Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀
Temp_Vars: V, W, X, Y, Z
Locations: l0, l1, l2, l3, l4, l5, l6
Transitions:
t₁₂: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l1(X₀, X₁, X₂, W, X₄, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, 0) :|: W ≤ 0
t₁₃: l0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l1(X₀, X₁, X₂, W, X₄, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, 0) :|: 1 ≤ W
t₁₁: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l1(X₀+1, X₁, 1, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, V, X₁₆, X₁₆, 1, 1, 0, X₂₀) :|: V ≤ 0 ∧ X₀+1 ≤ X₁
t₉: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l2(X₀, X₁, Z, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, V, X₁₆, 0, Z, Z, 0, X₂₀) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ 2 ≤ Z
t₁₀: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l2(X₀, X₁, Z, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, V, X₁₆, 0, Z, Z, 0, X₂₀) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0
t₀: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₁ ≤ X₀
t₈: l1(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: 1 ≤ V ∧ X₀+1 ≤ X₁
t₁: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: 3 ≤ X₂
t₂: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₂ ≤ 1
t₇: l2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l3(X₀, X₁, 2, X₃, X₄+1, X₅, X₇, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₂ ≤ 2 ∧ 2 ≤ X₂
t₅: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l4(X₀, X₁, X₂, X₃, 0, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: 1 ≤ X₃
t₆: l3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l4(X₀, X₁, X₂, X₃, 0, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₃ ≤ 0
t₃: l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l4(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀)
t₄: l6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → l5(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀)

Preprocessing

Cut unreachable locations [l5; l6] from the program graph

Eliminate variables {X,Y,X₄,X₅,X₆,X₇,X₈,X₉,X₁₀,X₁₁,X₁₂,X₁₃,X₁₄,X₁₅,X₁₆,X₁₇,X₁₈,X₁₉,X₂₀} that do not contribute to the problem

Found invariant 1+X₀ ≤ X₁ for location l2

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: V, W, Z
Locations: l0, l1, l2, l3, l4
Transitions:
t₃₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, W) :|: W ≤ 0
t₃₁: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, W) :|: 1 ≤ W
t₃₆: l1(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, 1, X₃) :|: V ≤ 0 ∧ X₀+1 ≤ X₁
t₃₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ 2 ≤ Z
t₃₅: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0
t₃₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₃₃: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ V ∧ X₀+1 ≤ X₁
t₃₇: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 3 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₈: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1 ∧ 1+X₀ ≤ X₁
t₃₉: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, 2, X₃) :|: X₂ ≤ 2 ∧ 2 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₄₀: l3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃
t₄₁: l3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₄₂: l4(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)

Analysing control-flow refined program

Found invariant 1+X₀ ≤ X₁ for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location n_l1___2

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

Found invariant 1+X₀ ≤ X₁ for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location n_l1___2

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

Time-Bound by TWN-Loops:

TWN-Loops: t₁₀₉ 4⋅X₀+4⋅X₁+13 {O(n)}

TWN-Loops:

entry: t₁₁₁: l1(X₀, X₁, X₂, X₃) → n_l1___1(X₀+1, X₁, 1, X₃) :|: 1 ≤ X₃ ∧ 1+X₀ ≤ X₁
results in twn-loop: twn:Inv: [1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁] , (X₀,X₁,X₂,X₃) -> (X₀+1,X₁,1,X₃) :|: 1 ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁
X₂: [[n == 0]] * X₂ + [[n != 0]]
X₃: X₃

Termination: true
Formula:

1 < 0 ∧ 1 < X₃
∨ 1 < 0 ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 < 0 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1 < 0 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₃
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < X₃
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 1 < X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₃
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ 1 < X₃
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 < X₃ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1 ∧ 1 < X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ X₃ ≤ 1

Stabilization-Threshold for: 1+X₀ ≤ X₁
alphas_abs: 1+X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}
Stabilization-Threshold for: X₀ ≤ X₁
alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₁₁₁:
X₀: X₀+1 {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁₁₁: 1 {O(1)}
Results in: 4⋅X₀+4⋅X₁+13 {O(n)}

4⋅X₀+4⋅X₁+13 {O(n)}

Found invariant 1+X₀ ≤ X₁ for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location n_l1___2

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

Time-Bound by TWN-Loops:

TWN-Loops: t₁₁₀ 4⋅X₀+4⋅X₁+13 {O(n)}

TWN-Loops:

entry: t₁₁₂: l1(X₀, X₁, X₂, X₃) → n_l1___2(X₀+1, X₁, 1, X₃) :|: X₃ ≤ 0 ∧ 1+X₀ ≤ X₁
results in twn-loop: twn:Inv: [X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁] , (X₀,X₁,X₂,X₃) -> (X₀+1,X₁,1,X₃) :|: X₃ ≤ 0 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁
order: [X₀; X₁; X₂; X₃]
closed-form:
X₀: X₀ + [[n != 0]] * n^1
X₁: X₁
X₂: [[n == 0]] * X₂ + [[n != 0]]
X₃: X₃

