KoAT2 Proof Output

Initial Problem

Start: f0
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: f0, f13, f37, f45, f52, f54, f57
Transitions:
t₁₂: f0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f13(X₀, X₁, X₂, W, X₄, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, 0) :|: W ≤ 0
t₁₃: f0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f13(X₀, X₁, X₂, W, X₄, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, 0) :|: 1 ≤ W
t₁₁: f13(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f13(1+X₀, X₁, 1, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, V, X₁₆, X₁₆, 1, 1, 0, X₂₀) :|: 1+X₀ ≤ X₁ ∧ V ≤ 0
t₉: f13(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f37(X₀, X₁, Z, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, V, X₁₆, 0, Z, Z, 0, X₂₀) :|: 1+X₀ ≤ X₁ ∧ 2 ≤ Z ∧ V ≤ 0
t₁₀: f13(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f37(X₀, X₁, Z, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, V, X₁₆, 0, Z, Z, 0, X₂₀) :|: 1+X₀ ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0
t₀: f13(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₁ ≤ X₀
t₈: f13(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, V, W, X, Y, X₄, V, V, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: 1 ≤ V ∧ 1+X₀ ≤ X₁
t₁: f37(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: 3 ≤ X₂
t₂: f37(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₂ ≤ 1
t₇: f37(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f45(X₀, X₁, 2, X₃, 1+X₄, X₅, X₇, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₂ ≤ 2 ∧ 2 ≤ X₂
t₅: f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f52(X₀, X₁, X₂, X₃, 0, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: 1 ≤ X₃
t₆: f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f52(X₀, X₁, X₂, X₃, 0, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) :|: X₃ ≤ 0
t₃: f52(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f52(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀)
t₄: f54(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀) → f57(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 [f54; f57] 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 f37

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: V, W, Z
Locations: f0, f13, f37, f45, f52
Transitions:
t₂₉: f0(X₀, X₁, X₂, X₃) → f13(X₀, X₁, X₂, W) :|: W ≤ 0
t₃₀: f0(X₀, X₁, X₂, X₃) → f13(X₀, X₁, X₂, W) :|: 1 ≤ W
t₃₁: f13(X₀, X₁, X₂, X₃) → f13(1+X₀, X₁, 1, X₃) :|: 1+X₀ ≤ X₁ ∧ V ≤ 0
t₃₂: f13(X₀, X₁, X₂, X₃) → f37(X₀, X₁, Z, X₃) :|: 1+X₀ ≤ X₁ ∧ 2 ≤ Z ∧ V ≤ 0
t₃₃: f13(X₀, X₁, X₂, X₃) → f37(X₀, X₁, Z, X₃) :|: 1+X₀ ≤ X₁ ∧ V ≤ 0 ∧ Z ≤ 0
t₃₄: f13(X₀, X₁, X₂, X₃) → f45(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₃₅: f13(X₀, X₁, X₂, X₃) → f45(X₀, X₁, X₂, X₃) :|: 1 ≤ V ∧ 1+X₀ ≤ X₁
t₃₆: f37(X₀, X₁, X₂, X₃) → f45(X₀, X₁, X₂, X₃) :|: 3 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₇: f37(X₀, X₁, X₂, X₃) → f45(X₀, X₁, X₂, X₃) :|: X₂ ≤ 1 ∧ 1+X₀ ≤ X₁
t₃₈: f37(X₀, X₁, X₂, X₃) → f45(X₀, X₁, 2, X₃) :|: X₂ ≤ 2 ∧ 2 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₉: f45(X₀, X₁, X₂, X₃) → f52(X₀, X₁, X₂, X₃) :|: 1 ≤ X₃
t₄₀: f45(X₀, X₁, X₂, X₃) → f52(X₀, X₁, X₂, X₃) :|: X₃ ≤ 0
t₄₁: f52(X₀, X₁, X₂, X₃) → f52(X₀, X₁, X₂, X₃)

MPRF for transition t₃₁: f13(X₀, X₁, X₂, X₃) → f13(1+X₀, X₁, 1, X₃) :|: 1+X₀ ≤ X₁ ∧ V ≤ 0 of depth 1:

new bound:

2⋅X₀+2⋅X₁ {O(n)}

MPRF:

• f13: [X₁-X₀]

Found invariant 1+X₀ ≤ X₁ for location f37

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)}