KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₄+X₅ ∧ 1 ≤ X₃+X₅ ∧ X₃ ≤ X₅ ∧ 1 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ for location eval_foo_stop

Found invariant 1 ≤ X₅ ∧ 2 ≤ X₄+X₅ ∧ 2 ≤ X₃+X₅ ∧ X₃ ≤ X₅ ∧ 2 ≤ X₂+X₅ ∧ X₂ ≤ X₅ ∧ 2 ≤ X₁+X₅ ∧ X₁ ≤ X₅ ∧ 2 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₄ ∧ 2 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 1 ≤ X₃ ∧ 2 ≤ X₂+X₃ ∧ X₂ ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₀, X₁, X₄, X₅) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₃ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₂+X₅ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₃+X₅ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₁+X₅ ∧ 2 ≤ X₄+X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₂+X₅ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₃+X₅ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₁+X₅ ∧ 2 ≤ X₄+X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃-1, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₁+X₅ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₂+X₅ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₃+X₅ ∧ 2 ≤ X₄+X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₀, X₁, X₄, X₅) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅

MPRF for transition t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₂+X₅ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₃+X₅ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₁+X₅ ∧ 2 ≤ X₄+X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [1+X₃]
• eval_foo_bb3_in: [X₂]

MPRF for transition t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₃ ≤ 0 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₂+X₅ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₃+X₅ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₁+X₅ ∧ 2 ≤ X₄+X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

2⋅X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [2⋅X₁]
• eval_foo_bb2_in: [1+X₃]
• eval_foo_bb3_in: [X₂]

MPRF for transition t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃-1, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 1 ≤ X₄ ∧ 1 ≤ X₅ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₃ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₀+X₅ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₁+X₅ ∧ 2 ≤ X₂+X₃ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₂+X₅ ∧ 2 ≤ X₃+X₄ ∧ 2 ≤ X₃+X₅ ∧ 2 ≤ X₄+X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₃]
• eval_foo_bb3_in: [X₂]

All Bounds

Timebounds

Overall timebound:4⋅X₅+8 {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: X₅+1 {O(n)}
t₇: 2⋅X₅ {O(n)}
t₈: X₅ {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₅+8 {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: X₅+1 {O(n)}
t₇: 2⋅X₅ {O(n)}
t₈: X₅ {O(n)}
t₉: 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: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: X₄ {O(n)}
t₂, X₁: X₅ {O(n)}
t₂, X₂: X₄ {O(n)}
t₂, X₃: X₅ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₄ {O(n)}
t₃, X₁: X₅ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: X₄ {O(n)}
t₄, X₁: X₅ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: X₄ {O(n)}
t₅, X₁: X₅ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}
t₆, X₀: X₄ {O(n)}
t₆, X₁: X₅ {O(n)}
t₆, X₂: X₄ {O(n)}
t₆, X₃: X₅ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₇, X₀: 0 {O(1)}
t₇, X₁: 0 {O(1)}
t₇, X₂: 0 {O(1)}
t₇, X₃: 0 {O(1)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: X₄ {O(n)}
t₈, X₁: X₅ {O(n)}
t₈, X₂: X₄ {O(n)}
t₈, X₃: X₅ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₉, X₀: 3⋅X₄ {O(n)}
t₉, X₁: 3⋅X₅ {O(n)}
t₉, X₂: 3⋅X₂ {O(n)}
t₉, X₃: 3⋅X₃ {O(n)}
t₉, X₄: 4⋅X₄ {O(n)}
t₉, X₅: 4⋅X₅ {O(n)}