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(X₄, 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, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀ ∧ 1 ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₁ ≤ 0
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₁, X₂, X₂, X₃, X₄, X₅, X₆) :|: 1+X₂ ≤ X₁
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₁, X₂, X₂, X₃, X₄, X₅, X₆) :|: X₁ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₁ ∧ X₁ ≤ X₂
t₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂-X₁, X₃, 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 [X₃; X₅] that do not contribute to the problem

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

Found invariant X₁ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₄ for location eval_foo_stop

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

Found invariant X₁ ≤ X₄ for location eval_foo_bb4_in

Cut unsatisfiable transition [t₂₅: eval_foo_bb2_in→eval_foo_bb1_in]

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄)
t₂₁: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₀, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₄
t₂₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ 0 ∧ X₁ ≤ X₄
t₂₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₁ ≤ X₄
t₂₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₁, X₂, X₂, X₃, X₄) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₄
t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₄
t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂-X₁, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₄
t₂₈: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₄
t₂₉: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂₁: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₀, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

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

MPRF for transition t₂₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₁, X₂, X₂, X₃, X₄) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

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

MPRF for transition t₂₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₁ ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₂ ∧ 1 ≤ X₁+X₂ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₄ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₃+X₄+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁-1]
• eval_foo_bb2_in: [X₁+X₂-1]
• eval_foo_bb3_in: [X₁+X₂-2]

MPRF for transition t₂₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂-X₁, X₃, X₄) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₃+X₄+1 {O(n)}

MPRF:

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

All Bounds

Timebounds

Overall timebound:2⋅X₃+4⋅X₄+7 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: X₄ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: X₄ {O(n)}
t₂₆: X₃+X₄+1 {O(n)}
t₂₇: X₃+X₄+1 {O(n)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃+4⋅X₄+7 {O(n)}
t₂₀: 1 {O(1)}
t₂₁: X₄ {O(n)}
t₂₂: 1 {O(1)}
t₂₃: 1 {O(1)}
t₂₄: X₄ {O(n)}
t₂₆: X₃+X₄+1 {O(n)}
t₂₇: X₃+X₄+1 {O(n)}
t₂₈: 1 {O(1)}
t₂₉: 1 {O(1)}

Sizebounds

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