KoAT2 Proof Output

Initial Problem

Start: evalfstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalfbb1in, evalfbb2in, evalfbb3in, evalfbb4in, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₆: evalfbb1in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₂-1)
t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₀ ≤ X₂
t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀
t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(X₀, X₁-1, X₂)
t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₀) :|: 1 ≤ X₁
t₃: evalfbb4in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb4in(X₁, X₀, X₂)
t₈: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂)
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

Preprocessing

Found invariant X₁ ≤ 0 for location evalfreturnin

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ for location evalfbb1in

Found invariant X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ for location evalfbb2in

Found invariant 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ for location evalfbb3in

Found invariant X₁ ≤ 0 for location evalfstop

Problem after Preprocessing

Start: evalfstart
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalfbb1in, evalfbb2in, evalfbb3in, evalfbb4in, evalfentryin, evalfreturnin, evalfstart, evalfstop
Transitions:
t₆: evalfbb1in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂
t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀
t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀
t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(X₀, X₁-1, X₂) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁
t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₀) :|: 1 ≤ X₁
t₃: evalfbb4in(X₀, X₁, X₂) → evalfreturnin(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁: evalfentryin(X₀, X₁, X₂) → evalfbb4in(X₁, X₀, X₂)
t₈: evalfreturnin(X₀, X₁, X₂) → evalfstop(X₀, X₁, X₂) :|: X₁ ≤ 0
t₀: evalfstart(X₀, X₁, X₂) → evalfentryin(X₀, X₁, X₂)

MPRF for transition t₂: evalfbb4in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₀) :|: 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁-1]
• evalfbb2in: [X₁-1]
• evalfbb3in: [X₁-1]
• evalfbb4in: [X₁]

MPRF for transition t₄: evalfbb2in(X₀, X₁, X₂) → evalfbb1in(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁+X₂-1-X₀]
• evalfbb2in: [X₁+X₂-X₀]
• evalfbb3in: [X₁+X₂-X₀]
• evalfbb4in: [X₁]

MPRF for transition t₅: evalfbb2in(X₀, X₁, X₂) → evalfbb3in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁-1]
• evalfbb4in: [X₁]

MPRF for transition t₆: evalfbb1in(X₀, X₁, X₂) → evalfbb2in(X₀, X₁, X₂-1) :|: 1 ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁+X₂-X₀]
• evalfbb3in: [X₁+X₂-X₀]
• evalfbb4in: [X₁]

MPRF for transition t₇: evalfbb3in(X₀, X₁, X₂) → evalfbb4in(X₀, X₁-1, X₂) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalfbb1in: [X₁]
• evalfbb2in: [X₁]
• evalfbb3in: [X₁]
• evalfbb4in: [X₁]

All Bounds

Timebounds

Overall timebound:5⋅X₀+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: X₀ {O(n)}
t₅: X₀ {O(n)}
t₆: X₀ {O(n)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₀+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: X₀ {O(n)}
t₅: X₀ {O(n)}
t₆: X₀ {O(n)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₁ {O(n)}
t₂, X₁: X₀ {O(n)}
t₂, X₂: 2⋅X₁ {O(n)}
t₃, X₀: 2⋅X₁ {O(n)}
t₃, X₁: 2⋅X₀ {O(n)}
t₃, X₂: 2⋅X₁+X₂+1 {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: 2⋅X₁ {O(n)}
t₅, X₀: X₁ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 2⋅X₁+1 {O(n)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: 2⋅X₁+1 {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₀ {O(n)}
t₇, X₂: 2⋅X₁+1 {O(n)}
t₈, X₀: 2⋅X₁ {O(n)}
t₈, X₁: 2⋅X₀ {O(n)}
t₈, X₂: 2⋅X₁+X₂+1 {O(n)}