Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+X₁, X₁+X₂, X₂-1, X₃) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃-1) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₃, X₃, X₃) :|: 1 ≤ X₃

Preprocessing

Found invariant X₀ ≤ 0 for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, X₃)
t₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+X₁, X₁+X₂, X₂-1, X₃) :|: 1 ≤ X₀
t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃-1) :|: X₀ ≤ 0
t₃: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₃, X₃, X₃) :|: 1 ≤ X₃ ∧ X₀ ≤ 0

MPRF for transition t₃: l2(X₀, X₁, X₂, X₃) → l1(X₃, X₃, X₃, X₃) :|: 1 ≤ X₃ ∧ X₀ ≤ 0 of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

l2 [X₃ ]
l1 [X₃-1 ]

MPRF for transition t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃-1) :|: X₀ ≤ 0 of depth 1:

new bound:

X₃+2 {O(n)}

MPRF:

l1 [1 ]
l2 [0 ]

Analysing control-flow refined program

Cut unsatisfiable transition t₄₁: n_l1___1→l2

Found invariant X₀ ≤ 0 for location l2

Found invariant X₃ ≤ 1+X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₃ ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location n_l1___1

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃-1) :|: X₀ ≤ 0

knowledge_propagation leads to new time bound X₃+2 {O(n)} for transition t₃₇: l1(X₀, X₁, X₂, X₃) → n_l1___1(X₀+X₁, X₁+X₂, X₂-1, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀+X₂ ∧ 1 ≤ X₀

knowledge_propagation leads to new time bound X₃+2 {O(n)} for transition t₃₈: l1(X₀, X₁, X₂, X₃) → n_l1___2(X₀+X₁, X₁+X₂, X₂-1, X₃) :|: 1 ≤ X₀

knowledge_propagation leads to new time bound X₃+2 {O(n)} for transition t₃₅: n_l1___1(X₀, X₁, X₂, X₃) → n_l1___2(X₀+X₁, X₁+X₂, X₂-1, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₀ ∧ X₁ ≤ X₀+X₂ ∧ 1 ≤ X₀ ∧ X₃ ≤ 1+X₂ ∧ 1+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1+X₂ ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₃ ∧ 2+X₂ ≤ X₁ ∧ 2+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀

MPRF for transition t₄₂: n_l1___2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃-1) :|: X₀ ≤ 0 of depth 1:

new bound:

X₃+2 {O(n)}

MPRF:

l1 [1 ]
n_l1___1 [1 ]
n_l1___2 [1 ]
l2 [0 ]

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: inf {Infinity}
t₂: X₃+2 {O(n)}
t₃: X₃+1 {O(n)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: inf {Infinity}
t₂: X₃+2 {O(n)}
t₃: X₃+1 {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₃: 3⋅X₃+2 {O(n)}
t₂, X₃: 3⋅X₃+2 {O(n)}
t₃, X₀: 3⋅X₃+2 {O(n)}
t₃, X₁: 3⋅X₃+2 {O(n)}
t₃, X₂: 3⋅X₃+2 {O(n)}
t₃, X₃: 3⋅X₃+2 {O(n)}