Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, 10, X₂, X₃) :|: X₀ ≤ 10
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 10 < X₀
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₁, X₂, X₃)
t₅: l3(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1
t₄: l3(X₀, X₁, X₂, X₃) → l6(X₀, X₁, X₂, X₃) :|: 1 < X₁
t₈: l4(X₀, X₁, X₂, X₃) → l7(X₀, X₁, X₂, X₃)
t₇: l5(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, X₂, X₃)
t₆: l6(X₀, X₁, X₂, X₃) → l3(X₀, X₁-1, X₂, X₃)
Preprocessing
Eliminate variables {X₃} that do not contribute to the problem
Found invariant X₂ ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10 for location l6
Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location l7
Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10 for location l5
Found invariant X₂ ≤ X₀ for location l1
Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location l4
Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 for location l3
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5, l6, l7
Transitions:
t₁₇: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁₈: l1(X₀, X₁, X₂) → l3(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀
t₁₉: l1(X₀, X₁, X₂) → l4(X₀, X₁, X₂) :|: 10 < X₀ ∧ X₂ ≤ X₀
t₂₀: l2(X₀, X₁, X₂) → l1(X₂, X₁, X₂)
t₂₁: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10
t₂₂: l3(X₀, X₁, X₂) → l6(X₀, X₁, X₂) :|: 1 < X₁ ∧ X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10
t₂₃: l4(X₀, X₁, X₂) → l7(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 11 ≤ X₀
t₂₄: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10
t₂₅: l6(X₀, X₁, X₂) → l3(X₀, X₁-1, X₂) :|: X₂ ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10
MPRF for transition t₁₈: l1(X₀, X₁, X₂) → l3(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀ of depth 1:
new bound:
X₂+11 {O(n)}
MPRF for transition t₂₁: l3(X₀, X₁, X₂) → l5(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 of depth 1:
new bound:
X₂+11 {O(n)}
MPRF for transition t₂₄: l5(X₀, X₁, X₂) → l1(X₀+1, X₁, X₂) :|: X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10 of depth 1:
new bound:
X₂+11 {O(n)}
TWN: t₂₂: l3→l6
cycle: [t₂₂: l3→l6; t₂₅: l6→l3]
loop: (1 < X₁,(X₁) -> (X₁-1)
order: [X₁]
closed-form:
X₁: X₁ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ 1 < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1
Stabilization-Threshold for: 1 < X₁
alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}
TWN - Lifting for t₂₂: l3→l6 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₁₈:
X₁: 10 {O(1)}
Runtime-bound of t₁₈: X₂+11 {O(n)}
Results in: 26⋅X₂+286 {O(n)}
TWN: t₂₅: l6→l3
TWN - Lifting for t₂₅: l6→l3 of 2⋅X₁+6 {O(n)}
relevant size-bounds w.r.t. t₁₈:
X₁: 10 {O(1)}
Runtime-bound of t₁₈: X₂+11 {O(n)}
Results in: 26⋅X₂+286 {O(n)}
All Bounds
Timebounds
Overall timebound:55⋅X₂+609 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₂+11 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂+11 {O(n)}
t₂₂: 26⋅X₂+286 {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₂+11 {O(n)}
t₂₅: 26⋅X₂+286 {O(n)}
Costbounds
Overall costbound: 55⋅X₂+609 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₂+11 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 1 {O(1)}
t₂₁: X₂+11 {O(n)}
t₂₂: 26⋅X₂+286 {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₂+11 {O(n)}
t₂₅: 26⋅X₂+286 {O(n)}
Sizebounds
t₁₇, X₀: X₀ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₀: 2⋅X₂+11 {O(n)}
t₁₈, X₁: 10 {O(1)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₀: 3⋅X₂+11 {O(n)}
t₁₉, X₁: X₁+9 {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: 2⋅X₂+11 {O(n)}
t₂₁, X₁: 9 {O(1)}
t₂₁, X₂: X₂ {O(n)}
t₂₂, X₀: 2⋅X₂+11 {O(n)}
t₂₂, X₁: 10 {O(1)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: 3⋅X₂+11 {O(n)}
t₂₃, X₁: X₁+9 {O(n)}
t₂₃, X₂: 2⋅X₂ {O(n)}
t₂₄, X₀: 2⋅X₂+11 {O(n)}
t₂₄, X₁: 9 {O(1)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: 2⋅X₂+11 {O(n)}
t₂₅, X₁: 9 {O(1)}
t₂₅, X₂: X₂ {O(n)}