KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂) → eval_start_1(X₀, X₁, X₂)
t₃: eval_start_1(X₀, X₁, X₂) → eval_start_2(X₀, X₁, X₂)
t₄: eval_start_2(X₀, X₁, X₂) → eval_start_3(X₀, X₁, X₂)
t₅: eval_start_3(X₀, X₁, X₂) → eval_start_4(X₀, X₁, X₂)
t₆: eval_start_4(X₀, X₁, X₂) → eval_start_5(X₀, X₁, X₂)
t₇: eval_start_5(X₀, X₁, X₂) → eval_start_bb1_in(X₂, 1, X₂)
t₁: eval_start_bb0_in(X₀, X₁, X₂) → eval_start_0(X₀, X₁, X₂)
t₈: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₁
t₉: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 0
t₁₀: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀-1, 1, X₂) :|: 1 ≤ X₀
t₁₁: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 1, X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ 0
t₁₂: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀-1, 0, X₂) :|: 1 ≤ X₀ ∧ X₀ ≤ 0
t₁₃: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 0, X₂) :|: X₀ ≤ 0
t₁₄: eval_start_bb3_in(X₀, X₁, X₂) → eval_start_stop(X₀, X₁, X₂)
t₀: eval_start_start(X₀, X₁, X₂) → eval_start_bb0_in(X₀, X₁, X₂)

Preprocessing

Cut unsatisfiable transition [t₁₁: eval_start_bb2_in→eval_start_bb1_in; t₁₂: eval_start_bb2_in→eval_start_bb1_in]

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 0 ≤ X₁ for location eval_start_bb1_in

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location eval_start_stop

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ for location eval_start_bb2_in

Found invariant X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location eval_start_bb3_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁, X₂) → eval_start_1(X₀, X₁, X₂)
t₃: eval_start_1(X₀, X₁, X₂) → eval_start_2(X₀, X₁, X₂)
t₄: eval_start_2(X₀, X₁, X₂) → eval_start_3(X₀, X₁, X₂)
t₅: eval_start_3(X₀, X₁, X₂) → eval_start_4(X₀, X₁, X₂)
t₆: eval_start_4(X₀, X₁, X₂) → eval_start_5(X₀, X₁, X₂)
t₇: eval_start_5(X₀, X₁, X₂) → eval_start_bb1_in(X₂, 1, X₂)
t₁: eval_start_bb0_in(X₀, X₁, X₂) → eval_start_0(X₀, X₁, X₂)
t₈: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb2_in(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₉: eval_start_bb1_in(X₀, X₁, X₂) → eval_start_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₁ ≤ 1 ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₁₀: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀-1, 1, X₂) :|: 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂
t₁₃: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂
t₁₄: eval_start_bb3_in(X₀, X₁, X₂) → eval_start_stop(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
t₀: eval_start_start(X₀, X₁, X₂) → eval_start_bb0_in(X₀, X₁, X₂)

MPRF for transition t₁₀: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀-1, 1, X₂) :|: 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_start_bb1_in: [X₀]
• eval_start_bb2_in: [X₀]

MPRF for transition t₁₃: eval_start_bb2_in(X₀, X₁, X₂) → eval_start_bb1_in(X₀, 0, X₂) :|: X₀ ≤ 0 ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ of depth 1:

new bound:

1 {O(1)}

MPRF:

• eval_start_bb1_in: [X₁]
• eval_start_bb2_in: [1]

TWN: t₁₀: eval_start_bb2_in→eval_start_bb1_in

cycle: [t₁₀: eval_start_bb2_in→eval_start_bb1_in; t₈: eval_start_bb1_in→eval_start_bb2_in]
original loop: (1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂,(X₀,X₁,X₂) -> (X₀-1,1,X₂))
transformed loop: (1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂,(X₀,X₁,X₂) -> (X₀-1,1,X₂))
loop: (1 ≤ X₁ ∧ X₁ ≤ 1 ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀ ∧ X₁ ≤ 1 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂,(X₀,X₁,X₂) -> (X₀-1,1,X₂))
order: [X₂; X₁; X₀]
closed-form:

X₂: X₂
X₁: [[n == 0]]⋅X₁ + [[n != 0]]
X₀: X₀ + [[n != 0]]⋅-1⋅n^1

Termination: true
Formula:

0 ≤ 1 ∧ X₀ ≤ 1 ∧ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
∨ X₁ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁

Stabilization-Threshold for: 1 ≤ X₀

alphas_abs: X₀
M: 0
N: 1
Bound: 2⋅X₀+2 {O(n)}

TWN - Lifting for [8: eval_start_bb1_in->eval_start_bb2_in; 10: eval_start_bb2_in->eval_start_bb1_in] of 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₇: eval_start_5→eval_start_bb1_in:
X₀: X₂ {O(n)}
Runtime-bound of t₇: 1 {O(1)}
Results in: 2⋅X₂+4 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₂+15 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+4 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₂ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+15 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+4 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₂ {O(n)}
t₁₃: 1 {O(1)}
t₁₄: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {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: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: 1 {O(1)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: 1 {O(1)}
t₈, X₂: X₂ {O(n)}
t₉, X₀: X₂ {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₂: X₂ {O(n)}
t₁₀, X₀: X₂ {O(n)}
t₁₀, X₁: 1 {O(1)}
t₁₀, X₂: X₂ {O(n)}
t₁₃, X₀: X₂ {O(n)}
t₁₃, X₁: 0 {O(1)}
t₁₃, X₂: X₂ {O(n)}
t₁₄, X₀: X₂ {O(n)}
t₁₄, X₁: 0 {O(1)}
t₁₄, X₂: X₂ {O(n)}