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:
t₂₉: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, W) :|: W ≤ 0
t₃₀: l0(X₀, X₁, X₂, X₃) → l1(X₀, X₁, X₂, W) :|: 1 ≤ W
t₃₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, 1, X₃) :|: V ≤ 0 ∧ X₀+1 ≤ X₁
t₃₂: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ 2 ≤ Z
t₃₃: l1(X₀, X₁, X₂, X₃) → l2(X₀, X₁, Z, X₃) :|: X₀+1 ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0
t₃₄: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₃₅: l1(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 1 ≤ V ∧ X₀+1 ≤ X₁
t₃₆: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: 3 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₇: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1 ∧ 1+X₀ ≤ X₁
t₃₈: l2(X₀, X₁, X₂, X₃) → l3(X₀, X₁, 2, X₃) :|: X₂ ≤ 2 ∧ 2 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₉: l3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃
t₄₀: l3(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₄₁: l4(X₀, X₁, X₂, X₃) → l4(X₀, X₁, X₂, X₃)

MPRF for transition t₃₁: l1(X₀, X₁, X₂, X₃) → l1(X₀+1, X₁, 1, X₃) :|: V ≤ 0 ∧ X₀+1 ≤ X₁ of depth 1:

new bound:

2⋅X₀+2⋅X₁ {O(n)}

Analysing control-flow refined program

Found invariant 1+X₀ ≤ X₁ for location l2

CFR did not improve the program. Rolling back

CFR did not improve the program. Rolling back

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₂₉: 1 {O(1)}
t₃₀: 1 {O(1)}
t₃₁: 2⋅X₀+2⋅X₁ {O(n)}
t₃₂: 1 {O(1)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}
t₃₅: 1 {O(1)}
t₃₆: 1 {O(1)}
t₃₇: 1 {O(1)}
t₃₈: 1 {O(1)}
t₃₉: 1 {O(1)}
t₄₀: 1 {O(1)}
t₄₁: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₂₉: 1 {O(1)}
t₃₀: 1 {O(1)}
t₃₁: 2⋅X₀+2⋅X₁ {O(n)}
t₃₂: 1 {O(1)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}
t₃₅: 1 {O(1)}
t₃₆: 1 {O(1)}
t₃₇: 1 {O(1)}
t₃₈: 1 {O(1)}
t₃₉: 1 {O(1)}
t₄₀: 1 {O(1)}
t₄₁: inf {Infinity}

Sizebounds

t₂₉, X₀: X₀ {O(n)}
t₂₉, X₁: X₁ {O(n)}
t₂₉, X₂: X₂ {O(n)}
t₃₀, X₀: X₀ {O(n)}
t₃₀, X₁: X₁ {O(n)}
t₃₀, X₂: X₂ {O(n)}
t₃₁, X₀: 2⋅X₁+4⋅X₀ {O(n)}
t₃₁, X₁: 2⋅X₁ {O(n)}
t₃₁, X₂: 1 {O(1)}
t₃₂, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₂, X₁: 4⋅X₁ {O(n)}
t₃₃, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₃, X₁: 4⋅X₁ {O(n)}
t₃₄, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₄, X₁: 4⋅X₁ {O(n)}
t₃₄, X₂: 2⋅X₂+1 {O(n)}
t₃₅, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₅, X₁: 4⋅X₁ {O(n)}
t₃₅, X₂: 2⋅X₂+1 {O(n)}
t₃₆, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₆, X₁: 4⋅X₁ {O(n)}
t₃₇, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₇, X₁: 4⋅X₁ {O(n)}
t₃₈, X₀: 2⋅X₁+6⋅X₀ {O(n)}
t₃₈, X₁: 4⋅X₁ {O(n)}
t₃₈, X₂: 2 {O(1)}
t₃₉, X₀: 10⋅X₁+30⋅X₀ {O(n)}
t₃₉, X₁: 20⋅X₁ {O(n)}
t₄₀, X₀: 10⋅X₁+30⋅X₀ {O(n)}
t₄₀, X₁: 20⋅X₁ {O(n)}
t₄₁, X₀: 20⋅X₁+60⋅X₀ {O(n)}
t₄₁, X₁: 40⋅X₁ {O(n)}