KoAT2 Proof Output

Initial Problem

Start: eval_cousot9_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_cousot9_bb0_in, eval_cousot9_bb1_in, eval_cousot9_bb2_in, eval_cousot9_bb3_in, eval_cousot9_start, eval_cousot9_stop
Transitions:
t₁: eval_cousot9_bb0_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₃, X₁, X₁, X₃)
t₂: eval_cousot9_bb1_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂
t₃: eval_cousot9_bb1_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0
t₄: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀
t₅: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₁, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ 0
t₆: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₀-1, X₁, X₂-1, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ 0
t₇: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0
t₈: eval_cousot9_bb3_in(X₀, X₁, X₂, X₃) → eval_cousot9_stop(X₀, X₁, X₂, X₃)
t₀: eval_cousot9_start(X₀, X₁, X₂, X₃) → eval_cousot9_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Cut unsatisfiable transition [t₅: eval_cousot9_bb2_in→eval_cousot9_bb1_in; t₆: eval_cousot9_bb2_in→eval_cousot9_bb1_in]

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location eval_cousot9_bb3_in

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location eval_cousot9_stop

Found invariant X₂ ≤ X₁ for location eval_cousot9_bb1_in

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_cousot9_bb2_in

Problem after Preprocessing

Start: eval_cousot9_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_cousot9_bb0_in, eval_cousot9_bb1_in, eval_cousot9_bb2_in, eval_cousot9_bb3_in, eval_cousot9_start, eval_cousot9_stop
Transitions:
t₁: eval_cousot9_bb0_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₃, X₁, X₁, X₃)
t₂: eval_cousot9_bb1_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ X₂ ≤ X₁
t₃: eval_cousot9_bb1_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb3_in(X₀, X₁, X₂, X₃) :|: X₂ ≤ 0 ∧ X₂ ≤ X₁
t₄: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁
t₇: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁
t₈: eval_cousot9_bb3_in(X₀, X₁, X₂, X₃) → eval_cousot9_stop(X₀, X₁, X₂, X₃) :|: X₂ ≤ X₁ ∧ X₂ ≤ 0
t₀: eval_cousot9_start(X₀, X₁, X₂, X₃) → eval_cousot9_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₇: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₁, X₁, X₂-1, X₃) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

• eval_cousot9_bb1_in: [X₂]
• eval_cousot9_bb2_in: [X₂]

MPRF for transition t₄: eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁⋅X₁+X₁+X₃+1 {O(n^2)}

MPRF:

• eval_cousot9_bb1_in: [1+X₀]
• eval_cousot9_bb2_in: [1+X₀]

knowledge_propagation leads to new time bound X₁⋅X₁+2⋅X₁+X₃+2 {O(n^2)} for transition t₂: eval_cousot9_bb1_in(X₀, X₁, X₂, X₃) → eval_cousot9_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₂ ∧ X₂ ≤ X₁

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_cousot9_bb1_in_v2

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_cousot9_bb1_in_v3

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_cousot9_bb1_in_v1

Found invariant 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 1+X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_cousot9_bb2_in_v4

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location eval_cousot9_bb1_in_v4

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location eval_cousot9_stop

Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₂ for location eval_cousot9_bb1_in

Found invariant X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₁ for location eval_cousot9_bb2_in_v1

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_cousot9_bb2_in_v2

Found invariant X₂ ≤ 0 ∧ X₂ ≤ X₁ for location eval_cousot9_bb3_in

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location eval_cousot9_bb2_in_v3

Found invariant 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 2+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location eval_cousot9_bb2_in_v5

All Bounds

Timebounds

Overall timebound:2⋅X₁⋅X₁+2⋅X₃+4⋅X₁+7 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁⋅X₁+2⋅X₁+X₃+2 {O(n^2)}
t₃: 1 {O(1)}
t₄: X₁⋅X₁+X₁+X₃+1 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁⋅X₁+2⋅X₃+4⋅X₁+7 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁⋅X₁+2⋅X₁+X₃+2 {O(n^2)}
t₃: 1 {O(1)}
t₄: X₁⋅X₁+X₁+X₃+1 {O(n^2)}
t₇: X₁ {O(n)}
t₈: 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: X₁ {O(n)}
t₁, X₂: X₁ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₁+X₃ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₁ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₁+X₃ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₁ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₁+X₃ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₁ {O(n)}
t₄, X₃: X₃ {O(n)}
t₇, X₀: X₁ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₁ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₁+X₃ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₁ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}