KoAT2 Proof Output

Initial Problem

Start: start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval1, eval2, start
Transitions:
t₀: eval1(X₀, X₁, X₂) → eval2(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀
t₂: eval2(X₀, X₁, X₂) → eval1(X₀-1, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ X₂ ≤ X₁
t₁: eval2(X₀, X₁, X₂) → eval2(X₀, X₁, X₂-1) :|: 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂
t₃: start(X₀, X₁, X₂) → eval1(X₀, X₁, X₂)

Preprocessing

Found invariant 1+X₁ ≤ X₀ for location eval2

Problem after Preprocessing

MPRF for transition t₀: eval1(X₀, X₁, X₂) → eval2(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• eval1: [X₀-X₁]
• eval2: [X₀-1-X₁]

MPRF for transition t₁: eval2(X₀, X₁, X₂) → eval2(X₀, X₁, X₂-1) :|: 1+X₁ ≤ X₀ ∧ 1+X₁ ≤ X₂ of depth 1:

new bound:

X₁+X₂ {O(n)}

MPRF:

• eval1: [X₂-X₁]
• eval2: [X₂-X₁]

MPRF for transition t₂: eval2(X₀, X₁, X₂) → eval1(X₀-1, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₀+X₁ {O(n)}

MPRF:

• eval1: [X₀-X₁]
• eval2: [X₀-X₁]

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)}