KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₇: eval_foo_0(X₀, X₁, X₂, X₃) → eval_foo_1(X₀, X₁, X₂, X₃)
t₈: eval_foo_1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1, X₂, X₃) :|: 1 ≤ X₀
t₉: eval_foo_1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 0, X₂, X₃) :|: X₀ ≤ 0
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₂, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ X₁ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_0(X₀, X₁, X₂, X₃)
t₁₀: 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

Found invariant X₀ ≤ X₃ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ for location eval_foo_0

Found invariant X₀ ≤ X₃ for location eval_foo_1

Found invariant X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location eval_foo_stop

Found invariant X₀ ≤ X₃ ∧ X₁ ≤ 0 ∧ 0 ≤ X₁ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_0, eval_foo_1, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₇: eval_foo_0(X₀, X₁, X₂, X₃) → eval_foo_1(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃
t₈: eval_foo_1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃
t₉: eval_foo_1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 0, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₂, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ X₀ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₁ ∧ X₀ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ X₁ ≤ 0 ∧ X₀ ≤ X₃
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_0(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃
t₁₀: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₃ ∧ 0 ≤ X₁ ∧ X₁ ≤ 0
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ X₀ ≤ X₃

MPRF for transition t₈: eval_foo_1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_foo_0: [X₀]
• eval_foo_1: [X₀]
• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₉: eval_foo_1(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 0, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_0: [1]
• eval_foo_1: [1]
• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [1]

TWN: t₇: eval_foo_0→eval_foo_1

cycle: [t₇: eval_foo_0→eval_foo_1; t₉: eval_foo_1→eval_foo_bb1_in; 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_0]
loop: (1+X₁ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∨ 1 ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃,(X₀,X₁,X₃) -> (X₀-1,0,X₃))
order: [X₃; X₁; X₀]
closed-form:

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

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃
∨ 1 ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃

loop: (1+X₁ ≤ 0 ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃ ∨ 1 ≤ X₁ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃,(X₀,X₁,X₃) -> (X₀-1,0,X₃))
order: [X₃; X₁; X₀]
closed-form:

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

Termination: true
Formula:

0 ≤ 1 ∧ 1 ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃
∨ 1 ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₃

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_0; 7: eval_foo_0->eval_foo_1; 9: eval_foo_1->eval_foo_bb1_in] of 2 {O(1)}

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

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_0; 7: eval_foo_0->eval_foo_1; 9: eval_foo_1->eval_foo_bb1_in] of 2 {O(1)}

relevant size-bounds w.r.t. t₈: eval_foo_1→eval_foo_bb1_in:
Runtime-bound of t₈: X₃ {O(n)}
Results in: 2⋅X₃ {O(n)}

All Bounds

Timebounds

Overall timebound:7⋅X₃+X₂+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 2⋅X₃+2 {O(n)}
t₄: 1 {O(1)}
t₅: 2⋅X₃+2 {O(n)}
t₇: 2⋅X₃+2 {O(n)}
t₈: X₃ {O(n)}
t₉: X₂+1 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 7⋅X₃+X₂+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 2⋅X₃+2 {O(n)}
t₄: 1 {O(1)}
t₅: 2⋅X₃+2 {O(n)}
t₇: 2⋅X₃+2 {O(n)}
t₈: X₃ {O(n)}
t₉: X₂+1 {O(n)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₃ {O(n)}
t₂, X₁: X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₃ {O(n)}
t₃, X₁: X₂+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 3⋅X₃+1 {O(n)}
t₄, X₁: 0 {O(1)}
t₄, X₂: 3⋅X₂ {O(n)}
t₄, X₃: 3⋅X₃ {O(n)}
t₅, X₀: 2⋅X₃ {O(n)}
t₅, X₁: 2⋅X₂+1 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₇, X₀: 2⋅X₃ {O(n)}
t₇, X₁: 2⋅X₂+1 {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₈, X₀: 2⋅X₃ {O(n)}
t₈, X₁: 1 {O(1)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₉, X₀: 2⋅X₃+1 {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₂: 2⋅X₂ {O(n)}
t₉, X₃: 2⋅X₃ {O(n)}
t₁₀, X₀: 3⋅X₃+1 {O(n)}
t₁₀, X₁: 0 {O(1)}
t₁₀, X₂: 3⋅X₂ {O(n)}
t₁₀, X₃: 3⋅X₃ {O(n)}