Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₀+X₁
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, -2⋅X₁, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location l5

Found invariant X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location l4

Found invariant 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l3, l4, l5
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 0 < X₀+X₁ ∧ X₀ ≤ X₂
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₄: l3(X₀, X₁, X₂, X₃) → l1(X₀-1, -2⋅X₁, X₂, X₃) :|: 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0

Analysing control-flow refined program

Found invariant X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ for location n_l3___3

Found invariant 1 ≤ X₂+X₃ ∧ 1+X₀ ≤ X₂ for location n_l1___2

Found invariant X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location l5

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location l1

Found invariant X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 for location l4

Found invariant 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ for location n_l3___1

MPRF for transition t₃₈: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₁ ≤ 2⋅X₀ ∧ 1+X₀ ≤ X₂ ∧ 0 < X₀+X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₀ ≤ X₂ of depth 1:

new bound:

3⋅X₂+3 {O(n)}

MPRF:

n_l3___1 [3⋅X₀-3 ]
n_l1___2 [3⋅X₀ ]

MPRF for transition t₄₀: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___2(X₀-1, -2⋅X₁, X₂, X₃) :|: 1+X₀ ≤ X₂ ∧ X₁ ≤ 2⋅X₀ ∧ 0 < X₀+X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁ of depth 1:

new bound:

3⋅X₂+3 {O(n)}

MPRF:

n_l3___1 [3⋅X₀ ]
n_l1___2 [3⋅X₀ ]

CFR: Improvement to new bound with the following program:

new bound:

6⋅X₂+6 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: l0, l1, l2, l4, l5, n_l1___2, n_l3___1, n_l3___3
Transitions:
t₀: l0(X₀, X₁, X₂, X₃) → l2(X₀, X₁, X₂, X₃)
t₃: l1(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₃₉: l1(X₀, X₁, X₂, X₃) → n_l3___3(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 0 < X₀+X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₁: l2(X₀, X₁, X₂, X₃) → l1(X₂, X₃, X₂, X₃)
t₅: l4(X₀, X₁, X₂, X₃) → l5(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₀+X₁ ≤ 0
t₄₅: n_l1___2(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₀+X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₀ ≤ X₂
t₃₈: n_l1___2(X₀, X₁, X₂, X₃) → n_l3___1(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₁ ≤ 2⋅X₀ ∧ 1+X₀ ≤ X₂ ∧ 0 < X₀+X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₀ ≤ X₂
t₄₀: n_l3___1(X₀, X₁, X₂, X₃) → n_l1___2(X₀-1, -2⋅X₁, X₂, X₃) :|: 1+X₀ ≤ X₂ ∧ X₁ ≤ 2⋅X₀ ∧ 0 < X₀+X₁ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁
t₄₁: n_l3___3(X₀, X₁, X₂, X₃) → n_l1___2(X₀-1, -2⋅X₁, X₂, X₃) :|: 0 < X₀+X₃ ∧ X₁ ≤ X₃ ∧ X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀+X₁

All Bounds

Timebounds

Overall timebound:6⋅X₂+13 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₃₉: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 1 {O(1)}
t₃₈: 3⋅X₂+3 {O(n)}
t₄₅: 1 {O(1)}
t₄₀: 3⋅X₂+3 {O(n)}
t₄₁: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₂+13 {O(n)}
t₀: 1 {O(1)}
t₃: 1 {O(1)}
t₃₉: 1 {O(1)}
t₁: 1 {O(1)}
t₅: 1 {O(1)}
t₃₈: 3⋅X₂+3 {O(n)}
t₄₅: 1 {O(1)}
t₄₀: 3⋅X₂+3 {O(n)}
t₄₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃₉, X₀: X₂ {O(n)}
t₃₉, X₁: X₃ {O(n)}
t₃₉, X₂: X₂ {O(n)}
t₃₉, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₅, X₀: 3⋅X₂+2 {O(n)}
t₅, X₁: 2⋅2^(3⋅X₂+3)⋅X₃+3⋅X₃ {O(EXP)}
t₅, X₂: 3⋅X₂ {O(n)}
t₅, X₃: 3⋅X₃ {O(n)}
t₃₈, X₀: X₂+1 {O(n)}
t₃₈, X₁: 2⋅2^(3⋅X₂+3)⋅X₃ {O(EXP)}
t₃₈, X₂: X₂ {O(n)}
t₃₈, X₃: X₃ {O(n)}
t₄₅, X₀: 2⋅X₂+2 {O(n)}
t₄₅, X₁: 2⋅2^(3⋅X₂+3)⋅X₃+2⋅X₃ {O(EXP)}
t₄₅, X₂: 2⋅X₂ {O(n)}
t₄₅, X₃: 2⋅X₃ {O(n)}
t₄₀, X₀: X₂+1 {O(n)}
t₄₀, X₁: 2⋅2^(3⋅X₂+3)⋅X₃ {O(EXP)}
t₄₀, X₂: X₂ {O(n)}
t₄₀, X₃: X₃ {O(n)}
t₄₁, X₀: X₂+1 {O(n)}
t₄₁, X₁: 2⋅X₃ {O(n)}
t₄₁, X₂: X₂ {O(n)}
t₄₁, X₃: X₃ {O(n)}