n)}
t₂₁, X₂: 6⋅X₆+6⋅X₇ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₁, X₆: X₆ {O(n)}
t₂₁, X₇: X₇ {O(n)}
t₂₂, X₀: 2⋅X₅ {O(n)}
t₂₂, X₁: 2⋅X₇+7⋅X₆ {O(n)}
t₂₂, X₂: 6⋅X₆+7⋅X₇ {O(n)}
t₂₂, X₃: 11⋅X₇+20⋅X₆ {O(n)}
t₂₂, X₄: 16⋅X₇+19⋅X₆ {O(n)}
t₂₂, X₅: 2⋅X₅ {O(n)}
t₂₂, X₆: 2⋅X₆ {O(n)}
t₂₂, X₇: 2⋅X₇ {O(n)}
t₂₃, X₀: 4⋅X₅ {O(n)}
t₂₃, X₁: 14⋅X₆+4⋅X₇ {O(n)}
t₂₃, X₂: 12⋅X₆+14⋅X₇ {O(n)}
t₂₃, X₃: 13⋅X₇+27⋅X₆ {O(n)}
t₂₃, X₄: 23⋅X₇+25⋅X₆ {O(n)}
t₂₃, X₅: 4⋅X₅ {O(n)}
t₂₃, X₆: 4⋅X₆ {O(n)}
t₂₃, X₇: 4⋅X₇ {O(n)}
t₂₄, X₀: 2⋅X₅ {O(n)}
t₂₄, X₁: 2⋅X₇+7⋅X₆ {O(n)}
t₂₄, X₂: 6⋅X₆+7⋅X₇ {O(n)}
t₂₄, X₃: 11⋅X₇+20⋅X₆ {O(n)}
t₂₄, X₄: 16⋅X₇+19⋅X₆ {O(n)}
t₂₄, X₅: 2⋅X₅ {O(n)}
t₂₄, X₆: 2⋅X₆ {O(n)}
t₂₄, X₇: 2⋅X₇ {O(n)}
t₂₅, X₀: 4⋅X₅ {O(n)}
t₂₅, X₁: 14⋅X₆+4⋅X₇ {O(n)}
t₂₅, X₂: 12⋅X₆+14⋅X₇ {O(n)}
t₂₅, X₃: 13⋅X₇+27⋅X₆ {O(n)}
t₂₅, X₄: 23⋅X₇+25⋅X₆ {O(n)}
t₂₅, X₅: 4⋅X₅ {O(n)}
t₂₅, X₆: 4⋅X₆ {O(n)}
t₂₅, X₇: 4⋅X₇ {O(n)}