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
Temp_Vars: A1, B1, C1, D1, E1, F1, G1, H1, I1, J1, K1, U, V, W, X, Y, Z
Locations: f0, f101, f14, f57, f6
Transitions:
t₀: f0(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f6(0, 0, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉)
t₂: f14(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f14(X₀, X₁, X₂-1, U+V, W, X+Y, Z, A1+B1, C1, D1+E1, F1, D1+E1+U+V, U+V-D1-E1, A1+B1+X+Y, X+Y-A1-B1, G1, H1, I1+J1, J1+K1, J1) :|: 0 ≤ X₂
t₅: f14(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f57(X₀, X₁, 7, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) :|: 1+X₂ ≤ 0
t₄: f57(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f101(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) :|: 1+X₂ ≤ 0
t₃: f57(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f57(X₀, X₁, X₂-1, U+V, W, X+Y, Z, A1+B1, C1, D1+E1, F1, D1+E1+U+V, U+V-D1-E1, A1+B1+X+Y, X+Y-A1-B1, G1, H1, I1+J1, J1+K1, J1) :|: 0 ≤ X₂
t₆: f6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f14(X₀, X₁, 7, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) :|: 64 ≤ X₁
t₁: f6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) → f6(U, 1+X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₄, X₁₅, X₁₆, X₁₇, X₁₈, X₁₉) :|: X₁ ≤ 63

Preprocessing

Eliminate variables [A1; B1; C1; D1; E1; F1; G1; H1; I1; J1; K1; U; V; W; X; Y; Z; X₀; X₃; X₄; X₅; X₆; X₇; X₈; X₉; X₁₀; X₁₁; X₁₂; X₁₃; X₁₄; X₁₅; X₁₆; X₁₇; X₁₈; X₁₉] that do not contribute to the problem

Found invariant X₁ ≤ 7 ∧ 57+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 71 ∧ 0 ≤ 1+X₁ ∧ 63 ≤ X₀+X₁ ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ 64 ≤ X₀ for location f14

Found invariant X₀ ≤ 64 ∧ 0 ≤ X₀ for location f6

Found invariant 1+X₁ ≤ 0 ∧ 65+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 63 ∧ 0 ≤ 1+X₁ ∧ 63 ≤ X₀+X₁ ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ 64 ≤ X₀ for location f101

Found invariant X₁ ≤ 7 ∧ 57+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 71 ∧ 0 ≤ 1+X₁ ∧ 63 ≤ X₀+X₁ ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ 64 ≤ X₀ for location f57

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f0, f101, f14, f57, f6
Transitions:
t₁₆: f0(X₀, X₁) → f6(0, X₁)
t₁₇: f14(X₀, X₁) → f14(X₀, X₁-1) :|: 0 ≤ X₁ ∧ X₀+X₁ ≤ 71 ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ X₁ ≤ 7 ∧ 0 ≤ 1+X₁ ∧ 57+X₁ ≤ X₀ ∧ 63 ≤ X₀+X₁ ∧ 64 ≤ X₀
t₁₈: f14(X₀, X₁) → f57(X₀, 7) :|: 1+X₁ ≤ 0 ∧ X₀+X₁ ≤ 71 ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ X₁ ≤ 7 ∧ 0 ≤ 1+X₁ ∧ 57+X₁ ≤ X₀ ∧ 63 ≤ X₀+X₁ ∧ 64 ≤ X₀
t₁₉: f57(X₀, X₁) → f101(X₀, X₁) :|: 1+X₁ ≤ 0 ∧ X₀+X₁ ≤ 71 ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ X₁ ≤ 7 ∧ 0 ≤ 1+X₁ ∧ 57+X₁ ≤ X₀ ∧ 63 ≤ X₀+X₁ ∧ 64 ≤ X₀
t₂₀: f57(X₀, X₁) → f57(X₀, X₁-1) :|: 0 ≤ X₁ ∧ X₀+X₁ ≤ 71 ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ X₁ ≤ 7 ∧ 0 ≤ 1+X₁ ∧ 57+X₁ ≤ X₀ ∧ 63 ≤ X₀+X₁ ∧ 64 ≤ X₀
t₂₁: f6(X₀, X₁) → f14(X₀, 7) :|: 64 ≤ X₀ ∧ X₀ ≤ 64 ∧ 0 ≤ X₀
t₂₂: f6(X₀, X₁) → f6(1+X₀, X₁) :|: X₀ ≤ 63 ∧ X₀ ≤ 64 ∧ 0 ≤ X₀

MPRF for transition t₂₂: f6(X₀, X₁) → f6(1+X₀, X₁) :|: X₀ ≤ 63 ∧ X₀ ≤ 64 ∧ 0 ≤ X₀ of depth 1:

new bound:

65 {O(1)}

MPRF:

• f6: [65-X₀]

MPRF for transition t₁₇: f14(X₀, X₁) → f14(X₀, X₁-1) :|: 0 ≤ X₁ ∧ X₀+X₁ ≤ 71 ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ X₁ ≤ 7 ∧ 0 ≤ 1+X₁ ∧ 57+X₁ ≤ X₀ ∧ 63 ≤ X₀+X₁ ∧ 64 ≤ X₀ of depth 1:

new bound:

8 {O(1)}

MPRF:

• f14: [1+X₁]

MPRF for transition t₂₀: f57(X₀, X₁) → f57(X₀, X₁-1) :|: 0 ≤ X₁ ∧ X₀+X₁ ≤ 71 ∧ X₀ ≤ 65+X₁ ∧ X₀ ≤ 64 ∧ X₁ ≤ 7 ∧ 0 ≤ 1+X₁ ∧ 57+X₁ ≤ X₀ ∧ 63 ≤ X₀+X₁ ∧ 64 ≤ X₀ of depth 1:

new bound:

8 {O(1)}

MPRF:

• f57: [1+X₁]

All Bounds

Timebounds

Overall timebound:85 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 8 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 8 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 65 {O(1)}

Costbounds

Overall costbound: 85 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: 8 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 8 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 65 {O(1)}

Sizebounds

t₁₆, X₀: 0 {O(1)}
t₁₆, X₁: X₁ {O(n)}
t₁₇, X₀: 64 {O(1)}
t₁₇, X₁: 6 {O(1)}
t₁₈, X₀: 64 {O(1)}
t₁₈, X₁: 7 {O(1)}
t₁₉, X₀: 64 {O(1)}
t₁₉, X₁: 1 {O(1)}
t₂₀, X₀: 64 {O(1)}
t₂₀, X₁: 6 {O(1)}
t₂₁, X₀: 64 {O(1)}
t₂₁, X₁: 7 {O(1)}
t₂₂, X₀: 64 {O(1)}
t₂₂, X₁: X₁ {O(n)}