KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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) → eval_foo_bb1_in(X₂, X₄, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₁-1, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₁ ≤ 0
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀, X₂, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₁ ≤ 0
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄) :|: X₁ ≤ 0
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Cut unsatisfiable transition [t₅: eval_foo_bb2_in→eval_foo_bb1_in; t₆: eval_foo_bb2_in→eval_foo_bb1_in]

Eliminate variables [X₃] that do not contribute to the problem

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₂ ∧ 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:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂
t₂₃: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃) :|: X₁ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂⋅X₂+X₂+X₃+1 {O(n^2)}

MPRF:

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

knowledge_propagation leads to new time bound X₂⋅X₂+2⋅X₂+X₃+2 {O(n^2)} for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ X₂

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

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb1_in_v3

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v5

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 0 for location eval_foo_bb3_in

Found invariant X₂ ≤ 1+X₁ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v4

Found invariant 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1+X₁ ≤ X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v2

Found invariant 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb1_in_v4

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

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v1

Found invariant X₂ ≤ X₁ ∧ 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in_v3

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+2⋅X₃+4⋅X₂+7 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: X₂⋅X₂+2⋅X₂+X₃+2 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: X₂⋅X₂+X₂+X₃+1 {O(n^2)}
t₂₁: X₂ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+2⋅X₃+4⋅X₂+7 {O(n^2)}
t₁₇: 1 {O(1)}
t₁₈: X₂⋅X₂+2⋅X₂+X₃+2 {O(n^2)}
t₁₉: 1 {O(1)}
t₂₀: X₂⋅X₂+X₂+X₃+1 {O(n^2)}
t₂₁: X₂ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₃ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₈, X₀: X₂ {O(n)}
t₁₈, X₁: X₂+X₃ {O(n)}
t₁₈, X₂: X₂ {O(n)}
t₁₈, X₃: X₃ {O(n)}
t₁₉, X₀: 2⋅X₂ {O(n)}
t₁₉, X₁: X₂+X₃ {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₁₉, X₃: 2⋅X₃ {O(n)}
t₂₀, X₀: X₂ {O(n)}
t₂₀, X₁: X₂+X₃ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₁, X₀: X₂ {O(n)}
t₂₁, X₁: X₂ {O(n)}
t₂₁, X₂: X₂ {O(n)}
t₂₁, X₃: X₃ {O(n)}
t₂₂, X₀: 2⋅X₂ {O(n)}
t₂₂, X₁: X₂+X₃ {O(n)}
t₂₂, X₂: 2⋅X₂ {O(n)}
t₂₂, X₃: 2⋅X₃ {O(n)}
t₂₃, X₀: X₀ {O(n)}
t₂₃, X₁: X₁ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}