Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: l0, l1, l2, l3
Transitions:
t₀: l0(X₀, X₁, X₂) → l1(D, 0, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁+1, X₂) :|: X₁ ≤ 9
t₄: l1(X₀, X₁, X₂) → l2(X₀, X₁, 0) :|: 10 ≤ X₁
t₂: l2(X₀, X₁, X₂) → l2(X₀, X₁, X₂+1) :|: X₂ ≤ 9
t₃: l2(X₀, X₁, X₂) → l3(X₀, X₁, X₂) :|: 10 ≤ X₂

Preprocessing

Eliminate variables {D,X₀} that do not contribute to the problem

Found invariant X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 0 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ X₀ ≤ 10+X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀ for location l2

Found invariant X₀ ≤ 10 ∧ 0 ≤ X₀ for location l1

Found invariant X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 10 ≤ X₁ ∧ 20 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀ for location l3

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: l0, l1, l2, l3
Transitions:
t₁₁: l0(X₀, X₁) → l1(0, X₁)
t₁₂: l1(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 9 ∧ X₀ ≤ 10 ∧ 0 ≤ X₀
t₁₃: l1(X₀, X₁) → l2(X₀, 0) :|: 10 ≤ X₀ ∧ X₀ ≤ 10 ∧ 0 ≤ X₀
t₁₄: l2(X₀, X₁) → l2(X₀, X₁+1) :|: X₁ ≤ 9 ∧ X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 0 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ X₀ ≤ 10+X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀
t₁₅: l2(X₀, X₁) → l3(X₀, X₁) :|: 10 ≤ X₁ ∧ X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 0 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ X₀ ≤ 10+X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀

MPRF for transition t₁₂: l1(X₀, X₁) → l1(X₀+1, X₁) :|: X₀ ≤ 9 ∧ X₀ ≤ 10 ∧ 0 ≤ X₀ of depth 1:

new bound:

10 {O(1)}

MPRF for transition t₁₄: l2(X₀, X₁) → l2(X₀, X₁+1) :|: X₁ ≤ 9 ∧ X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 0 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ X₀ ≤ 10+X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀ of depth 1:

new bound:

11 {O(1)}

All Bounds

Timebounds

Overall timebound:24 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 10 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 11 {O(1)}
t₁₅: 1 {O(1)}

Costbounds

Overall costbound: 24 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 10 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 11 {O(1)}
t₁₅: 1 {O(1)}

Sizebounds

t₁₁, X₀: 0 {O(1)}
t₁₁, X₁: X₁ {O(n)}
t₁₂, X₀: 10 {O(1)}
t₁₂, X₁: X₁ {O(n)}
t₁₃, X₀: 10 {O(1)}
t₁₃, X₁: 0 {O(1)}
t₁₄, X₀: 10 {O(1)}
t₁₄, X₁: 10 {O(1)}
t₁₅, X₀: 10 {O(1)}
t₁₅, X₁: 10 {O(1)}