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(X₃, 100, 1, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-X₁, X₁-X₂, -X₂, X₃, X₄, X₅)
t₅: 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 [X₄; X₅] that do not contribute to the problem

Found invariant 0 ≤ X₃ ∧ 0 ≤ 1+X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 99 ≤ X₁+X₃ ∧ X₁ ≤ 100+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ 99+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 101 ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ 1+X₂ ∧ 98 ≤ X₁+X₂ ∧ X₁ ≤ 100+X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₀ ∧ 99 ≤ X₁ ∧ 99 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ 99+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 101 ∧ 0 ≤ 1+X₂ ∧ 98 ≤ X₁+X₂ ∧ X₁ ≤ 100+X₂ ∧ X₁ ≤ 100 ∧ 98 ≤ X₁ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ 99+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 101 ∧ X₀+X₂ ≤ 0 ∧ 0 ≤ 1+X₂ ∧ 98 ≤ X₁+X₂ ∧ X₁ ≤ 100+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 100 ∧ X₀+X₁ ≤ 99 ∧ 99 ≤ X₁ ∧ 100+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₃ ∧ X₂ ≤ 1 ∧ 99+X₂ ≤ X₁ ∧ X₁+X₂ ≤ 101 ∧ X₀+X₂ ≤ 0 ∧ 0 ≤ 1+X₂ ∧ 98 ≤ X₁+X₂ ∧ X₁ ≤ 100+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 100 ∧ X₀+X₁ ≤ 99 ∧ 99 ≤ X₁ ∧ 100+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb3_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:
t₁₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, 100, 1, X₃)
t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁+X₂ ≤ 101 ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₂ ∧ 0 ≤ 1+X₂ ∧ X₂ ≤ 1 ∧ 98 ≤ X₁ ∧ 98 ≤ X₁+X₂ ∧ 99+X₂ ≤ X₁ ∧ X₀ ≤ X₃
t₁₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0 ∧ X₁+X₂ ≤ 101 ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₂ ∧ 0 ≤ 1+X₂ ∧ X₂ ≤ 1 ∧ 98 ≤ X₁ ∧ 98 ≤ X₁+X₂ ∧ 99+X₂ ≤ X₁ ∧ X₀ ≤ X₃
t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, X₁-X₂, -X₂, X₃) :|: X₁+X₂ ≤ 101 ∧ X₁ ≤ 100+X₀ ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₂ ∧ X₁ ≤ 100+X₃ ∧ 0 ≤ 1+X₀+X₂ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₂+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ 98 ≤ X₁+X₂ ∧ 99 ≤ X₀+X₁ ∧ 99 ≤ X₁ ∧ 99+X₂ ≤ X₁ ∧ 99 ≤ X₁+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₁+X₂ ≤ 101 ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₂ ∧ X₀+X₁ ≤ 99 ∧ 0 ≤ 1+X₂ ∧ X₂ ≤ 1 ∧ 1+X₀ ≤ 0 ∧ 98 ≤ X₁+X₂ ∧ 99 ≤ X₁ ∧ 99+X₂ ≤ X₁ ∧ 100+X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀+X₂ ≤ 0 ∧ X₀ ≤ X₃
t₁₆: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁+X₂ ≤ 101 ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₂ ∧ 0 ≤ 1+X₂ ∧ X₂ ≤ 1 ∧ 98 ≤ X₁ ∧ 98 ≤ X₁+X₂ ∧ 99+X₂ ≤ X₁ ∧ X₀ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [X₀+98⋅X₁-9800]

MPRF for transition t₁₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, X₁-X₂, -X₂, X₃) :|: X₁+X₂ ≤ 101 ∧ X₁ ≤ 100+X₀ ∧ X₁ ≤ 100 ∧ X₁ ≤ 100+X₂ ∧ X₁ ≤ 100+X₃ ∧ 0 ≤ 1+X₀+X₂ ∧ X₂ ≤ 1+X₀ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₂+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ 98 ≤ X₁+X₂ ∧ 99 ≤ X₀+X₁ ∧ 99 ≤ X₁ ∧ 99+X₂ ≤ X₁ ∧ 99 ≤ X₁+X₃ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₃+197 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁-48-49⋅X₂]
• eval_foo_bb2_in: [50+X₀-49⋅X₂]

All Bounds

Timebounds

Overall timebound:2⋅X₃+202 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₃+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃+197 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃+202 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₃+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃+197 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₃ {O(n)}
t₁₁, X₁: 100 {O(1)}
t₁₁, X₂: 1 {O(1)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: X₃+100 {O(n)}
t₁₂, X₁: 100 {O(1)}
t₁₂, X₂: 1 {O(1)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₀: 2⋅X₃+100 {O(n)}
t₁₃, X₁: 100 {O(1)}
t₁₃, X₂: 1 {O(1)}
t₁₃, X₃: 2⋅X₃ {O(n)}
t₁₄, X₀: X₃+100 {O(n)}
t₁₄, X₁: 100 {O(1)}
t₁₄, X₂: 1 {O(1)}
t₁₄, X₃: X₃ {O(n)}
t₁₅, X₀: 2⋅X₃+100 {O(n)}
t₁₅, X₁: 100 {O(1)}
t₁₅, X₂: 1 {O(1)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}
t₁₆, X₃: X₃ {O(n)}