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, X₂₂
Temp_Vars: A1, B1, X, Y, Z
Locations: f0, f11, f37, f45, f53, f55, f58
Transitions:
t₁₂: f0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f11(X₀, X₁, X₂, X₃, X₄, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, 0)
t₁₁: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f11(1+X₀, X₁, 1, X₃, X₄, X₅, X₆, Y, X, Z, A1, X₃, Y, Y, Y, Y, X₁₇, X₁₇, 1, 1, 1, 0, X₂₂) :|: 1+X₀ ≤ X₁ ∧ Y ≤ 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, X₂₂) → f37(X₀, X₁, B1, X₃, X₄, X₅, X₆, Y, X, Z, A1, X₃, Y, Y, Y, Y, X₁₇, 0, B1, B1, B1, 0, X₂₂) :|: 1+X₀ ≤ X₁ ∧ 2 ≤ B1 ∧ Y ≤ 0
t₁₀: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f37(X₀, X₁, B1, X₃, X₄, X₅, X₆, Y, X, Z, A1, X₃, Y, Y, Y, Y, X₁₇, 0, B1, B1, B1, 0, X₂₂) :|: 1+X₀ ≤ X₁ ∧ B1 ≤ 0 ∧ Y ≤ 0
t₀: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: X₁ ≤ X₀
t₈: f11(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, Y, X, Z, A1, X₃, Y, Y, Y, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: 1 ≤ Y ∧ 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, X₂₁, X₂₂) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: 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, X₂₁, X₂₂) → f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: 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, X₂₁, X₂₂) → f45(X₀, X₁, 2, 1+X₃, X₄, X₅, X₇, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: 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, X₂₁, X₂₂) → f53(X₀, X₁, X₂, 0, X, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: X ≤ 0
t₆: f45(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f53(X₀, X₁, X₂, 0, X, 0, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) :|: 1 ≤ X
t₃: f53(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f53(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂)
t₄: f55(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂) → f58(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉, X₂₀, X₂₁, X₂₂)

Preprocessing

Cut unreachable locations [f55; f58] from the program graph

Eliminate variables [A1; Z; X₃; X₄; X₅; X₆; X₇; X₈; X₉; X₁₀; X₁₁; X₁₂; X₁₃; X₁₄; X₁₅; X₁₆; X₁₇; X₁₈; X₁₉; X₂₀; X₂₁; X₂₂] 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: B1, X, Y
Locations: f0, f11, f37, f45, f53
Transitions:
t₂₈: f0(X₀, X₁, X₂) → f11(X₀, X₁, X₂)
t₂₉: f11(X₀, X₁, X₂) → f11(1+X₀, X₁, 1) :|: 1+X₀ ≤ X₁ ∧ Y ≤ 0
t₃₀: f11(X₀, X₁, X₂) → f37(X₀, X₁, B1) :|: 1+X₀ ≤ X₁ ∧ 2 ≤ B1 ∧ Y ≤ 0
t₃₁: f11(X₀, X₁, X₂) → f37(X₀, X₁, B1) :|: 1+X₀ ≤ X₁ ∧ B1 ≤ 0 ∧ Y ≤ 0
t₃₂: f11(X₀, X₁, X₂) → f45(X₀, X₁, X₂) :|: X₁ ≤ X₀
t₃₃: f11(X₀, X₁, X₂) → f45(X₀, X₁, X₂) :|: 1 ≤ Y ∧ 1+X₀ ≤ X₁
t₃₄: f37(X₀, X₁, X₂) → f45(X₀, X₁, X₂) :|: 3 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₅: f37(X₀, X₁, X₂) → f45(X₀, X₁, X₂) :|: X₂ ≤ 1 ∧ 1+X₀ ≤ X₁
t₃₆: f37(X₀, X₁, X₂) → f45(X₀, X₁, 2) :|: X₂ ≤ 2 ∧ 2 ≤ X₂ ∧ 1+X₀ ≤ X₁
t₃₇: f45(X₀, X₁, X₂) → f53(X₀, X₁, X₂) :|: X ≤ 0
t₃₈: f45(X₀, X₁, X₂) → f53(X₀, X₁, X₂) :|: 1 ≤ X
t₃₉: f53(X₀, X₁, X₂) → f53(X₀, X₁, X₂)

MPRF for transition t₂₉: f11(X₀, X₁, X₂) → f11(1+X₀, X₁, 1) :|: 1+X₀ ≤ X₁ ∧ Y ≤ 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}
t₂₈: 1 {O(1)}
t₂₉: X₀+X₁ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}
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: inf {Infinity}

Costbounds

Overall costbound: inf {Infinity}
t₂₈: 1 {O(1)}
t₂₉: X₀+X₁ {O(n)}
t₃₀: 1 {O(1)}
t₃₁: 1 {O(1)}
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: inf {Infinity}

Sizebounds

t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}
t₂₉, X₀: 2⋅X₀+X₁ {O(n)}
t₂₉, X₁: X₁ {O(n)}
t₂₉, X₂: 1 {O(1)}
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: X₂+1 {O(n)}
t₃₃, X₀: 3⋅X₀+X₁ {O(n)}
t₃₃, X₁: 2⋅X₁ {O(n)}
t₃₃, X₂: X₂+1 {O(n)}
t₃₄, X₀: 3⋅X₀+X₁ {O(n)}
t₃₄, X₁: 2⋅X₁ {O(n)}
t₃₅, X₀: 3⋅X₀+X₁ {O(n)}
t₃₅, X₁: 2⋅X₁ {O(n)}
t₃₆, X₀: 3⋅X₀+X₁ {O(n)}
t₃₆, X₁: 2⋅X₁ {O(n)}
t₃₆, X₂: 2 {O(1)}
t₃₇, X₀: 15⋅X₀+5⋅X₁ {O(n)}
t₃₇, X₁: 10⋅X₁ {O(n)}
t₃₈, X₀: 15⋅X₀+5⋅X₁ {O(n)}
t₃₈, X₁: 10⋅X₁ {O(n)}
t₃₉, X₀: 10⋅X₁+30⋅X₀ {O(n)}
t₃₉, X₁: 20⋅X₁ {O(n)}