KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₅ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location eval_foo_bb5_in

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

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₅ ∧ X₁ ≤ X₄ ∧ X₀ ≤ X₃ for location eval_foo_stop

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

Found invariant 1+X₅ ≤ X₃ ∧ 1+X₅ ≤ X₀ ∧ X₂ ≤ X₅ ∧ X₂ ≤ X₄ ∧ X₁ ≤ X₄ ∧ 1+X₂ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ 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_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₃, X₄, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: 1+X₅ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₅ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₅ ≤ X₂ ∧ 1+X₅ ≤ X₀ ∧ 1+X₅ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ 1+X₅ ≤ X₀ ∧ 1+X₅ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂-1, X₃, X₄, X₅) :|: 1+X₅ ≤ X₀ ∧ 1+X₅ ≤ X₁ ∧ 1+X₅ ≤ X₂ ∧ 1+X₅ ≤ X₃ ∧ 1+X₅ ≤ X₄ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₀ ∧ 1+X₅ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₅ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅
t₈: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₃ ∧ X₀ ≤ X₅ ∧ X₁ ≤ X₄
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₁, X₃, X₄, X₅) :|: 1+X₅ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₄ of depth 1:

new bound:

X₃+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-X₅]
• eval_foo_bb2_in: [X₀-1-X₅]
• eval_foo_bb3_in: [X₀-1-X₅]
• eval_foo_bb4_in: [X₀-1-X₅]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₅ ≤ X₂ ∧ 1+X₅ ≤ X₀ ∧ 1+X₅ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁-X₅]
• eval_foo_bb2_in: [X₂-X₅]
• eval_foo_bb3_in: [X₂-1-X₅]
• eval_foo_bb4_in: [X₂-X₅]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₂ ≤ X₅ ∧ 1+X₅ ≤ X₀ ∧ 1+X₅ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ of depth 1:

new bound:

X₃+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-X₅]
• eval_foo_bb2_in: [X₀-X₅]
• eval_foo_bb3_in: [X₀-X₅]
• eval_foo_bb4_in: [X₀-1-X₅]

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

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁-X₅]
• eval_foo_bb2_in: [X₂-X₅]
• eval_foo_bb3_in: [X₂-X₅]
• eval_foo_bb4_in: [X₂-X₅]

MPRF for transition t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀-1, X₂, X₂, X₃, X₄, X₅) :|: 1+X₂ ≤ X₀ ∧ 1+X₅ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₅ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₂ ≤ X₄ ∧ X₂ ≤ X₅ of depth 1:

new bound:

X₃+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-X₅]
• eval_foo_bb2_in: [X₀-X₅]
• eval_foo_bb3_in: [X₀-X₅]
• eval_foo_bb4_in: [X₀-X₅]

All Bounds

Timebounds

Overall timebound:2⋅X₄+3⋅X₃+5⋅X₅+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+X₅ {O(n)}
t₃: 1 {O(1)}
t₄: X₄+X₅ {O(n)}
t₅: X₃+X₅ {O(n)}
t₆: X₄+X₅ {O(n)}
t₇: X₃+X₅ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₄+3⋅X₃+5⋅X₅+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+X₅ {O(n)}
t₃: 1 {O(1)}
t₄: X₄+X₅ {O(n)}
t₅: X₃+X₅ {O(n)}
t₆: X₄+X₅ {O(n)}
t₇: X₃+X₅ {O(n)}
t₈: 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: X₃ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: 2⋅X₃+X₅ {O(n)}
t₂, X₁: 3⋅X₄+X₅ {O(n)}
t₂, X₂: 2⋅X₄+X₅ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 3⋅X₃+X₅ {O(n)}
t₃, X₁: 3⋅X₄+X₅ {O(n)}
t₃, X₂: 2⋅X₄+X₂+X₅ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₄, X₀: 2⋅X₃+X₅ {O(n)}
t₄, X₁: 3⋅X₄+X₅ {O(n)}
t₄, X₂: 2⋅X₄+X₅ {O(n)}
t₄, X₃: X₃ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₀: 2⋅X₃+X₅ {O(n)}
t₅, X₁: 2⋅X₅+6⋅X₄ {O(n)}
t₅, X₂: 2⋅X₄+X₅ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₀: 2⋅X₃+X₅ {O(n)}
t₆, X₁: 3⋅X₄+X₅ {O(n)}
t₆, X₂: 2⋅X₄+X₅ {O(n)}
t₆, X₃: X₃ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₇, X₀: 2⋅X₃+X₅ {O(n)}
t₇, X₁: 2⋅X₄+X₅ {O(n)}
t₇, X₂: 2⋅X₄+X₅ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₈, X₀: 3⋅X₃+X₅ {O(n)}
t₈, X₁: 3⋅X₄+X₅ {O(n)}
t₈, X₂: 2⋅X₄+X₂+X₅ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₈, X₄: 2⋅X₄ {O(n)}
t₈, X₅: 2⋅X₅ {O(n)}