KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
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, X₆) → eval_foo_bb1_in(X₅, X₆, X₂, X₃, X₄, X₅, X₆)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₀, X₁, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₀ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ 0
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₃
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃-1, X₄, X₅, X₆)
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

Preprocessing

Eliminate variables [X₄] that do not contribute to the problem

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:
t₂₀: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂₁: 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
t₂₂: 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
t₂₃: 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
t₂₄: 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
t₂₅: 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
t₂₆: 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
t₂₇: 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
t₂₈: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₉: 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 t₂₁: 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 t₂₅: 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:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁]
• eval_foo_bb2_in: [1+X₃]
• eval_foo_bb3_in: [X₂]

MPRF for transition t₂₆: 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:

X₄+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₂]
• eval_foo_bb3_in: [X₃]

MPRF for transition t₂₇: 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:

X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₂]
• eval_foo_bb3_in: [X₂]

All Bounds

Timebounds

Overall timebound:3⋅X₄+X₅+8 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: X₄+X₅ {O(n)}
t₂₆: X₄+1 {O(n)}
t₂₇: X₄ {O(n)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₄+X₅+8 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: X₄+X₅ {O(n)}
t₂₆: X₄+1 {O(n)}
t₂₇: X₄ {O(n)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}

Sizebounds

t₂₀, X₀: X₄ {O(n)}
t₂₀, X₁: X₅ {O(n)}
t₂₀, X₂: X₂ {O(n)}
t₂₀, X₃: X₃ {O(n)}
t₂₀, X₄: X₄ {O(n)}
t₂₀, X₅: X₅ {O(n)}
t₂₁, X₀: X₄ {O(n)}
t₂₁, X₁: X₅ {O(n)}
t₂₁, X₂: X₄ {O(n)}
t₂₁, X₃: X₅ {O(n)}
t₂₁, X₄: X₄ {O(n)}
t₂₁, X₅: X₅ {O(n)}
t₂₂, X₀: X₄ {O(n)}
t₂₂, X₁: X₅ {O(n)}
t₂₂, X₂: X₂ {O(n)}
t₂₂, X₃: X₃ {O(n)}
t₂₂, X₄: X₄ {O(n)}
t₂₂, X₅: X₅ {O(n)}
t₂₃, X₀: X₄ {O(n)}
t₂₃, X₁: X₅ {O(n)}
t₂₃, X₂: X₂ {O(n)}
t₂₃, X₃: X₃ {O(n)}
t₂₃, X₄: X₄ {O(n)}
t₂₃, X₅: X₅ {O(n)}
t₂₄, X₀: X₄ {O(n)}
t₂₄, X₁: X₅ {O(n)}
t₂₄, X₂: X₂ {O(n)}
t₂₄, X₃: X₃ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₄, X₅: 2⋅X₅ {O(n)}
t₂₅, X₀: 0 {O(1)}
t₂₅, X₁: 0 {O(1)}
t₂₅, X₂: 0 {O(1)}
t₂₅, X₃: 0 {O(1)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₆, X₀: X₄ {O(n)}
t₂₆, X₁: X₅ {O(n)}
t₂₆, X₂: X₄ {O(n)}
t₂₆, X₃: X₅ {O(n)}
t₂₆, X₄: X₄ {O(n)}
t₂₆, X₅: X₅ {O(n)}
t₂₇, X₀: X₄ {O(n)}
t₂₇, X₁: X₅ {O(n)}
t₂₇, X₂: X₄ {O(n)}
t₂₇, X₃: X₅ {O(n)}
t₂₇, X₄: X₄ {O(n)}
t₂₇, X₅: X₅ {O(n)}
t₂₈, X₀: 3⋅X₄ {O(n)}
t₂₈, X₁: 3⋅X₅ {O(n)}
t₂₈, X₂: 3⋅X₂ {O(n)}
t₂₈, X₃: 3⋅X₃ {O(n)}
t₂₈, X₄: 4⋅X₄ {O(n)}
t₂₈, X₅: 4⋅X₅ {O(n)}
t₂₉, X₀: X₀ {O(n)}
t₂₉, X₁: X₁ {O(n)}
t₂₉, X₂: X₂ {O(n)}
t₂₉, X₃: X₃ {O(n)}
t₂₉, X₄: X₄ {O(n)}
t₂₉, X₅: X₅ {O(n)}