Initial Problem

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval1, eval2, start
Transitions:
t₀: eval1(X₀, X₁) → eval2(1+X₀, 1) :|: 0 ≤ X₀
t₂: eval2(X₀, X₁) → eval1(X₀-2, X₁) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₁: eval2(X₀, X₁) → eval2(X₀, 1+X₁) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀
t₃: start(X₀, X₁) → eval1(X₀, X₁)

Preprocessing

Found invariant X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ 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(1+X₀, 1) :|: 0 ≤ X₀
t₂: eval2(X₀, X₁) → eval1(X₀-2, X₁) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 2 ≤ X₀+X₁
t₁: eval2(X₀, X₁) → eval2(X₀, 1+X₁) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 2 ≤ X₀+X₁
t₃: start(X₀, X₁) → eval1(X₀, X₁)

MPRF for transition t₀: eval1(X₀, X₁) → eval2(1+X₀, 1) :|: 0 ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

• eval1: [1+X₀]
• eval2: [X₀-1]

MPRF for transition t₂: eval2(X₀, X₁) → eval1(X₀-2, X₁) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

X₀+2 {O(n)}

• eval1: [2+X₀]
• eval2: [1+X₀]

MPRF for transition t₁: eval2(X₀, X₁) → eval2(X₀, 1+X₁) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀ ∧ 2 ≤ X₀+X₁ of depth 1:

new bound:

2⋅X₀⋅X₀+7⋅X₀+5 {O(n^2)}

• eval1: [1+X₀]
• eval2: [1+X₀-X₁]

Cut unsatisfiable transition [t₂: eval2→eval1; t₂₃: eval2→eval1]

Found invariant X₁ ≤ 1 ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval2

Found invariant X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval2_v1

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+9⋅X₀+9 {O(n^2)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅X₀+7⋅X₀+5 {O(n^2)}
t₂: X₀+2 {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+9⋅X₀+9 {O(n^2)}
t₀: X₀+1 {O(n)}
t₁: 2⋅X₀⋅X₀+7⋅X₀+5 {O(n^2)}
t₂: X₀+2 {O(n)}
t₃: 1 {O(1)}

Sizebounds

t₀, X₀: 2⋅X₀+1 {O(n)}
t₀, X₁: 1 {O(1)}
t₁, X₀: 2⋅X₀+1 {O(n)}
t₁, X₁: 2⋅X₀⋅X₀+7⋅X₀+6 {O(n^2)}
t₂, X₀: 2⋅X₀+1 {O(n)}
t₂, X₁: 2⋅X₀⋅X₀+7⋅X₀+6 {O(n^2)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}