Initial Problem

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1) :|: X₁+1 ≤ X₀ ∧ X₁+1 ≤ X₂
t₂: l1(X₀, X₁, X₂) → l2(X₀-1, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₁
t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀

Preprocessing

Found invariant 1+X₁ ≤ X₀ for location l1

Problem after Preprocessing

Start: l0
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: l0, l1, l2
Transitions:
t₃: l0(X₀, X₁, X₂) → l2(X₀, X₁, X₂)
t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1) :|: X₁+1 ≤ X₀ ∧ X₁+1 ≤ X₂ ∧ 1+X₁ ≤ X₀
t₂: l1(X₀, X₁, X₂) → l2(X₀-1, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₁ ≤ X₀
t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀

MPRF for transition t₀: l2(X₀, X₁, X₂) → l1(X₀, X₁, X₂) :|: X₁+1 ≤ X₀ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF for transition t₁: l1(X₀, X₁, X₂) → l1(X₀, X₁, X₂-1) :|: X₁+1 ≤ X₀ ∧ X₁+1 ≤ X₂ ∧ 1+X₁ ≤ X₀ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF for transition t₂: l1(X₀, X₁, X₂) → l2(X₀-1, X₁, X₂) :|: X₁+1 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₁ ≤ X₀ of depth 1:

new bound:

X₀+X₁ {O(n)}

All Bounds

Timebounds

Overall timebound:2⋅X₀+3⋅X₁+X₂+1 {O(n)}
t₀: X₀+X₁ {O(n)}
t₁: X₁+X₂ {O(n)}
t₂: X₀+X₁ {O(n)}
t₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀+3⋅X₁+X₂+1 {O(n)}
t₀: X₀+X₁ {O(n)}
t₁: X₁+X₂ {O(n)}
t₂: X₀+X₁ {O(n)}
t₃: 1 {O(1)}

Sizebounds

t₀, X₀: 2⋅X₀+X₁ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: 2⋅X₂+X₁ {O(n)}
t₁, X₀: 2⋅X₀+X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 2⋅X₂+X₁ {O(n)}
t₂, X₀: 2⋅X₀+X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: 2⋅X₂+X₁ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}