Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂-1) :|: X₁+1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ X₁
t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
Preprocessing
Found invariant X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂-1) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀
t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
knowledge_propagation leads to new time bound 1 {O(1)} for transition t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
Found invariant X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ for location l1
Time-Bound by TWN-Loops:
TWN-Loops: t₁ 2⋅X₀+2⋅X₁+6 {O(n)}
TWN-Loops:
entry: t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
results in twn-loop: twn:Inv: [X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀] , (X₀,X₁,X₂) -> (X₀-1,X₁,X₂-1) :|: X₁+1 ≤ X₀
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀ + [[n != 0]] * -1 * n^1
X₁: X₁
X₂: X₂ + [[n != 0]] * -1 * n^1
Termination: true
Formula:
1 < 0
∨ X₁+1 < X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
Stabilization-Threshold for: X₁+1 ≤ X₀
alphas_abs: X₁+1+X₀
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}
relevant size-bounds w.r.t. t₀:
X₀: X₀ {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₀: 1 {O(1)}
Results in: 2⋅X₀+2⋅X₁+6 {O(n)}
2⋅X₀+2⋅X₁+6 {O(n)}
Found invariant 1 ≤ 0 for location l2
Found invariant 1 ≤ 0 for location l1
Found invariant X₂ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location l1
Time-Bound by TWN-Loops:
TWN-Loops: t₂ 12⋅X₀+12⋅X₁+42 {O(n)}
TWN-Loops:
entry: t₁: l1(X₀, X₁, X₂) → l1(X₀-1, X₁, X₂-1) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀
results in twn-loop: twn:Inv: [X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁,X₂) :|: X₀ ≤ X₁ ∧ X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
entry: t₃: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
results in twn-loop: twn:Inv: [X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀] , (X₀,X₁,X₂) -> (X₀,X₁,X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₁
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂
Termination: true
Formula:
X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 < X₀ ∧ X₀ < X₁
∨ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ < X₁
∨ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀ ∧ X₀ < X₁
∨ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ < X₁
∨ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 < X₀ ∧ X₀ < X₁
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ < X₁
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀ ∧ X₀ < X₁
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ < X₁
∨ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀
relevant size-bounds w.r.t. t₁:
Runtime-bound of t₁: 2⋅X₀+2⋅X₁+6 {O(n)}
Results in: 12⋅X₀+12⋅X₁+36 {O(n)}
order: [X₀; X₁; X₂]
closed-form:
X₀: X₀
X₁: X₁
X₂: X₂
Termination: true
Formula:
X₀ < X₁ ∧ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 < X₀
∨ X₀ < X₁ ∧ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ < X₁ ∧ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀
∨ X₀ < X₁ ∧ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 < X₀
∨ X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀
∨ X₀ < X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 < X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < X₂ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ < X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 < X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ < X₀ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 < X₀
∨ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₁+1 ≤ X₀ ∧ X₀ ≤ X₁+1
relevant size-bounds w.r.t. t₃:
Runtime-bound of t₃: 1 {O(1)}
Results in: 6 {O(1)}
12⋅X₀+12⋅X₁+42 {O(n)}
All Bounds
Timebounds
Overall timebound:14⋅X₀+14⋅X₁+50 {O(n)}
t₃: 1 {O(1)}
t₁: 2⋅X₀+2⋅X₁+6 {O(n)}
t₂: 12⋅X₀+12⋅X₁+42 {O(n)}
t₀: 1 {O(1)}
Costbounds
Overall costbound: 14⋅X₀+14⋅X₁+50 {O(n)}
t₃: 1 {O(1)}
t₁: 2⋅X₀+2⋅X₁+6 {O(n)}
t₂: 12⋅X₀+12⋅X₁+42 {O(n)}
t₀: 1 {O(1)}
Sizebounds
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₁, X₀: 2⋅X₁+3⋅X₀+6 {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₀+2⋅X₁+X₂+6 {O(n)}
t₂, X₀: 2⋅X₁+3⋅X₀+6 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₀+2⋅X₁+X₂+6 {O(n)}
t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}