KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1+X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₄+X₅ ∧ 3 ≤ X₂+X₅ ∧ 1 ≤ X₁+X₅ ∧ 1+X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_local_alloc_bb4_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 0 ≤ X₀ for location eval_local_alloc_bb2_in

Found invariant X₄ ≤ X₀ ∧ 0 ≤ X₀ for location eval_local_alloc_bb6_in

Found invariant 0 ≤ X₀ for location eval_local_alloc_bb1_in

Found invariant 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_local_alloc_bb3_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 0 ≤ X₀ for location eval_local_alloc_bb5_in

Found invariant X₄ ≤ X₀ ∧ 0 ≤ X₀ for location eval_local_alloc_stop

Problem after Preprocessing

Start: eval_local_alloc_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_local_alloc_bb0_in, eval_local_alloc_bb1_in, eval_local_alloc_bb2_in, eval_local_alloc_bb3_in, eval_local_alloc_bb4_in, eval_local_alloc_bb5_in, eval_local_alloc_bb6_in, eval_local_alloc_start, eval_local_alloc_stop
Transitions:
t₁: eval_local_alloc_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb1_in(0, X₁, X₂, X₃, X₄, X₃)
t₂: eval_local_alloc_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₄ ∧ 0 ≤ X₀
t₃: eval_local_alloc_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₄ ≤ X₀ ∧ 0 ≤ X₀
t₄: eval_local_alloc_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb3_in(X₀, 0, X₂, X₃, X₄, X₅) :|: 1+X₅ ≤ X₂ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀
t₅: eval_local_alloc_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀
t₆: eval_local_alloc_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₅ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₇: eval_local_alloc_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₈: eval_local_alloc_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb3_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀+X₄ ∧ 1 ≤ X₀+X₅ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₁+X₅ ∧ 1+X₁ ≤ X₅ ∧ 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₄+X₅ ∧ 3 ≤ X₂+X₄ ∧ 3 ≤ X₂+X₅ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁
t₉: eval_local_alloc_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb1_in(1+X₀, X₁, X₂, X₃, X₄, 0) :|: 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀
t₁₀: eval_local_alloc_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ X₄ ≤ X₀
t₀: eval_local_alloc_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_local_alloc_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₄ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_local_alloc_bb1_in: [1+X₄-X₀]
• eval_local_alloc_bb2_in: [X₄-X₀]
• eval_local_alloc_bb3_in: [X₄-X₀]
• eval_local_alloc_bb4_in: [X₄-X₀]
• eval_local_alloc_bb5_in: [X₄-X₀]

MPRF for transition t₄: eval_local_alloc_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb3_in(X₀, 0, X₂, X₃, X₄, X₅) :|: 1+X₅ ≤ X₂ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_local_alloc_bb1_in: [X₄-X₀]
• eval_local_alloc_bb2_in: [X₄-X₀]
• eval_local_alloc_bb3_in: [X₄-1-X₀]
• eval_local_alloc_bb4_in: [X₄-1-X₀]
• eval_local_alloc_bb5_in: [X₄-1-X₀]

MPRF for transition t₅: eval_local_alloc_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_local_alloc_bb1_in: [X₄-X₀]
• eval_local_alloc_bb2_in: [X₄-X₀]
• eval_local_alloc_bb3_in: [X₄-X₀]
• eval_local_alloc_bb4_in: [X₄-X₀]
• eval_local_alloc_bb5_in: [X₄-1-X₀]

MPRF for transition t₇: eval_local_alloc_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₅ ≤ X₁ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

MPRF for transition t₉: eval_local_alloc_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb1_in(1+X₀, X₁, X₂, X₃, X₄, 0) :|: 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

MPRF for transition t₆: eval_local_alloc_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₅ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₄+X₂ {O(n^2)}

MPRF:

• eval_local_alloc_bb1_in: [X₂]
• eval_local_alloc_bb2_in: [X₂]
• eval_local_alloc_bb3_in: [X₂-1-X₁]
• eval_local_alloc_bb4_in: [X₂-2-X₁]
• eval_local_alloc_bb5_in: [X₂]

