KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(1, X₄, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(0, 1+X₁, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₁
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₅
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

Preprocessing

Eliminate variables [X₂; X₃] that do not contribute to the problem

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ for location eval_foo_bb1_in

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

Found invariant X₂ ≤ X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ X₀ for location eval_foo_bb3_in

Cut unsatisfiable transition [t₁₆: eval_foo_bb1_in→eval_foo_bb2_in]

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₅: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1, X₂, X₂, X₃)
t₁₇: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₀ ≤ 1 ∧ X₂ ≤ X₁
t₁₉: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, 1+X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, 1+X₁, X₂, X₃) :|: 1+X₁ ≤ X₃ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁
t₂₁: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₂ ≤ X₁
t₂₂: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₁₉: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, 1+X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

1 {O(1)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [1]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, 1+X₁, X₂, X₃) :|: 1+X₁ ≤ X₃ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₃-X₁]
• eval_foo_bb2_in: [X₃-X₁]

TWN: t₂₀: eval_foo_bb2_in→eval_foo_bb1_in

cycle: [t₂₀: eval_foo_bb2_in→eval_foo_bb1_in; t₁₇: eval_foo_bb1_in→eval_foo_bb2_in]
original loop: (1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁,(X₀,X₁,X₂,X₃) -> (X₀,1+X₁,X₂,X₃))
transformed loop: (1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁,(X₀,X₁,X₂,X₃) -> (X₀,1+X₁,X₂,X₃))
loop: (1 ≤ X₀ ∧ X₀ ≤ 1 ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ ∧ X₂ ≤ X₁,(X₀,X₁,X₂,X₃) -> (X₀,1+X₁,X₂,X₃))
order: [X₃; X₂; X₁; X₀]
closed-form:

X₃: X₃
X₂: X₂
X₁: X₁ + [[n != 0]]⋅n^1
X₀: X₀

Termination: true
Formula:

0 ≤ 1 ∧ X₀ ≤ 1 ∧ X₃ ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 2+X₁ ≤ X₃ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁
∨ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 0 ≤ X₀ ∧ X₂ ≤ X₁

Stabilization-Threshold for: 1+X₁ ≤ X₃

alphas_abs: X₁+X₃
M: 0
N: 1
Bound: 2⋅X₁+2⋅X₃+2 {O(n)}

TWN - Lifting for [17: eval_foo_bb1_in->eval_foo_bb2_in; 20: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₁+2⋅X₃+4 {O(n)}

relevant size-bounds w.r.t. t₁₅: eval_foo_bb0_in→eval_foo_bb1_in:
X₁: X₂ {O(n)}
X₃: X₃ {O(n)}
Runtime-bound of t₁₅: 1 {O(1)}
Results in: 2⋅X₂+2⋅X₃+4 {O(n)}

All Bounds

Timebounds

Overall timebound:3⋅X₂+3⋅X₃+9 {O(n)}
t₁₅: 1 {O(1)}
t₁₇: 2⋅X₂+2⋅X₃+4 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: X₂+X₃ {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₂+3⋅X₃+9 {O(n)}
t₁₅: 1 {O(1)}
t₁₇: 2⋅X₂+2⋅X₃+4 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: X₂+X₃ {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}

Sizebounds

t₁₅, X₀: 1 {O(1)}
t₁₅, X₁: X₂ {O(n)}
t₁₅, X₂: X₂ {O(n)}
t₁₅, X₃: X₃ {O(n)}
t₁₇, X₀: 1 {O(1)}
t₁₇, X₁: 2⋅X₂+X₃ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: 0 {O(1)}
t₁₈, X₁: 2⋅X₂+X₃+1 {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 0 {O(1)}
t₁₉, X₁: 2⋅X₂+X₃+1 {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₂₀, X₀: 1 {O(1)}
t₂₀, X₁: 2⋅X₂+X₃ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: 0 {O(1)}
t₂₁, X₁: 2⋅X₂+X₃+1 {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: X₀ {O(n)}
t₂₂, X₁: X₁ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}