KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalspeedpldi2bb2in

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalspeedpldi2bb3in

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location evalspeedpldi2bb5in

Problem after Preprocessing

Start: evalspeedpldi2start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: evalspeedpldi2bb2in, evalspeedpldi2bb3in, evalspeedpldi2bb5in, evalspeedpldi2entryin, evalspeedpldi2returnin, evalspeedpldi2start, evalspeedpldi2stop
Transitions:
t₆: evalspeedpldi2bb2in(X₀, X₁, X₂) → evalspeedpldi2bb3in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₇: evalspeedpldi2bb2in(X₀, X₁, X₂) → evalspeedpldi2bb5in(X₀, 0, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₈: evalspeedpldi2bb3in(X₀, X₁, X₂) → evalspeedpldi2bb5in(X₀, 1+X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁
t₄: evalspeedpldi2bb5in(X₀, X₁, X₂) → evalspeedpldi2bb2in(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂
t₅: evalspeedpldi2bb5in(X₀, X₁, X₂) → evalspeedpldi2returnin(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂
t₁: evalspeedpldi2entryin(X₀, X₁, X₂) → evalspeedpldi2bb5in(X₁, 0, X₀) :|: 1 ≤ X₁ ∧ 0 ≤ X₀
t₂: evalspeedpldi2entryin(X₀, X₁, X₂) → evalspeedpldi2returnin(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₃: evalspeedpldi2entryin(X₀, X₁, X₂) → evalspeedpldi2returnin(X₀, X₁, X₂) :|: X₁ ≤ 0
t₉: evalspeedpldi2returnin(X₀, X₁, X₂) → evalspeedpldi2stop(X₀, X₁, X₂)
t₀: evalspeedpldi2start(X₀, X₁, X₂) → evalspeedpldi2entryin(X₀, X₁, X₂)

MPRF for transition t₄: evalspeedpldi2bb5in(X₀, X₁, X₂) → evalspeedpldi2bb2in(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₀+1 {O(n)}

MPRF:

• evalspeedpldi2bb2in: [X₁+2⋅X₂]
• evalspeedpldi2bb3in: [X₁+2⋅X₂]
• evalspeedpldi2bb5in: [1+X₁+2⋅X₂]

MPRF for transition t₆: evalspeedpldi2bb2in(X₀, X₁, X₂) → evalspeedpldi2bb3in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalspeedpldi2bb2in: [X₂]
• evalspeedpldi2bb3in: [X₂-1]
• evalspeedpldi2bb5in: [X₂]

MPRF for transition t₇: evalspeedpldi2bb2in(X₀, X₁, X₂) → evalspeedpldi2bb5in(X₀, 0, X₂) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

2⋅X₀+1 {O(n)}

MPRF:

• evalspeedpldi2bb2in: [2⋅X₁+2⋅X₂-1]
• evalspeedpldi2bb3in: [2⋅X₁+2⋅X₂-1]
• evalspeedpldi2bb5in: [2⋅X₁+2⋅X₂-1]

MPRF for transition t₈: evalspeedpldi2bb3in(X₀, X₁, X₂) → evalspeedpldi2bb5in(X₀, 1+X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• evalspeedpldi2bb2in: [X₂]
• evalspeedpldi2bb3in: [X₂]
• evalspeedpldi2bb5in: [X₂]

All Bounds

Timebounds

Overall timebound:6⋅X₀+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 2⋅X₀+1 {O(n)}
t₅: 1 {O(1)}
t₆: X₀ {O(n)}
t₇: 2⋅X₀+1 {O(n)}
t₈: X₀ {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₀+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 2⋅X₀+1 {O(n)}
t₅: 1 {O(1)}
t₆: X₀ {O(n)}
t₇: 2⋅X₀+1 {O(n)}
t₈: X₀ {O(n)}
t₉: 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: 0 {O(1)}
t₁, X₂: X₀ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₄, X₀: X₁ {O(n)}
t₄, X₁: X₀ {O(n)}
t₄, X₂: X₀ {O(n)}
t₅, X₀: 2⋅X₁ {O(n)}
t₅, X₁: X₀ {O(n)}
t₅, X₂: 0 {O(1)}
t₆, X₀: X₁ {O(n)}
t₆, X₁: X₀ {O(n)}
t₆, X₂: X₀ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: 0 {O(1)}
t₇, X₂: X₀ {O(n)}
t₈, X₀: X₁ {O(n)}
t₈, X₁: X₀ {O(n)}
t₈, X₂: X₀ {O(n)}
t₉, X₀: 2⋅X₀+2⋅X₁ {O(n)}
t₉, X₁: 2⋅X₁+X₀ {O(n)}
t₉, X₂: 2⋅X₂ {O(n)}