KoAT2 Proof Output

Initial Problem

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:
t₁: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(X₁, X₁)
t₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1+X₀ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1 ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀-1, X₁) :|: 1 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 0
t₇: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁)
t₀: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

Preprocessing

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

Found invariant X₀ ≤ 0 ∧ 0 ≤ 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:
t₁: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(X₁, X₁)
t₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1+X₀ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1 ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀-1, X₁) :|: 1 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 0
t₇: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₀: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1+X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

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

MPRF for transition t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 0 of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

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

TWN: t₂: eval_foo_bb1_in→eval_foo_bb2_in

cycle: [t₂: eval_foo_bb1_in→eval_foo_bb2_in; t₃: eval_foo_bb1_in→eval_foo_bb2_in; t₅: eval_foo_bb2_in→eval_foo_bb1_in]
loop: (1+X₀ ≤ 0 ∧ 1 ≤ X₀ ∨ 1 ≤ X₀ ∧ 1 ≤ X₀,(X₀) -> (X₀-1))
order: [X₀]
closed-form:

X₀: X₀ + [[n != 0]]⋅-1⋅n^1

Termination: true
Formula:

0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀
∨ 0 ≤ 1 ∧ X₀ ≤ 1 ∧ 1 ≤ 0 ∧ 1 ≤ X₀ ∧ 1+X₀ ≤ 0
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 2 ≤ X₀
∨ 0 ≤ 1 ∧ 1 ≤ 0 ∧ 2 ≤ X₀
∨ 1 ≤ 0
∨ 1 ≤ 0 ∧ 1+X₀ ≤ 0

Stabilization-Threshold for: 1 ≤ X₀

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

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

X₀: X₀ + [[n != 0]]⋅-1⋅n^1

Termination: true
Formula:

Stabilization-Threshold for: 1 ≤ X₀

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

TWN - Lifting for [2: eval_foo_bb1_in->eval_foo_bb2_in; 3: eval_foo_bb1_in->eval_foo_bb2_in; 5: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₁: eval_foo_bb0_in→eval_foo_bb1_in:
X₀: X₁ {O(n)}
Runtime-bound of t₁: 1 {O(1)}
Results in: 2⋅X₁+4 {O(n)}

TWN - Lifting for [2: eval_foo_bb1_in->eval_foo_bb2_in; 3: eval_foo_bb1_in->eval_foo_bb2_in; 5: eval_foo_bb2_in->eval_foo_bb1_in] of 2⋅X₀+4 {O(n)}

relevant size-bounds w.r.t. t₆: eval_foo_bb2_in→eval_foo_bb1_in:
X₀: X₁+1 {O(n)}
Runtime-bound of t₆: X₁+1 {O(n)}
Results in: 2⋅X₁⋅X₁+8⋅X₁+6 {O(n^2)}

Found invariant 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location eval_foo_bb1_in_v1

Found invariant 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v2

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

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

Found invariant 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0 for location eval_foo_bb2_in_v2

Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_bb1_in

Found invariant 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

Analysing control-flow refined program

MPRF for transition t₅₂: eval_foo_bb2_in_v1(X₀, X₁) → eval_foo_bb1_in_v2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_foo_bb1_in_v2: [X₀]
• eval_foo_bb2_in_v1: [X₀]

MPRF for transition t₅₄: eval_foo_bb1_in_v2(X₀, X₁) → eval_foo_bb2_in_v1(X₀, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_foo_bb1_in_v2: [1+X₀]
• eval_foo_bb2_in_v1: [X₀]

MPRF for transition t₄₉: eval_foo_bb2_in_v2(X₀, X₁) → eval_foo_bb1_in_v1(1+X₀, X₁) :|: X₀ ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• eval_foo_bb1_in_v1: [1-X₀]
• eval_foo_bb2_in_v2: [1-X₀]