Termination: true
Formula:

1 < 0 ∧ X₃ < 0
∨ 1 < 0 ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1 < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ < 0
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1 < 0 ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ < 0
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1 < 0 ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ X₃ < 0
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1 < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ < 0
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1+X₀ < X₁ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ < 0
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1+X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ X₃ < 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1 ≤ 0 ∧ 0 ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ < 0
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ X₀ < X₁ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ < 0
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ < 0 ∧ X₃ ≤ 0 ∧ 0 ≤ X₃
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃ ∧ X₃ < 0
∨ 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ 0 ∧ 0 ≤ 1 ∧ X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 0 ≤ X₃

Stabilization-Threshold for: 1+X₀ ≤ X₁
alphas_abs: 1+X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+4 {O(n)}
Stabilization-Threshold for: X₀ ≤ X₁
alphas_abs: X₀+X₁
M: 0
N: 1
Bound: 2⋅X₀+2⋅X₁+2 {O(n)}

relevant size-bounds w.r.t. t₁₁₂:
X₀: X₀+1 {O(n)}
X₁: X₁ {O(n)}
Runtime-bound of t₁₁₂: 1 {O(1)}
Results in: 4⋅X₀+4⋅X₁+13 {O(n)}

4⋅X₀+4⋅X₁+13 {O(n)}

CFR: Improvement to new bound with the following program:

new bound:

8⋅X₀+8⋅X₁+26 {O(n)}

cfr-program:

Start: l0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: V, W, Z
Locations: l0, l1, l2, l3, l4, n_l1___1, n_l1___2
Transitions:
t₃₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, W) :|: W ≤ 0
t₃₁: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, W) :|: 1 ≤ W
t₃₄: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ 2 ≤ Z
t₃₅: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0
t₃₂: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₃₃: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ V ∧ X₀+1 ≤ X₁
t₁₁₁: l1(X₀, X₁, X₂, X₃) → n_l1___1(X₀+1, X₁, 1, X₃) :|: 1 ≤ X₃ ∧ 1+X₀ ≤ X₁
t₁₁₂: l1(X₀, X₁, X₂, X₃) → n_l1___2(X₀+1, X₁, 1, X₃) :|: X₃ ≤ 0 ∧ 1+X₀ ≤ X₁
t₃₇: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 3 ≤ X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁
t₃₈: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1 ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁
t₃₉: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, 2, X₃) :|: X₂ ≤ 2 ∧ 2 ≤ X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁
t₄₀: l3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃
t₄₁: l3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₄₂: l4(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)
t₁₁₈: n_l1___1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ 2 ≤ Z ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₂₀: n_l1___1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0 ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₂₂: n_l1___1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₂₄: n_l1___1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ V ∧ X₀+1 ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₀₉: n_l1___1(X₀, X₁, X₂, X₃) → n_l1___1(X₀+1, X₁, 1, X₃) :|: 1 ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₁₉: n_l1___2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ 2 ≤ Z ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₂₁: n_l1___2(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0 ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₂₃: n_l1___2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₂₅: n_l1___2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ V ∧ X₀+1 ≤ X₁ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁
t₁₁₀: n_l1___2(X₀, X₁, X₂, X₃) → n_l1___2(X₀+1, X₁, 1, X₃) :|: X₃ ≤ 0 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁

Analysing control-flow refined program

Found invariant 1+X₀ ≤ X₁ for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location n_l1___2

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

Found invariant 1+X₀ ≤ X₁ for location l2

Found invariant X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₂+X₃ ≤ 1 ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₀ ≤ X₁ for location n_l1___2

