Initial Problem

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

Preprocessing

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

Problem after Preprocessing

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

MPRF for transition t₁: eval2(X₀, X₁) → eval2(X₀-1, X₁-1) :|: 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀+1 {O(n)}

• eval1: [1+X₀]
• eval2: [1+X₀]

MPRF for transition t₂: eval2(X₀, X₁) → eval1(X₀, X₁) :|: X₀ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

• eval1: [X₀]
• eval2: [1]

All Bounds

Timebounds

Overall timebound:2⋅X₀+3 {O(n)}
t₀: 1 {O(1)}
t₁: X₀+1 {O(n)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+3 {O(n)}
t₀: 1 {O(1)}
t₁: X₀+1 {O(n)}
t₂: X₀ {O(n)}
t₃: 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: 0 {O(1)}
t₂, X₁: 0 {O(1)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}