Initial Problem
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-2, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀+1 ≤ X₁
t₀: l2(X₀, X₁) → l1(X₀+1, 1) :|: 0 ≤ X₀
Preprocessing
Found invariant X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁) → l2(X₀, X₁)
t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-2, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀+1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀+1, 1) :|: 0 ≤ X₀
MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-2, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀+1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l2 [X₀+1 ]
l1 [X₀ ]
MPRF for transition t₀: l2(X₀, X₁) → l1(X₀+1, 1) :|: 0 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l2 [X₀+1 ]
l1 [X₀-1 ]
MPRF for transition t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
MPRF:
l1 [2⋅X₀-X₁ ]
l2 [X₀+1 ]
Analysing control-flow refined program
Cut unsatisfiable transition t₂: l1→l2
Found invariant X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1
Found invariant X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1
knowledge_propagation leads to new time bound X₀+1 {O(n)} for transition t₃₃: l1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀
MPRF for transition t₃₂: n_l1___1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 1 ≤ X₁ ∧ 2 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
2⋅X₀⋅X₀+4⋅X₀+2 {O(n^2)}
MPRF:
l1 [X₀-X₁ ]
n_l1___1 [X₀+1-X₁ ]
l2 [-X₁ ]
MPRF for transition t₃₆: n_l1___1(X₀, X₁) → l2(X₀-2, X₁) :|: 0 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀+1 ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:
new bound:
X₀+1 {O(n)}
MPRF:
l1 [X₀ ]
n_l1___1 [X₀ ]
l2 [X₀+1 ]
CFR did not improve the program. Rolling back
All Bounds
Timebounds
Overall timebound:4⋅X₀⋅X₀+9⋅X₀+6 {O(n^2)}
t₃: 1 {O(1)}
t₁: 4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
t₂: X₀+1 {O(n)}
t₀: X₀+1 {O(n)}
Costbounds
Overall costbound: 4⋅X₀⋅X₀+9⋅X₀+6 {O(n^2)}
t₃: 1 {O(1)}
t₁: 4⋅X₀⋅X₀+7⋅X₀+3 {O(n^2)}
t₂: X₀+1 {O(n)}
t₀: X₀+1 {O(n)}
Sizebounds
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₁, X₀: 2⋅X₀+1 {O(n)}
t₁, X₁: 4⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}
t₂, X₀: 2⋅X₀+1 {O(n)}
t₂, X₁: 4⋅X₀⋅X₀+7⋅X₀+4 {O(n^2)}
t₀, X₀: 2⋅X₀+1 {O(n)}
t₀, X₁: 1 {O(1)}