KoAT2 Proof Output

Initial Problem

Start: eval_wise_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_wise_0, eval_wise_1, eval_wise_2, eval_wise__critedge_in, eval_wise_bb0_in, eval_wise_bb1_in, eval_wise_bb2_in, eval_wise_start, eval_wise_stop
Transitions:
t₂: eval_wise_0(X₀, X₁, X₂, X₃) → eval_wise_1(X₀, X₁, X₂, X₃)
t₃: eval_wise_1(X₀, X₁, X₂, X₃) → eval_wise_2(X₀, X₁, X₂, X₃)
t₆: eval_wise_2(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(X₂, X₃, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₃
t₄: eval_wise_2(X₀, X₁, X₂, X₃) → eval_wise_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ 0
t₅: eval_wise_2(X₀, X₁, X₂, X₃) → eval_wise_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0
t₁₀: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁
t₁₁: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₁₂: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(1+X₀, 1+X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ X₀
t₁₃: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(X₀, 1+X₁, X₂, X₃) :|: X₁ ≤ X₀
t₁: eval_wise_bb0_in(X₀, X₁, X₂, X₃) → eval_wise_0(X₀, X₁, X₂, X₃)
t₇: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise__critedge_in(X₀, X₁, X₂, X₃) :|: 3+X₁ ≤ X₀
t₈: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise__critedge_in(X₀, X₁, X₂, X₃) :|: 3+X₀ ≤ X₁
t₉: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2+X₀ ∧ X₀ ≤ 2+X₁
t₁₄: eval_wise_bb2_in(X₀, X₁, X₂, X₃) → eval_wise_stop(X₀, X₁, X₂, X₃)
t₀: eval_wise_start(X₀, X₁, X₂, X₃) → eval_wise_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Cut unsatisfiable transition [t₁₁: eval_wise__critedge_in→eval_wise_bb1_in; t₁₂: eval_wise__critedge_in→eval_wise_bb1_in]

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_wise__critedge_in

Found invariant X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_wise_bb1_in

Problem after Preprocessing

Start: eval_wise_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_wise_0, eval_wise_1, eval_wise_2, eval_wise__critedge_in, eval_wise_bb0_in, eval_wise_bb1_in, eval_wise_bb2_in, eval_wise_start, eval_wise_stop
Transitions:
t₂: eval_wise_0(X₀, X₁, X₂, X₃) → eval_wise_1(X₀, X₁, X₂, X₃)
t₃: eval_wise_1(X₀, X₁, X₂, X₃) → eval_wise_2(X₀, X₁, X₂, X₃)
t₆: eval_wise_2(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(X₂, X₃, X₂, X₃) :|: 0 ≤ X₂ ∧ 0 ≤ X₃
t₄: eval_wise_2(X₀, X₁, X₂, X₃) → eval_wise_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ 0
t₅: eval_wise_2(X₀, X₁, X₂, X₃) → eval_wise_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0
t₁₀: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₃: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(X₀, 1+X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁: eval_wise_bb0_in(X₀, X₁, X₂, X₃) → eval_wise_0(X₀, X₁, X₂, X₃)
t₇: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise__critedge_in(X₀, X₁, X₂, X₃) :|: 3+X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₈: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise__critedge_in(X₀, X₁, X₂, X₃) :|: 3+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₉: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise_bb2_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 2+X₀ ∧ X₀ ≤ 2+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃
t₁₄: eval_wise_bb2_in(X₀, X₁, X₂, X₃) → eval_wise_stop(X₀, X₁, X₂, X₃)
t₀: eval_wise_start(X₀, X₁, X₂, X₃) → eval_wise_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₇: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise__critedge_in(X₀, X₁, X₂, X₃) :|: 3+X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+X₃+2 {O(n)}

MPRF:

• eval_wise__critedge_in: [X₀-3-X₁]
• eval_wise_bb1_in: [X₀-2-X₁]

MPRF for transition t₈: eval_wise_bb1_in(X₀, X₁, X₂, X₃) → eval_wise__critedge_in(X₀, X₁, X₂, X₃) :|: 3+X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+X₃+4 {O(n)}

MPRF:

• eval_wise__critedge_in: [3+X₁-X₀]
• eval_wise_bb1_in: [4+X₁-X₀]

MPRF for transition t₁₀: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_wise__critedge_in: [1+X₁-X₀]
• eval_wise_bb1_in: [1+X₁-X₀]

MPRF for transition t₁₃: eval_wise__critedge_in(X₀, X₁, X₂, X₃) → eval_wise_bb1_in(X₀, 1+X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₀+X₃ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_wise__critedge_in: [1+X₀-X₁]
• eval_wise_bb1_in: [1+X₀-X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₂+4⋅X₃+17 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₂+X₃+2 {O(n)}
t₈: X₂+X₃+4 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₂+X₃+1 {O(n)}
t₁₃: X₂+X₃+1 {O(n)}
t₁₄: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₂+4⋅X₃+17 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₂+X₃+2 {O(n)}
t₈: X₂+X₃+4 {O(n)}
t₉: 1 {O(1)}
t₁₀: X₂+X₃+1 {O(n)}
t₁₃: X₂+X₃+1 {O(n)}
t₁₄: 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: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₀: X₂ {O(n)}
t₆, X₁: X₃ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: 2⋅X₃+X₂+1 {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: 2⋅X₂+X₃+1 {O(n)}
t₈, X₁: X₃ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: 4⋅X₂+X₃+1 {O(n)}
t₉, X₁: 4⋅X₃+X₂+1 {O(n)}
t₉, X₂: 3⋅X₂ {O(n)}
t₉, X₃: 3⋅X₃ {O(n)}
t₁₀, X₀: 2⋅X₂+X₃+1 {O(n)}
t₁₀, X₁: X₃ {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₃, X₀: X₂ {O(n)}
t₁₃, X₁: 2⋅X₃+X₂+1 {O(n)}
t₁₃, X₂: X₂ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₄, X₀: 2⋅X₀+4⋅X₂+X₃+1 {O(n)}
t₁₄, X₁: 2⋅X₁+4⋅X₃+X₂+1 {O(n)}
t₁₄, X₂: 5⋅X₂ {O(n)}
t₁₄, X₃: 5⋅X₃ {O(n)}