Initial Problem

Start: f0
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: f0, f19, f4, f7
Transitions:
t₀: f0(X₀, X₁, X₂, X₃) → f4(0, X₁, X₂, X₃)
t₆: f4(X₀, X₁, X₂, X₃) → f19(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀
t₁: f4(X₀, X₁, X₂, X₃) → f7(X₀, X₁, 1+X₀, X₃) :|: 1+X₀ ≤ X₁
t₅: f7(X₀, X₁, X₂, X₃) → f4(1+X₀, X₁, X₂, X₃) :|: X₁ ≤ X₂
t₂: f7(X₀, X₁, X₂, X₃) → f7(X₀, X₁, 1+X₂, 0) :|: 1+X₂ ≤ X₁
t₃: f7(X₀, X₁, X₂, X₃) → f7(X₀, X₁-1, X₂, E) :|: 1+E ≤ 0 ∧ 1+X₂ ≤ X₁
t₄: f7(X₀, X₁, X₂, X₃) → f7(X₀, X₁-1, X₂, E) :|: 1 ≤ E ∧ 1+X₂ ≤ X₁

Preprocessing

Eliminate variables [X₃] that do not contribute to the problem

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f19

Found invariant 0 ≤ X₀ for location f4

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location f7

Problem after Preprocessing

Start: f0
Program_Vars: X₀, X₁, X₂
Temp_Vars: E
Locations: f0, f19, f4, f7
Transitions:
t₁₄: f0(X₀, X₁, X₂) → f4(0, X₁, X₂)
t₁₅: f4(X₀, X₁, X₂) → f19(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ 0 ≤ X₀
t₁₆: f4(X₀, X₁, X₂) → f7(X₀, X₁, 1+X₀) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀
t₁₇: f7(X₀, X₁, X₂) → f4(1+X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₁₈: f7(X₀, X₁, X₂) → f7(X₀, X₁, 1+X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₁₉: f7(X₀, X₁, X₂) → f7(X₀, X₁-1, X₂) :|: 1+E ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₂₀: f7(X₀, X₁, X₂) → f7(X₀, X₁-1, X₂) :|: 1 ≤ E ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁

MPRF for transition t₁₆: f4(X₀, X₁, X₂) → f7(X₀, X₁, 1+X₀) :|: 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• f4: [X₁-X₀]
• f7: [X₁-1-X₀]

MPRF for transition t₁₇: f7(X₀, X₁, X₂) → f4(1+X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• f4: [X₁-X₀]
• f7: [X₁-X₀]

MPRF for transition t₁₉: f7(X₀, X₁, X₂) → f7(X₀, X₁-1, X₂) :|: 1+E ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• f4: [X₁]
• f7: [X₁]

MPRF for transition t₂₀: f7(X₀, X₁, X₂) → f7(X₀, X₁-1, X₂) :|: 1 ≤ E ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• f4: [X₁]
• f7: [X₁]

MPRF for transition t₁₈: f7(X₀, X₁, X₂) → f7(X₀, X₁, 1+X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

2⋅X₁⋅X₁+X₁ {O(n^2)}

MPRF:

• f4: [X₁-X₀]
• f7: [1+X₁-X₂]

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₀ for location f19

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location f7_v1

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 2+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location f7_v2

Found invariant 0 ≤ X₀ for location f4

Found invariant X₂ ≤ X₁ ∧ X₂ ≤ 1+X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location f7

All Bounds

Timebounds

Overall timebound:2⋅X₁⋅X₁+5⋅X₁+2 {O(n^2)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: X₁ {O(n)}
t₁₇: X₁ {O(n)}
t₁₈: 2⋅X₁⋅X₁+X₁ {O(n^2)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+5⋅X₁+2 {O(n^2)}
t₁₄: 1 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: X₁ {O(n)}
t₁₇: X₁ {O(n)}
t₁₈: 2⋅X₁⋅X₁+X₁ {O(n^2)}
t₁₉: X₁ {O(n)}
t₂₀: X₁ {O(n)}

Sizebounds

t₁₄, X₀: 0 {O(1)}
t₁₄, X₁: X₁ {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₅, X₀: X₁ {O(n)}
t₁₅, X₁: 2⋅X₁ {O(n)}
t₁₅, X₂: 6⋅X₁⋅X₁+7⋅X₁+X₂+8 {O(n^2)}
t₁₆, X₀: X₁ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₁+2 {O(n)}
t₁₇, X₀: X₁ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: 6⋅X₁⋅X₁+7⋅X₁+8 {O(n^2)}
t₁₈, X₀: X₁ {O(n)}
t₁₈, X₁: X₁ {O(n)}
t₁₈, X₂: 2⋅X₁⋅X₁+2⋅X₁+2 {O(n^2)}
t₁₉, X₀: X₁ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: 2⋅X₁⋅X₁+2⋅X₁+2 {O(n^2)}
t₂₀, X₀: X₁ {O(n)}
t₂₀, X₁: X₁ {O(n)}
t₂₀, X₂: 2⋅X₁⋅X₁+2⋅X₁+2 {O(n^2)}