KoAT2 Proof Output

Initial Problem

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:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, 0, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃) :|: X₀ ≤ 50
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-1, X₂, X₃) :|: 51 ≤ X₀
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

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

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

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

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

Found invariant 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁
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:
t₁₃: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(0, 0)
t₁₄: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 0 ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₁₅: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₁₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, 1+X₁) :|: X₀ ≤ 50 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₇: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁-1) :|: 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₈: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 0 ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₀+X₁
t₁₉: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

MPRF for transition t₁₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, 1+X₁) :|: X₀ ≤ 50 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

51 {O(1)}

MPRF:

• eval_foo_bb1_in: [51-X₀]
• eval_foo_bb2_in: [51-X₀]

TWN: t₁₇: eval_foo_bb2_in→eval_foo_bb1_in

cycle: [t₁₇: eval_foo_bb2_in→eval_foo_bb1_in; t₁₄: eval_foo_bb1_in→eval_foo_bb2_in]
loop: (0 ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁,(X₀,X₁) -> (1+X₀,X₁-1))
order: [X₁; X₀]
closed-form:

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

Termination: true
Formula:

0 ≤ 1 ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ 0 ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
∨ 0 ≤ 1 ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
∨ 0 ≤ 1 ∧ 0 ≤ 1+X₁ ∧ 1 ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₁ ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₁ ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
∨ 1 ≤ 0 ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀

Stabilization-Threshold for: 0 ≤ X₁

alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}

Stabilization-Threshold for: 0 ≤ 1+X₁

alphas_abs: 2+X₁
M: 0
N: 1
Bound: 2⋅X₁+6 {O(n)}

loop: (0 ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁,(X₀,X₁) -> (1+X₀,X₁-1))
order: [X₁; X₀]
closed-form:

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

Termination: true
Formula:

Stabilization-Threshold for: 0 ≤ X₁

alphas_abs: 1+X₁
M: 0
N: 1
Bound: 2⋅X₁+4 {O(n)}

Stabilization-Threshold for: 0 ≤ 1+X₁

alphas_abs: 2+X₁
M: 0
N: 1
Bound: 2⋅X₁+6 {O(n)}

TWN - Lifting for [14: eval_foo_bb1_in->eval_foo_bb2_in; 17: eval_foo_bb2_in->eval_foo_bb1_in] of 4⋅X₁+12 {O(n)}

relevant size-bounds w.r.t. t₁₃: eval_foo_bb0_in→eval_foo_bb1_in:
X₁: 0 {O(1)}
Runtime-bound of t₁₃: 1 {O(1)}
Results in: 12 {O(1)}

TWN - Lifting for [14: eval_foo_bb1_in->eval_foo_bb2_in; 17: eval_foo_bb2_in->eval_foo_bb1_in] of 4⋅X₁+12 {O(n)}

relevant size-bounds w.r.t. t₁₆: eval_foo_bb2_in→eval_foo_bb1_in:
X₁: 51 {O(1)}
Runtime-bound of t₁₆: 51 {O(1)}
Results in: 11016 {O(1)}

All Bounds

Timebounds

Overall timebound:22111 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 11028 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 51 {O(1)}
t₁₇: 11028 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: 22111 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 11028 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 51 {O(1)}
t₁₇: 11028 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: 0 {O(1)}
t₁₃, X₁: 0 {O(1)}
t₁₄, X₀: 106 {O(1)}
t₁₄, X₁: 52 {O(1)}
t₁₅, X₀: 108 {O(1)}
t₁₅, X₁: 1 {O(1)}
t₁₆, X₀: 51 {O(1)}
t₁₆, X₁: 51 {O(1)}
t₁₇, X₀: 108 {O(1)}
t₁₇, X₁: 52 {O(1)}
t₁₈, X₀: 108 {O(1)}
t₁₈, X₁: 1 {O(1)}
t₁₉, X₀: X₀ {O(n)}
t₁₉, X₁: X₁ {O(n)}