Initial Problem
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₁: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₂+1 ≤ X₁
t₀: l1(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₁+1 ≤ X₀
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂+2 ≤ X₁ ∧ X₀ ≤ X₁
Preprocessing
Found invariant 1+X₂ ≤ X₁ for location l1
Problem after Preprocessing
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1
Transitions:
t₁: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₂+1 ≤ X₁
t₀: l1(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₁+1 ≤ X₀ ∧ 1+X₂ ≤ X₁
t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂+2 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁
MPRF for transition t₀: l1(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₁+1 ≤ X₀ ∧ 1+X₂ ≤ X₁ of depth 1:
new bound:
X₀+X₁ {O(n)}
MPRF:
l1 [X₀-X₁ ]
Found invariant 1 ≤ 0 for location l1
Found invariant 1+X₂ ≤ X₁ for location l1
MPRF for transition t₂: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂+1) :|: X₂+2 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ of depth 1:
new bound:
2⋅X₁⋅X₁+3⋅X₀⋅X₁+X₀⋅X₀+X₀⋅X₂+X₁⋅X₂+X₁+X₂ {O(n^2)}
MPRF:
l1 [X₁-X₂ ]
Analysing control-flow refined program
Found invariant 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location n_l1___2
Found invariant 1+X₂ ≤ X₁ for location l1
Found invariant 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ for location n_l1___1
MPRF for transition t₄₉: n_l1___2(X₀, X₁, X₂) → n_l1___2(X₀, X₁+1, X₂) :|: 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ X₁ ≤ X₀ of depth 1:
new bound:
X₀+X₁+2 {O(n)}
MPRF:
n_l1___2 [X₀+1-X₁ ]
MPRF for transition t₄₇: n_l1___1(X₀, X₁, X₂) → n_l1___1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ of depth 1:
new bound:
3⋅X₂+4⋅X₁+X₀+7 {O(n)}
MPRF:
n_l1___1 [X₁-X₂ ]
CFR: Improvement to new bound with the following program:
new bound:
2⋅X₀+3⋅X₂+5⋅X₁+9 {O(n)}
cfr-program:
Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, n_l1___1, n_l1___2
Transitions:
t₁: l0(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₂+1 ≤ X₁
t₅₀: l1(X₀, X₁, X₂) → n_l1___1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁
t₅₁: l1(X₀, X₁, X₂) → n_l1___2(X₀, X₁+1, X₂) :|: 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁
t₄₇: n_l1___1(X₀, X₁, X₂) → n_l1___1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ X₀ ≤ X₁
t₄₈: n_l1___2(X₀, X₁, X₂) → n_l1___1(X₀, X₁, X₂+1) :|: 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 2+X₂ ≤ X₁ ∧ X₀ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ X₁ ≤ X₀
t₄₉: n_l1___2(X₀, X₁, X₂) → n_l1___2(X₀, X₁+1, X₂) :|: 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ X₁ ≤ X₀
All Bounds
Timebounds
Overall timebound:2⋅X₀+3⋅X₂+5⋅X₁+13 {O(n)}
t₁: 1 {O(1)}
t₅₀: 1 {O(1)}
t₅₁: 1 {O(1)}
t₄₇: 3⋅X₂+4⋅X₁+X₀+7 {O(n)}
t₄₈: 1 {O(1)}
t₄₉: X₀+X₁+2 {O(n)}
Costbounds
Overall costbound: 2⋅X₀+3⋅X₂+5⋅X₁+13 {O(n)}
t₁: 1 {O(1)}
t₅₀: 1 {O(1)}
t₅₁: 1 {O(1)}
t₄₇: 3⋅X₂+4⋅X₁+X₀+7 {O(n)}
t₄₈: 1 {O(1)}
t₄₉: X₀+X₁+2 {O(n)}
Sizebounds
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₅₀, X₀: X₀ {O(n)}
t₅₀, X₁: X₁ {O(n)}
t₅₀, X₂: X₂+1 {O(n)}
t₅₁, X₀: X₀ {O(n)}
t₅₁, X₁: X₁+1 {O(n)}
t₅₁, X₂: X₂ {O(n)}
t₄₇, X₀: 3⋅X₀ {O(n)}
t₄₇, X₁: 4⋅X₁+X₀+4 {O(n)}
t₄₇, X₂: 4⋅X₁+6⋅X₂+X₀+10 {O(n)}
t₄₈, X₀: 2⋅X₀ {O(n)}
t₄₈, X₁: 3⋅X₁+X₀+4 {O(n)}
t₄₈, X₂: 2⋅X₂+2 {O(n)}
t₄₉, X₀: X₀ {O(n)}
t₄₉, X₁: 2⋅X₁+X₀+3 {O(n)}
t₄₉, X₂: X₂ {O(n)}