KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_while2_bb3_in

Found invariant X₁ ≤ X₀ for location eval_while2_bb1_in

Found invariant X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_while2_bb2_in

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location eval_while2_bb5_in

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location eval_while2_stop

Found invariant X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_while2_bb4_in

Problem after Preprocessing

Start: eval_while2_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_while2_bb0_in, eval_while2_bb1_in, eval_while2_bb2_in, eval_while2_bb3_in, eval_while2_bb4_in, eval_while2_bb5_in, eval_while2_start, eval_while2_stop
Transitions:
t₁: eval_while2_bb0_in(X₀, X₁, X₂) → eval_while2_bb1_in(X₀, X₀, X₂)
t₂: eval_while2_bb1_in(X₀, X₁, X₂) → eval_while2_bb2_in(X₀, X₁, X₀) :|: 1 ≤ X₁ ∧ X₁ ≤ X₀
t₃: eval_while2_bb1_in(X₀, X₁, X₂) → eval_while2_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ 0 ∧ X₁ ≤ X₀
t₄: eval_while2_bb2_in(X₀, X₁, X₂) → eval_while2_bb3_in(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂
t₅: eval_while2_bb2_in(X₀, X₁, X₂) → eval_while2_bb4_in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂
t₆: eval_while2_bb3_in(X₀, X₁, X₂) → eval_while2_bb2_in(X₀, X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀
t₇: eval_while2_bb4_in(X₀, X₁, X₂) → eval_while2_bb1_in(X₀, X₁-1, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₈: eval_while2_bb5_in(X₀, X₁, X₂) → eval_while2_stop(X₀, X₁, X₂) :|: X₁ ≤ X₀ ∧ X₁ ≤ 0
t₀: eval_while2_start(X₀, X₁, X₂) → eval_while2_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂: eval_while2_bb1_in(X₀, X₁, X₂) → eval_while2_bb2_in(X₀, X₁, X₀) :|: 1 ≤ X₁ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_while2_bb1_in: [X₁]
• eval_while2_bb2_in: [X₁-1]
• eval_while2_bb3_in: [X₁-1]
• eval_while2_bb4_in: [X₁-1]

MPRF for transition t₅: eval_while2_bb2_in(X₀, X₁, X₂) → eval_while2_bb4_in(X₀, X₁, X₂) :|: X₂ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_while2_bb1_in: [X₁]
• eval_while2_bb2_in: [X₁]
• eval_while2_bb3_in: [X₁]
• eval_while2_bb4_in: [X₁-1]

MPRF for transition t₇: eval_while2_bb4_in(X₀, X₁, X₂) → eval_while2_bb1_in(X₀, X₁-1, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1+X₂ ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0 of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_while2_bb1_in: [X₁]
• eval_while2_bb2_in: [X₁]
• eval_while2_bb3_in: [X₁]
• eval_while2_bb4_in: [X₁]

MPRF for transition t₄: eval_while2_bb2_in(X₀, X₁, X₂) → eval_while2_bb3_in(X₀, X₁, X₂) :|: 1 ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₀⋅X₀+2⋅X₀+1 {O(n^2)}

MPRF:

• eval_while2_bb1_in: [1+X₀]
• eval_while2_bb2_in: [1+X₂]
• eval_while2_bb3_in: [X₂]
• eval_while2_bb4_in: [0]

MPRF for transition t₆: eval_while2_bb3_in(X₀, X₁, X₂) → eval_while2_bb2_in(X₀, X₁, X₂-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₀⋅X₀+X₀ {O(n^2)}

MPRF:

• eval_while2_bb1_in: [X₀]
• eval_while2_bb2_in: [X₂]
• eval_while2_bb3_in: [X₂]
• eval_while2_bb4_in: [X₂]

Cut unsatisfiable transition [t₅: eval_while2_bb2_in→eval_while2_bb4_in; t₄₅: eval_while2_bb2_in→eval_while2_bb4_in]

Found invariant 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_while2_bb2_in_v1

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_while2_bb3_in_v1

Found invariant X₁ ≤ X₀ for location eval_while2_bb1_in

Found invariant X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_while2_bb2_in

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location eval_while2_bb5_in

Found invariant 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_while2_bb3_in_v2

Found invariant X₁ ≤ 0 ∧ X₁ ≤ X₀ for location eval_while2_stop

All Bounds

Timebounds

Overall timebound:2⋅X₀⋅X₀+6⋅X₀+5 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: X₀⋅X₀+2⋅X₀+1 {O(n^2)}
t₅: X₀ {O(n)}
t₆: X₀⋅X₀+X₀ {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₀⋅X₀+6⋅X₀+5 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: X₀⋅X₀+2⋅X₀+1 {O(n^2)}
t₅: X₀ {O(n)}
t₆: X₀⋅X₀+X₀ {O(n^2)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Sizebounds

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