Initial Problem

Start: f0
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: f0, f19, f29, f8
Transitions:
t₀: f0(X₀, X₁, X₂) → f8(D, 0, X₂)
t₂: f19(X₀, X₁, X₂) → f19(X₀, X₁, 1+X₂) :|: X₂ ≤ 9
t₃: f19(X₀, X₁, X₂) → f29(X₀, X₁, X₂) :|: 10 ≤ X₂
t₄: f8(X₀, X₁, X₂) → f19(X₀, X₁, 0) :|: 10 ≤ X₁
t₁: f8(X₀, X₁, X₂) → f8(X₀, 1+X₁, X₂) :|: X₁ ≤ 9

Preprocessing

Eliminate variables [D; X₀] that do not contribute to the problem

Found invariant X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 10 ≤ X₁ ∧ 20 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀ for location f29

Found invariant X₀ ≤ 10 ∧ 0 ≤ X₀ for location f8

Found invariant X₁ ≤ 10 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 20 ∧ 0 ≤ X₁ ∧ 10 ≤ X₀+X₁ ∧ X₀ ≤ 10+X₁ ∧ X₀ ≤ 10 ∧ 10 ≤ X₀ for location f19

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁
Temp_Vars:
Locations: f0, f19, f29, f8
Transitions:
t₁₁: f0(X₀, X₁) → f8(0, X₁)
t₁₂: f19(X₀, X₁) → f19(X₀, 1+X₁) :|: X₁ ≤ 9 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 ∧ X₀ ≤ 10+X₁ ∧ X₁ ≤ 10 ∧ 10 ≤ X₀ ∧ 10 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₃: f19(X₀, X₁) → f29(X₀, X₁) :|: 10 ≤ X₁ ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 ∧ X₀ ≤ 10+X₁ ∧ X₁ ≤ 10 ∧ 10 ≤ X₀ ∧ 10 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₄: f8(X₀, X₁) → f19(X₀, 0) :|: 10 ≤ X₀ ∧ X₀ ≤ 10 ∧ 0 ≤ X₀
t₁₅: f8(X₀, X₁) → f8(1+X₀, X₁) :|: X₀ ≤ 9 ∧ X₀ ≤ 10 ∧ 0 ≤ X₀

MPRF for transition t₁₅: f8(X₀, X₁) → f8(1+X₀, X₁) :|: X₀ ≤ 9 ∧ X₀ ≤ 10 ∧ 0 ≤ X₀ of depth 1:

new bound:

11 {O(1)}

MPRF:

• f8: [11-X₀]

MPRF for transition t₁₂: f19(X₀, X₁) → f19(X₀, 1+X₁) :|: X₁ ≤ 9 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 ∧ X₀ ≤ 10+X₁ ∧ X₁ ≤ 10 ∧ 10 ≤ X₀ ∧ 10 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

11 {O(1)}

MPRF:

• f19: [1+X₀-X₁]

All Bounds

Timebounds

Overall timebound:25 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 11 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 11 {O(1)}

Costbounds

Overall costbound: 25 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 11 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}
t₁₅: 11 {O(1)}

Sizebounds

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