MPRF for transition t₈: eval_local_alloc_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_local_alloc_bb3_in(X₀, 1+X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀+X₄ ∧ 1 ≤ X₀+X₅ ∧ 1+X₀ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₁+X₅ ∧ 1+X₁ ≤ X₅ ∧ 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₂ ∧ 2 ≤ X₄+X₅ ∧ 3 ≤ X₂+X₄ ∧ 3 ≤ X₂+X₅ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₄+X₂ {O(n^2)}

MPRF:

• eval_local_alloc_bb1_in: [X₂]
• eval_local_alloc_bb2_in: [X₂]
• eval_local_alloc_bb3_in: [X₂-1-X₁]
• eval_local_alloc_bb4_in: [X₂-1-X₁]
• eval_local_alloc_bb5_in: [X₂]

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 0 ≤ X₀ for location eval_local_alloc_bb2_in

Found invariant X₄ ≤ X₀ ∧ 0 ≤ X₀ for location eval_local_alloc_bb6_in

Found invariant 0 ≤ X₀ for location eval_local_alloc_bb1_in

Found invariant 1+X₅ ≤ X₂ ∧ 1 ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_local_alloc_bb3_in

Found invariant 1+X₅ ≤ X₂ ∧ 2 ≤ X₅ ∧ 3 ≤ X₄+X₅ ∧ 5 ≤ X₂+X₅ ∧ 3 ≤ X₁+X₅ ∧ 1+X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ 1 ≤ X₄ ∧ 4 ≤ X₂+X₄ ∧ 2 ≤ X₁+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 3 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_local_alloc_bb4_in_v2

Found invariant 1+X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₄+X₅ ∧ 3 ≤ X₂+X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ 2 ≤ X₁+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_local_alloc_bb3_in_v1

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 0 ≤ X₀ for location eval_local_alloc_bb5_in

Found invariant 1+X₅ ≤ X₂ ∧ 1 ≤ X₅ ∧ 2 ≤ X₄+X₅ ∧ 3 ≤ X₂+X₅ ∧ 1 ≤ X₁+X₅ ∧ 1+X₁ ≤ X₅ ∧ 1 ≤ X₀+X₅ ∧ 1 ≤ X₄ ∧ 3 ≤ X₂+X₄ ∧ 1 ≤ X₁+X₄ ∧ 1+X₁ ≤ X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₀ ≤ X₄ ∧ 2 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_local_alloc_bb4_in_v1

Found invariant X₄ ≤ X₀ ∧ 0 ≤ X₀ for location eval_local_alloc_stop

All Bounds

Timebounds

Overall timebound:2⋅X₂⋅X₄+2⋅X₂+5⋅X₄+5 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₄+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: X₄ {O(n)}
t₆: X₂⋅X₄+X₂ {O(n^2)}
t₇: X₄ {O(n)}
t₈: X₂⋅X₄+X₂ {O(n^2)}
t₉: X₄ {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂⋅X₄+2⋅X₂+5⋅X₄+5 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₄+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: X₄ {O(n)}
t₆: X₂⋅X₄+X₂ {O(n^2)}
t₇: X₄ {O(n)}
t₈: X₂⋅X₄+X₂ {O(n^2)}
t₉: X₄ {O(n)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: 0 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₃ {O(n)}
t₂, X₀: X₄ {O(n)}
t₂, X₁: X₁+X₂+1 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₃ {O(n)}
t₃, X₀: X₄ {O(n)}
t₃, X₁: 2⋅X₁+X₂+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: X₃ {O(n)}
t₄, X₀: X₄ {O(n)}
t₄, X₁: 0 {O(1)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₃ {O(n)}
t₅, X₀: X₄ {O(n)}
t₅, X₁: X₁+X₂+1 {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₃ {O(n)}
t₆, X₀: X₄ {O(n)}
t₆, X₁: X₂+1 {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₃ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: X₂+1 {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: 2⋅X₃ {O(n)}
t₈, X₀: X₄ {O(n)}
t₈, X₁: X₂+1 {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₃ {O(n)}
t₉, X₀: X₄ {O(n)}
t₉, X₁: X₁+X₂+1 {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: 0 {O(1)}
t₁₀, X₀: X₄ {O(n)}
t₁₀, X₁: 2⋅X₁+X₂+1 {O(n)}
t₁₀, X₂: 2⋅X₂ {O(n)}
t₁₀, X₃: 2⋅X₃ {O(n)}
t₁₀, X₄: 2⋅X₄ {O(n)}
t₁₀, X₅: X₃ {O(n)}