KoAT2 Proof Output

Initial Problem

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

Preprocessing

Found invariant 1 ≤ X₀ for location eval2

Problem after Preprocessing

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

new bound:

X₀ {O(n)}

MPRF:

• eval1: [X₀]
• eval2: [X₀-1]

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

new bound:

X₁ {O(n)}

MPRF:

• eval1: [X₁]
• eval2: [X₁]

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

new bound:

X₀ {O(n)}

MPRF:

• eval1: [X₀]
• eval2: [X₀]

All Bounds

Timebounds

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

Costbounds

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