Initial Problem

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval1, eval2, eval3, start
Transitions:
t₀: eval1(X₀, X₁) → eval2(X₀, X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁
t₁: eval1(X₀, X₁) → eval3(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁
t₃: eval2(X₀, X₁) → eval1(X₀, X₁) :|: X₀ ≤ 0
t₂: eval2(X₀, X₁) → eval2(X₀-1, X₁) :|: 1 ≤ X₀
t₅: eval3(X₀, X₁) → eval1(X₀, X₁) :|: X₁ ≤ 0
t₄: eval3(X₀, X₁) → eval3(X₀, X₁-1) :|: 1 ≤ X₁
t₆: start(X₀, X₁) → eval1(X₀, X₁)

Preprocessing

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval2

Found invariant 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval3

Problem after Preprocessing

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval1, eval2, eval3, start
Transitions:
t₀: eval1(X₀, X₁) → eval2(X₀, X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁
t₁: eval1(X₀, X₁) → eval3(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁
t₃: eval2(X₀, X₁) → eval1(X₀, X₁) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₂: eval2(X₀, X₁) → eval2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₅: eval3(X₀, X₁) → eval1(X₀, X₁) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₄: eval3(X₀, X₁) → eval3(X₀, X₁-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₆: start(X₀, X₁) → eval1(X₀, X₁)

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₀: eval1(X₀, X₁) → eval2(X₀, X₁) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₁: eval1(X₀, X₁) → eval3(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₁

MPRF for transition t₂: eval2(X₀, X₁) → eval2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

• eval1: [1+X₀]
• eval2: [1+X₀]
• eval3: [1+X₀]

MPRF for transition t₃: eval2(X₀, X₁) → eval1(X₀, X₁) :|: X₀ ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

• eval1: [X₀]
• eval2: [1]
• eval3: [X₀]

MPRF for transition t₄: eval3(X₀, X₁) → eval3(X₀, X₁-1) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

2⋅X₁ {O(n)}

• eval1: [2⋅X₁]
• eval2: [2⋅X₁]
• eval3: [1+X₁]

MPRF for transition t₅: eval3(X₀, X₁) → eval1(X₀, X₁) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

• eval1: [X₁]
• eval2: [X₁]
• eval3: [1]

All Bounds

Timebounds

Overall timebound:2⋅X₀+3⋅X₁+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+1 {O(n)}
t₃: X₀ {O(n)}
t₄: 2⋅X₁ {O(n)}
t₅: X₁ {O(n)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+3⋅X₁+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀+1 {O(n)}
t₃: X₀ {O(n)}
t₄: 2⋅X₁ {O(n)}
t₅: X₁ {O(n)}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: 0 {O(1)}
t₃, X₁: X₁ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: 0 {O(1)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}