Found invariant X₃ ≤ 0 for location l4

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

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}
t₃₂: 1 {O(1)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}
t₃₅: 1 {O(1)}
t₁₁₁: 1 {O(1)}
t₁₁₂: 1 {O(1)}
t₃₇: 1 {O(1)}
t₃₈: 1 {O(1)}
t₃₉: 1 {O(1)}
t₄₀: 1 {O(1)}
t₄₁: 1 {O(1)}
t₄₂: inf {Infinity}
t₁₀₉: 4⋅X₀+4⋅X₁+13 {O(n)}
t₁₁₈: 1 {O(1)}
t₁₂₀: 1 {O(1)}
t₁₂₂: 1 {O(1)}
t₁₂₄: 1 {O(1)}
t₁₁₀: 4⋅X₀+4⋅X₁+13 {O(n)}
t₁₁₉: 1 {O(1)}
t₁₂₁: 1 {O(1)}
t₁₂₃: 1 {O(1)}
t₁₂₅: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}
t₃₂: 1 {O(1)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}
t₃₅: 1 {O(1)}
t₁₁₁: 1 {O(1)}
t₁₁₂: 1 {O(1)}
t₃₇: 1 {O(1)}
t₃₈: 1 {O(1)}
t₃₉: 1 {O(1)}
t₄₀: 1 {O(1)}
t₄₁: 1 {O(1)}
t₄₂: inf {Infinity}
t₁₀₉: 4⋅X₀+4⋅X₁+13 {O(n)}
t₁₁₈: 1 {O(1)}
t₁₂₀: 1 {O(1)}
t₁₂₂: 1 {O(1)}
t₁₂₄: 1 {O(1)}
t₁₁₀: 4⋅X₀+4⋅X₁+13 {O(n)}
t₁₁₉: 1 {O(1)}
t₁₂₁: 1 {O(1)}
t₁₂₃: 1 {O(1)}
t₁₂₅: 1 {O(1)}

Sizebounds

t₃₀, X₀: X₀ {O(n)}
t₃₀, X₁: X₁ {O(n)}
t₃₀, X₂: X₂ {O(n)}
t₃₁, X₀: X₀ {O(n)}
t₃₁, X₁: X₁ {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₂, X₀: 2⋅X₀ {O(n)}
t₃₂, X₁: 2⋅X₁ {O(n)}
t₃₂, X₂: 2⋅X₂ {O(n)}
t₃₃, X₀: 2⋅X₀ {O(n)}
t₃₃, X₁: 2⋅X₁ {O(n)}
t₃₃, X₂: 2⋅X₂ {O(n)}
t₃₄, X₀: 2⋅X₀ {O(n)}
t₃₄, X₁: 2⋅X₁ {O(n)}
t₃₅, X₀: 2⋅X₀ {O(n)}
t₃₅, X₁: 2⋅X₁ {O(n)}
t₁₁₁, X₀: X₀+1 {O(n)}
t₁₁₁, X₁: X₁ {O(n)}
t₁₁₁, X₂: 1 {O(1)}
t₁₁₂, X₀: X₀+1 {O(n)}
t₁₁₂, X₁: X₁ {O(n)}
t₁₁₂, X₂: 1 {O(1)}
t₃₇, X₀: 14⋅X₀+8⋅X₁+30 {O(n)}
t₃₇, X₁: 6⋅X₁ {O(n)}
t₃₈, X₀: 14⋅X₀+8⋅X₁+30 {O(n)}
t₃₈, X₁: 6⋅X₁ {O(n)}
t₃₉, X₀: 14⋅X₀+8⋅X₁+30 {O(n)}
t₃₉, X₁: 6⋅X₁ {O(n)}
t₃₉, X₂: 2 {O(1)}
t₄₀, X₀: 32⋅X₁+58⋅X₀+120 {O(n)}
t₄₀, X₁: 26⋅X₁ {O(n)}
t₄₁, X₀: 32⋅X₁+58⋅X₀+120 {O(n)}
t₄₁, X₁: 26⋅X₁ {O(n)}
t₄₂, X₀: 116⋅X₀+64⋅X₁+240 {O(n)}
t₄₂, X₁: 52⋅X₁ {O(n)}
t₁₀₉, X₀: 4⋅X₁+5⋅X₀+14 {O(n)}
t₁₀₉, X₁: X₁ {O(n)}
t₁₀₉, X₂: 1 {O(1)}
t₁₁₈, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₁₈, X₁: 2⋅X₁ {O(n)}
t₁₂₀, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₂₀, X₁: 2⋅X₁ {O(n)}
t₁₂₂, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₂₂, X₁: 2⋅X₁ {O(n)}
t₁₂₂, X₂: 1 {O(1)}
t₁₂₄, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₂₄, X₁: 2⋅X₁ {O(n)}
t₁₂₄, X₂: 1 {O(1)}
t₁₁₀, X₀: 4⋅X₁+5⋅X₀+14 {O(n)}
t₁₁₀, X₁: X₁ {O(n)}
t₁₁₀, X₂: 1 {O(1)}
t₁₁₉, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₁₉, X₁: 2⋅X₁ {O(n)}
t₁₂₁, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₂₁, X₁: 2⋅X₁ {O(n)}
t₁₂₃, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₂₃, X₁: 2⋅X₁ {O(n)}
t₁₂₃, X₂: 1 {O(1)}
t₁₂₅, X₀: 4⋅X₁+6⋅X₀+15 {O(n)}
t₁₂₅, X₁: 2⋅X₁ {O(n)}
t₁₂₅, X₂: 1 {O(1)}