MPRF for transition t₅₁: eval_foo_bb1_in_v1(X₀, X₁) → eval_foo_bb2_in_v2(X₀, X₁) :|: 1+X₀ ≤ 0 ∧ 0 ≤ 0 ∧ X₀ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ X₀ ≤ 0 of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_foo_bb1_in_v1: [1-X₀]
• eval_foo_bb2_in_v2: [-X₀]

CFR: Improvement to new bound with the following program:

method: PartialEvaluation new bound:

O(n)

cfr-program:

Start: eval_foo_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb1_in_v1, eval_foo_bb1_in_v2, eval_foo_bb2_in_v1, eval_foo_bb2_in_v2, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(X₁, X₁)
t₄₇: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in_v1(X₀, X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₄₈: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in_v2(X₀, X₁) :|: 1+X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₄₆: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₅₁: eval_foo_bb1_in_v1(X₀, X₁) → eval_foo_bb2_in_v2(X₀, X₁) :|: 1+X₀ ≤ 0 ∧ 0 ≤ 0 ∧ X₀ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ X₀ ≤ 0
t₅₀: eval_foo_bb1_in_v1(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ 0 ∧ X₀ ≤ 1 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ 1+X₁ ≤ 0
t₅₄: eval_foo_bb1_in_v2(X₀, X₁) → eval_foo_bb2_in_v1(X₀, X₁) :|: 1 ≤ X₀ ∧ 0 ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ 0 ≤ X₀
t₅₃: eval_foo_bb1_in_v2(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ 0 ≤ 0 ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁
t₅₂: eval_foo_bb2_in_v1(X₀, X₁) → eval_foo_bb1_in_v2(X₀-1, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁
t₄₉: eval_foo_bb2_in_v2(X₀, X₁) → eval_foo_bb1_in_v1(1+X₀, X₁) :|: X₀ ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀
t₇: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₀: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

All Bounds

Timebounds

Overall timebound:4⋅X₁+10 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 1 {O(1)}
t₇: 1 {O(1)}
t₄₆: 1 {O(1)}
t₄₇: 1 {O(1)}
t₄₈: 1 {O(1)}
t₄₉: X₁+1 {O(n)}
t₅₀: 1 {O(1)}
t₅₁: X₁ {O(n)}
t₅₂: X₁ {O(n)}
t₅₃: 1 {O(1)}
t₅₄: X₁ {O(n)}

Costbounds

Overall costbound: 4⋅X₁+10 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₄: 1 {O(1)}
t₇: 1 {O(1)}
t₄₆: 1 {O(1)}
t₄₇: 1 {O(1)}
t₄₈: 1 {O(1)}
t₄₉: X₁+1 {O(n)}
t₅₀: 1 {O(1)}
t₅₁: X₁ {O(n)}
t₅₂: X₁ {O(n)}
t₅₃: 1 {O(1)}
t₅₄: X₁ {O(n)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₄, X₀: 0 {O(1)}
t₄, X₁: 0 {O(1)}
t₇, X₀: 0 {O(1)}
t₇, X₁: 2⋅X₁ {O(n)}
t₄₆, X₀: 0 {O(1)}
t₄₆, X₁: 0 {O(1)}
t₄₇, X₀: X₁ {O(n)}
t₄₇, X₁: X₁ {O(n)}
t₄₈, X₀: X₁ {O(n)}
t₄₈, X₁: X₁ {O(n)}
t₄₉, X₀: X₁ {O(n)}
t₄₉, X₁: X₁ {O(n)}
t₅₀, X₀: 0 {O(1)}
t₅₀, X₁: X₁ {O(n)}
t₅₁, X₀: X₁ {O(n)}
t₅₁, X₁: X₁ {O(n)}
t₅₂, X₀: X₁ {O(n)}
t₅₂, X₁: X₁ {O(n)}
t₅₃, X₀: 0 {O(1)}
t₅₃, X₁: X₁ {O(n)}
t₅₄, X₀: X₁ {O(n)}
t₅₄, X₁: X₁ {O(n)}