KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb5_in

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb3_in

Found invariant 0 ≤ X₀ for location eval_textbook_ex2_bb1_in

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb2_in

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb4_in

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₀ for location eval_textbook_ex2_stop

Problem after Preprocessing

Start: eval_textbook_ex2_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_textbook_ex2_bb0_in, eval_textbook_ex2_bb1_in, eval_textbook_ex2_bb2_in, eval_textbook_ex2_bb3_in, eval_textbook_ex2_bb4_in, eval_textbook_ex2_bb5_in, eval_textbook_ex2_start, eval_textbook_ex2_stop
Transitions:
t₁: eval_textbook_ex2_bb0_in(X₀, X₁, X₂) → eval_textbook_ex2_bb1_in(0, X₁, X₂)
t₂: eval_textbook_ex2_bb1_in(X₀, X₁, X₂) → eval_textbook_ex2_bb2_in(X₀, 0, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀
t₃: eval_textbook_ex2_bb1_in(X₀, X₁, X₂) → eval_textbook_ex2_bb5_in(X₀, X₁, X₂) :|: X₂ ≤ X₀ ∧ 0 ≤ X₀
t₄: eval_textbook_ex2_bb2_in(X₀, X₁, X₂) → eval_textbook_ex2_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂
t₅: eval_textbook_ex2_bb2_in(X₀, X₁, X₂) → eval_textbook_ex2_bb4_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂
t₆: eval_textbook_ex2_bb3_in(X₀, X₁, X₂) → eval_textbook_ex2_bb2_in(X₀, 1+X₁, X₂) :|: 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₇: eval_textbook_ex2_bb4_in(X₀, X₁, X₂) → eval_textbook_ex2_bb1_in(1+X₀, X₁, X₂) :|: X₁ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₈: eval_textbook_ex2_bb5_in(X₀, X₁, X₂) → eval_textbook_ex2_stop(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ X₂ ≤ X₀
t₀: eval_textbook_ex2_start(X₀, X₁, X₂) → eval_textbook_ex2_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂: eval_textbook_ex2_bb1_in(X₀, X₁, X₂) → eval_textbook_ex2_bb2_in(X₀, 0, X₂) :|: 1+X₀ ≤ X₂ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_textbook_ex2_bb1_in: [X₂-X₀]
• eval_textbook_ex2_bb2_in: [X₂-1-X₀]
• eval_textbook_ex2_bb3_in: [X₂-1-X₀]
• eval_textbook_ex2_bb4_in: [X₂-X₁]

MPRF for transition t₅: eval_textbook_ex2_bb2_in(X₀, X₁, X₂) → eval_textbook_ex2_bb4_in(X₀, X₁, X₂) :|: 1+X₀ ≤ X₁ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_textbook_ex2_bb1_in: [X₂-X₀]
• eval_textbook_ex2_bb2_in: [X₂-X₀]
• eval_textbook_ex2_bb3_in: [X₂-X₀]
• eval_textbook_ex2_bb4_in: [X₂-1-X₀]

MPRF for transition t₇: eval_textbook_ex2_bb4_in(X₀, X₁, X₂) → eval_textbook_ex2_bb1_in(1+X₀, X₁, X₂) :|: X₁ ≤ 1+X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_textbook_ex2_bb1_in: [X₂-X₀]
• eval_textbook_ex2_bb2_in: [X₂-X₀]
• eval_textbook_ex2_bb3_in: [X₂-X₀]
• eval_textbook_ex2_bb4_in: [1+X₂-X₁]

MPRF for transition t₄: eval_textbook_ex2_bb2_in(X₀, X₁, X₂) → eval_textbook_ex2_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ ∧ X₁ ≤ X₂ of depth 1:

new bound:

X₂⋅X₂+2⋅X₂+1 {O(n^2)}

MPRF:

• eval_textbook_ex2_bb1_in: [1+X₂]
• eval_textbook_ex2_bb2_in: [1+X₂-X₁]
• eval_textbook_ex2_bb3_in: [X₂-X₁]
• eval_textbook_ex2_bb4_in: [X₂-X₁]

MPRF for transition t₆: eval_textbook_ex2_bb3_in(X₀, X₁, X₂) → eval_textbook_ex2_bb2_in(X₀, 1+X₁, X₂) :|: 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+X₂+1 {O(n^2)}

MPRF:

• eval_textbook_ex2_bb1_in: [1+X₀]
• eval_textbook_ex2_bb2_in: [1+X₀-X₁]
• eval_textbook_ex2_bb3_in: [1+X₀-X₁]
• eval_textbook_ex2_bb4_in: [X₀-X₁]

Cut unsatisfiable transition [t₅: eval_textbook_ex2_bb2_in→eval_textbook_ex2_bb4_in; t₄₅: eval_textbook_ex2_bb2_in→eval_textbook_ex2_bb4_in]

Found invariant 0 ≤ X₀ for location eval_textbook_ex2_bb1_in

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb2_in_v1

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₀ for location eval_textbook_ex2_stop

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb5_in

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb3_in_v1

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_textbook_ex2_bb2_in

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_textbook_ex2_bb3_in_v2

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₂+6⋅X₂+6 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₄: X₂⋅X₂+2⋅X₂+1 {O(n^2)}
t₅: X₂ {O(n)}
t₆: X₂⋅X₂+X₂+1 {O(n^2)}
t₇: X₂ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₂+6⋅X₂+6 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₄: X₂⋅X₂+2⋅X₂+1 {O(n^2)}
t₅: X₂ {O(n)}
t₆: X₂⋅X₂+X₂+1 {O(n^2)}
t₇: X₂ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: 0 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₁+X₂+2 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₂+2 {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₂+2 {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₂ {O(n)}
t₆, X₁: X₂+2 {O(n)}
t₆, X₂: X₂ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: X₂+2 {O(n)}
t₇, X₂: X₂ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: X₁+X₂+2 {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}