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) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁+1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁
t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 1 ≤ X₀

Preprocessing

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ 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) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁+1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀
t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 1 ≤ X₀

MPRF for transition t₂: l1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l2 [X₀ ]
l1 [X₀ ]

MPRF for transition t₀: l2(X₀, X₁) → l1(X₀, 0) :|: 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l2 [X₀ ]
l1 [X₀-1 ]

MPRF for transition t₁: l1(X₀, X₁) → l1(X₀, X₁+1) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁+1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

2⋅X₀⋅X₀ {O(n^2)}

MPRF:

l1 [2⋅X₀-X₁ ]
l2 [X₀ ]

Analysing control-flow refined program

Cut unsatisfiable transition t₂: l1→l2

Found invariant X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location l1

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location n_l1___1

knowledge_propagation leads to new time bound X₀ {O(n)} for transition t₃₃: l1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀

MPRF for transition t₃₂: n_l1___1(X₀, X₁) → n_l1___1(X₀, X₁+1) :|: 0 ≤ X₁ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀⋅X₀ {O(n^2)}

MPRF:

l1 [X₀-X₁ ]
n_l1___1 [X₀+1-X₁ ]
l2 [0 ]

MPRF for transition t₃₆: n_l1___1(X₀, X₁) → l2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

l1 [X₀ ]
n_l1___1 [X₀ ]
l2 [X₀ ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+2⋅X₀+1 {O(n^2)}
t₃: 1 {O(1)}
t₁: 2⋅X₀⋅X₀ {O(n^2)}
t₂: X₀ {O(n)}
t₀: X₀ {O(n)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+2⋅X₀+1 {O(n^2)}
t₃: 1 {O(1)}
t₁: 2⋅X₀⋅X₀ {O(n^2)}
t₂: X₀ {O(n)}
t₀: X₀ {O(n)}

Sizebounds

t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 2⋅X₀⋅X₀ {O(n^2)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: 2⋅X₀⋅X₀ {O(n^2)}
t₀, X₀: X₀ {O(n)}
t₀, X₁: 0 {O(1)}