KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇
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, X₇) → eval_foo_bb1_in(X₅, X₆, X₂, X₃, X₄, X₅, X₆, X₇)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb2_in(X₀, X₁, X₀, X₁, X₄, X₅, X₆, X₇) :|: 1+X₇ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₀ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₁ ≤ X₀
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₀ ≤ X₇
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅, X₆, X₇) :|: X₃ ≤ X₇
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) :|: 1+X₇ ≤ X₃
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃-1, X₄, X₅, X₆, X₇)
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇)

Preprocessing

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

Found invariant 1+X₆ ≤ X₅ ∧ 1+X₆ ≤ X₄ ∧ X₆ ≤ X₃ ∧ X₆ ≤ X₂ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₄ ∧ X₂ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ 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+X₆ ≤ X₅ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ X₃ ∧ 1+X₆ ≤ X₂ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ X₃ ≤ X₅ ∧ X₂ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₄ ∧ X₂ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ 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, 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:
t₂₀: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅, X₆)
t₂₁: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₀, X₁, X₄, X₅, X₆) :|: 1+X₆ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₂: 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 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₃: 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 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ X₆ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ X₆ ∧ 1+X₆ ≤ X₀ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ X₃ ≤ X₂ ∧ X₆ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₆ ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅
t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₆ ≤ X₃ ∧ 1+X₆ ≤ X₀ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ X₃ ≤ X₂ ∧ X₆ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₆ ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅
t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃-1, X₄, X₅, X₆) :|: 1+X₆ ≤ X₀ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₂ ∧ 1+X₆ ≤ X₃ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ 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, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₉: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

knowledge_propagation leads to new time bound 1 {O(1)} for transition t₂₁: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₀, X₁, X₄, X₅, X₆) :|: 1+X₆ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅

MPRF for transition t₂₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅, X₆) :|: X₃ ≤ X₆ ∧ 1+X₆ ≤ X₀ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ X₃ ≤ X₂ ∧ X₆ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₆ ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁-X₆]
• eval_foo_bb2_in: [1]
• eval_foo_bb3_in: [1]

MPRF for transition t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₆ ≤ X₃ ∧ 1+X₆ ≤ X₀ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ X₅ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₁ ≤ X₅ ∧ X₃ ≤ X₂ ∧ X₆ ≤ X₂ ∧ X₂ ≤ X₃ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ ∧ X₆ ≤ X₃ ∧ X₃ ≤ X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₄+X₆+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀-X₆]
• eval_foo_bb2_in: [1+X₂-X₆]
• eval_foo_bb3_in: [X₃-X₆]

MPRF for transition t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃-1, X₄, X₅, X₆) :|: 1+X₆ ≤ X₀ ∧ 1+X₆ ≤ X₁ ∧ 1+X₆ ≤ X₂ ∧ 1+X₆ ≤ X₃ ∧ 1+X₆ ≤ X₄ ∧ 1+X₆ ≤ 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+X₆ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁-X₆]
• eval_foo_bb2_in: [X₃-X₆]
• eval_foo_bb3_in: [X₂-X₆]

All Bounds

Timebounds

Overall timebound:2⋅X₅+3⋅X₆+X₄+8 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: X₅+X₆ {O(n)}
t₂₆: X₄+X₆+1 {O(n)}
t₂₇: X₅+X₆ {O(n)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₅+3⋅X₆+X₄+8 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: 1 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: X₅+X₆ {O(n)}
t₂₆: X₄+X₆+1 {O(n)}
t₂₇: X₅+X₆ {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)}
t₂₀, X₆: X₆ {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)}
t₂₁, X₆: X₆ {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)}
t₂₂, X₆: X₆ {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)}
t₂₃, X₆: X₆ {O(n)}
t₂₄, X₀: 3⋅X₄+X₆+1 {O(n)}
t₂₄, X₁: 3⋅X₅+X₆ {O(n)}
t₂₄, X₂: 2⋅X₄+X₂+X₆+1 {O(n)}
t₂₄, X₃: 2⋅X₅+X₃+X₆ {O(n)}
t₂₄, X₄: 2⋅X₄ {O(n)}
t₂₄, X₅: 2⋅X₅ {O(n)}
t₂₄, X₆: 2⋅X₆ {O(n)}
t₂₅, X₀: 2⋅X₄+X₆+1 {O(n)}
t₂₅, X₁: 2⋅X₅+X₆ {O(n)}
t₂₅, X₂: 2⋅X₄+X₆+1 {O(n)}
t₂₅, X₃: 2⋅X₅+X₆ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₅, X₆: X₆ {O(n)}
t₂₆, X₀: X₄ {O(n)}
t₂₆, X₁: X₅ {O(n)}
t₂₆, X₂: X₄+X₅+X₆ {O(n)}
t₂₆, X₃: 2⋅X₅+X₆ {O(n)}
t₂₆, X₄: X₄ {O(n)}
t₂₆, X₅: X₅ {O(n)}
t₂₆, X₆: X₆ {O(n)}
t₂₇, X₀: X₄ {O(n)}
t₂₇, X₁: X₅ {O(n)}
t₂₇, X₂: 2⋅X₄+X₆+1 {O(n)}
t₂₇, X₃: 2⋅X₅+X₆ {O(n)}
t₂₇, X₄: X₄ {O(n)}
t₂₇, X₅: X₅ {O(n)}
t₂₇, X₆: X₆ {O(n)}
t₂₈, X₀: 5⋅X₄+X₆+1 {O(n)}
t₂₈, X₁: 5⋅X₅+X₆ {O(n)}
t₂₈, X₂: 2⋅X₄+3⋅X₂+X₆+1 {O(n)}
t₂₈, X₃: 2⋅X₅+3⋅X₃+X₆ {O(n)}
t₂₈, X₄: 4⋅X₄ {O(n)}
t₂₈, X₅: 4⋅X₅ {O(n)}
t₂₈, X₆: 4⋅X₆ {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)}
t₂₉, X₆: X₆ {O(n)}