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, X₂₁
Temp_Vars: A1, W, X, Y, Z
Locations: f0, f11, f37, f51, f53, f56
Transitions:
t₁₀: f0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f11(X₀, X₁, 0, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, 0)
t₉: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f11(1+X₀, X₁, X₂, X₃, 1, X₅, W, X, Y, Z, X₃, W, W, W, W, X₁₆, X₁₆, 1, 1, 1, 0, X₂₁) :|: 1+X₀ ≤ X₁ ∧ W ≤ 0
t₇: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f37(X₀, X₁, X₂, X₃, A1, X₅, W, X, Y, Z, X₃, W, W, W, W, X₁₆, 0, A1, A1, A1, 0, X₂₁) :|: 1+X₀ ≤ X₁ ∧ 2 ≤ A1 ∧ W ≤ 0
t₈: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f37(X₀, X₁, X₂, X₃, A1, X₅, W, X, Y, Z, X₃, W, W, W, W, X₁₆, 0, A1, A1, A1, 0, X₂₁) :|: 1+X₀ ≤ X₁ ∧ A1 ≤ 0 ∧ W ≤ 0
t₂: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f51(X₀, X₁, 0, 0, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) :|: X₁ ≤ X₀
t₆: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f51(X₀, X₁, 0, 0, X₄, X₅, W, X, Y, Z, X₃, W, W, W, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) :|: 1 ≤ W ∧ 1+X₀ ≤ X₁
t₃: f37(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f51(X₀, X₁, 0, 0, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) :|: 3 ≤ X₄
t₄: f37(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f51(X₀, X₁, 0, 0, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) :|: X₄ ≤ 1
t₅: f37(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f51(X₀, X₁, 0, 0, 2, X₆, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) :|: X₄ ≤ 2 ∧ 2 ≤ X₄
t₀: f51(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f51(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁)
t₁: f53(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁) → f56(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁)

Preprocessing

Cut unreachable locations [f53; f56] from the program graph

Eliminate variables [X; Y; Z; X₂; X₃; X₅; X₆; X₇; X₈; X₉; X₁₀; X₁₁; X₁₂; X₁₃; X₁₄; X₁₅; X₁₆; X₁₇; X₁₈; X₁₉; X₂₀; X₂₁] 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
Temp_Vars: A1, W
Locations: f0, f11, f37, f51
Transitions:
t₂₄: f0(X₀, X₁, X₂) → f11(X₀, X₁, X₂)
t₂₅: f11(X₀, X₁, X₂) → f11(1+X₀, X₁, 1) :|: 1+X₀ ≤ X₁ ∧ W ≤ 0
t₂₆: f11(X₀, X₁, X₂) → f37(X₀, X₁, A1) :|: 1+X₀ ≤ X₁ ∧ 2 ≤ A1 ∧ W ≤ 0
t₂₇: f11(X₀, X₁, X₂) → f37(X₀, X₁, A1) :|: 1+X₀ ≤ X₁ ∧ A1 ≤ 0 ∧ W ≤ 0
t₂₈: f11(X₀, X₁, X₂) → f51(X₀, X₁, X₂) :|: X₁ ≤ X₀
t₂₉: f11(X₀, X₁, X₂) → f51(X₀, X₁, X₂) :|: 1 ≤ W ∧ 1+X₀ ≤ X₁
t₃₀: f37(X₀, X₁, X₂) → f51(X₀, X₁, X₂) :|: 3 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₁: f37(X₀, X₁, X₂) → f51(X₀, X₁, X₂) :|: X₂ ≤ 1 ∧ 1+X₀ ≤ X₁
t₃₂: f37(X₀, X₁, X₂) → f51(X₀, X₁, 2) :|: X₂ ≤ 2 ∧ 2 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₃: f51(X₀, X₁, X₂) → f51(X₀, X₁, X₂)

MPRF for transition t₂₅: f11(X₀, X₁, X₂) → f11(1+X₀, X₁, 1) :|: 1+X₀ ≤ X₁ ∧ W ≤ 0 of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• f11: [X₁-X₀]

Found invariant 1+X₀ ≤ X₁ for location f37

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₂₄: 1 {O(1)}
t₂₅: X₀+X₁ {O(n)}
t₂₆: 1 {O(1)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}
t₃₂: 1 {O(1)}
t₃₃: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₂₄: 1 {O(1)}
t₂₅: X₀+X₁ {O(n)}
t₂₆: 1 {O(1)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}
t₃₂: 1 {O(1)}
t₃₃: inf {Infinity}

Sizebounds

t₂₄, X₀: X₀ {O(n)}
t₂₄, X₁: X₁ {O(n)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: 2⋅X₀+X₁ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: 1 {O(1)}
t₂₆, X₀: 3⋅X₀+X₁ {O(n)}
t₂₆, X₁: 2⋅X₁ {O(n)}
t₂₇, X₀: 3⋅X₀+X₁ {O(n)}
t₂₇, X₁: 2⋅X₁ {O(n)}
t₂₈, X₀: 3⋅X₀+X₁ {O(n)}
t₂₈, X₁: 2⋅X₁ {O(n)}
t₂₈, X₂: X₂+1 {O(n)}
t₂₉, X₀: 3⋅X₀+X₁ {O(n)}
t₂₉, X₁: 2⋅X₁ {O(n)}
t₂₉, X₂: X₂+1 {O(n)}
t₃₀, X₀: 3⋅X₀+X₁ {O(n)}
t₃₀, X₁: 2⋅X₁ {O(n)}
t₃₁, X₀: 3⋅X₀+X₁ {O(n)}
t₃₁, X₁: 2⋅X₁ {O(n)}
t₃₂, X₀: 3⋅X₀+X₁ {O(n)}
t₃₂, X₁: 2⋅X₁ {O(n)}
t₃₂, X₂: 2 {O(1)}
t₃₃, X₀: 15⋅X₀+5⋅X₁ {O(n)}
t₃₃, X₁: 10⋅X₁ {O(n)}