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(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₁, X₁-1, X₂, X₃) :|: 0 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₁, X₁, X₂, X₃) :|: 1+X₁ ≤ 0
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

Found invariant X₁ ≤ X₃ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₃ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

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

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₁, X₁-1, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [1+X₁]

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃⋅X₃⋅X₃+4⋅X₃⋅X₃+X₂⋅X₃+2⋅X₂+7⋅X₃+6 {O(n^3)}

MPRF:

• eval_foo_bb1_in: [2+X₀]
• eval_foo_bb2_in: [1+X₀]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀+X₁, X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃⋅X₃⋅X₃+4⋅X₃⋅X₃+X₂⋅X₃+2⋅X₂+6⋅X₃+4 {O(n^3)}

MPRF:

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

Found invariant 1+X₃ ≤ 0 ∧ 1+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 2+X₁+X₃ ≤ 0 ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₃ ∧ 0 ≤ X₂ ∧ 1+X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ for location eval_foo_bb1_in_v1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ 1+X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ 1+X₀+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 2+X₀+X₁ ∧ 0 ≤ 1+X₀ for location eval_foo_bb1_in_v3

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ 1+X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_foo_bb3_in

Found invariant 1+X₃ ≤ 0 ∧ 2+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 2+X₁+X₃ ≤ 0 ∧ 1+X₃ ≤ X₀ ∧ X₁ ≤ X₃ ∧ 1 ≤ X₂ ∧ 2+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v4

Found invariant X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ 1+X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ 1+X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in_v3

All Bounds

Timebounds

Overall timebound:2⋅X₃⋅X₃⋅X₃+2⋅X₂⋅X₃+8⋅X₃⋅X₃+14⋅X₃+4⋅X₂+15 {O(n^3)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃⋅X₃⋅X₃+4⋅X₃⋅X₃+X₂⋅X₃+2⋅X₂+7⋅X₃+6 {O(n^3)}
t₃: 1 {O(1)}
t₄: X₃+1 {O(n)}
t₅: X₃⋅X₃⋅X₃+4⋅X₃⋅X₃+X₂⋅X₃+2⋅X₂+6⋅X₃+4 {O(n^3)}
t₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃⋅X₃⋅X₃+2⋅X₂⋅X₃+8⋅X₃⋅X₃+14⋅X₃+4⋅X₂+15 {O(n^3)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃⋅X₃⋅X₃+4⋅X₃⋅X₃+X₂⋅X₃+2⋅X₂+7⋅X₃+6 {O(n^3)}
t₃: 1 {O(1)}
t₄: X₃+1 {O(n)}
t₅: X₃⋅X₃⋅X₃+4⋅X₃⋅X₃+X₂⋅X₃+2⋅X₂+6⋅X₃+4 {O(n^3)}
t₆: 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: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₃⋅X₃+3⋅X₃+X₂+2 {O(n^2)}
t₂, X₁: X₃+1 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₃⋅X₃+2⋅X₂+3⋅X₃+2 {O(n^2)}
t₃, X₁: 2⋅X₃+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₃⋅X₃+3⋅X₃+X₂+2 {O(n^2)}
t₄, X₁: X₃+1 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₃⋅X₃+3⋅X₃+X₂+2 {O(n^2)}
t₅, X₁: X₃+1 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₃⋅X₃+2⋅X₂+3⋅X₃+2 {O(n^2)}
t₆, X₁: 2⋅X₃+1 {O(n)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}