KoAT2 Proof Output

Initial Problem

Start: start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval, start
Transitions:
t₀: eval(X₀, X₁) → eval(X₀-1, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀
t₁: eval(X₀, X₁) → eval(X₀-1, X₁) :|: 1 ≤ 2⋅X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₂: eval(X₀, X₁) → eval(X₀, X₁-1) :|: 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ X₁
t₃: eval(X₀, X₁) → eval(X₀, X₁-1) :|: 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₄: start(X₀, X₁) → eval(X₀, X₁)

Preprocessing

Cut unsatisfiable transition [t₃: eval→eval]

Problem after Preprocessing

MPRF for transition t₀: eval(X₀, X₁) → eval(X₀-1, X₁) :|: 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• eval: [X₀+X₁]

MPRF for transition t₁: eval(X₀, X₁) → eval(X₀-1, X₁) :|: 1 ≤ 2⋅X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ of depth 1:

new bound:

2⋅X₀ {O(n)}

MPRF:

• eval: [2⋅X₀]

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

new bound:

2⋅X₁ {O(n)}

MPRF:

• eval: [2⋅X₁]

All Bounds

Timebounds

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

Costbounds

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

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: 2⋅X₀ {O(n)}
t₁, X₁: 2⋅X₁ {O(n)}
t₂, X₀: 2⋅X₀ {O(n)}
t₂, X₁: 2⋅X₁ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}