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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₅, X₃, X₄, X₃, X₄, X₅, X₆)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₁ ≤ 100 ∧ X₂ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 101 ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₀ ≤ X₂
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₀-1, X₂, 1+X₁, X₃, X₄, X₅, X₆)
t₆: eval_foo_bb3_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 [X₆] that do not contribute to the problem

Found invariant X₂ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₀ ∧ X₁ ≤ 100 for location eval_foo_bb2_in

Found invariant X₀ ≤ X₅ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₅ for location eval_foo_stop

Found invariant X₀ ≤ X₅ for location eval_foo_bb3_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_start, eval_foo_stop
Transitions:
t₁₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₅, X₃, X₄, X₃, X₄, X₅)
t₁₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 100 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₅
t₁₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 101 ≤ X₁ ∧ X₀ ≤ X₅
t₁₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ X₂ ∧ X₀ ≤ X₅
t₁₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₂, 1+X₁, X₃, X₄, X₅) :|: X₁ ≤ 100 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₅
t₁₈: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₅
t₁₉: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₁₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 100 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₃+X₄+X₅+101 {O(n)}

MPRF:

• eval_foo_bb1_in: [101+X₀-X₁-X₂]
• eval_foo_bb2_in: [100+X₀-X₁-X₂]

MPRF for transition t₁₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₂, 1+X₁, X₃, X₄, X₅) :|: X₁ ≤ 100 ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₂ ≤ X₅ of depth 1:

new bound:

X₃+X₄+X₅+102 {O(n)}

MPRF:

• eval_foo_bb1_in: [102+X₀-X₁-X₂]
• eval_foo_bb2_in: [101+X₀-X₁-X₂]

All Bounds

Timebounds

Overall timebound:2⋅X₃+2⋅X₄+2⋅X₅+208 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃+X₄+X₅+101 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: X₃+X₄+X₅+102 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₃+2⋅X₄+2⋅X₅+208 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₃+X₄+X₅+101 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}
t₁₇: X₃+X₄+X₅+102 {O(n)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: X₅ {O(n)}
t₁₃, X₁: X₃ {O(n)}
t₁₃, X₂: X₄ {O(n)}
t₁₃, X₃: X₃ {O(n)}
t₁₃, X₄: X₄ {O(n)}
t₁₃, X₅: X₅ {O(n)}
t₁₄, X₀: 2⋅X₅+X₃+X₄+102 {O(n)}
t₁₄, X₁: 3⋅X₃+3⋅X₄+X₅+104 {O(n)}
t₁₄, X₂: 3⋅X₃+3⋅X₄+X₅+104 {O(n)}
t₁₄, X₃: X₃ {O(n)}
t₁₄, X₄: X₄ {O(n)}
t₁₄, X₅: X₅ {O(n)}
t₁₅, X₀: 4⋅X₅+X₃+X₄+2 {O(n)}
t₁₅, X₁: 2⋅X₅+3⋅X₄+4⋅X₃+4 {O(n)}
t₁₅, X₂: 2⋅X₅+3⋅X₃+4⋅X₄+4 {O(n)}
t₁₅, X₃: 2⋅X₃ {O(n)}
t₁₅, X₄: 2⋅X₄ {O(n)}
t₁₅, X₅: 2⋅X₅ {O(n)}
t₁₆, X₀: 4⋅X₅+X₃+X₄+2 {O(n)}
t₁₆, X₁: 2⋅X₅+3⋅X₄+4⋅X₃+4 {O(n)}
t₁₆, X₂: 2⋅X₅+3⋅X₃+4⋅X₄+4 {O(n)}
t₁₆, X₃: 2⋅X₃ {O(n)}
t₁₆, X₄: 2⋅X₄ {O(n)}
t₁₆, X₅: 2⋅X₅ {O(n)}
t₁₇, X₀: 3⋅X₅+X₃+X₄+2 {O(n)}
t₁₇, X₁: 2⋅X₅+3⋅X₃+3⋅X₄+4 {O(n)}
t₁₇, X₂: 2⋅X₅+3⋅X₃+3⋅X₄+4 {O(n)}
t₁₇, X₃: X₃ {O(n)}
t₁₇, X₄: X₄ {O(n)}
t₁₇, X₅: X₅ {O(n)}
t₁₈, X₀: 2⋅X₃+2⋅X₄+8⋅X₅+4 {O(n)}
t₁₈, X₁: 4⋅X₅+6⋅X₄+8⋅X₃+8 {O(n)}
t₁₈, X₂: 4⋅X₅+6⋅X₃+8⋅X₄+8 {O(n)}
t₁₈, X₃: 4⋅X₃ {O(n)}
t₁₈, X₄: 4⋅X₄ {O(n)}
t₁₈, X₅: 4⋅X₅ {O(n)}
t₁₉, X₀: X₀ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₁₉, X₃: X₃ {O(n)}
t₁₉, X₄: X₄ {O(n)}
t₁₉, X₅: X₅ {O(n)}