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_bb5_in, eval_foo_bb6_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, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1 ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0 ∧ 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 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ 0
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb5_in(X₀, X₁, X₀-1, X₁, X₄, X₅, X₆)
t₈: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb5_in(X₀, X₁, X₀, X₁-1, X₄, X₅, X₆) :|: 1 ≤ X₁
t₉: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb5_in(X₀, X₁, X₀, X₁, X₄, X₅, X₆) :|: X₁ ≤ 0
t₁₀: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅, X₆)
t₁₁: eval_foo_bb6_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 X₃ ≤ X₅ ∧ X₁ ≤ X₅ ∧ X₂ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ 1+X₂ for location eval_foo_bb5_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ ∧ X₁ ≤ 0 ∧ X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₁ ≤ X₅ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ ∧ X₁ ≤ 0 ∧ X₀+X₁ ≤ 0 ∧ X₀ ≤ 0 for location eval_foo_bb6_in

Found invariant X₁ ≤ X₅ ∧ X₀ ≤ X₄ ∧ X₀ ≤ 0 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_bb6_in, eval_foo_start, eval_foo_stop
Transitions:
t₂₃: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₁ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₂₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀-1, X₁, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₃₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀, X₁-1, X₄, X₅) :|: 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₃₁: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀, X₁, X₄, X₅) :|: X₁ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅
t₃₂: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₅ ∧ X₂ ≤ X₄ ∧ X₃ ≤ X₅
t₃₃: eval_foo_bb6_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀+X₁ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₅
t₃₄: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀-1]
• eval_foo_bb4_in: [X₀]
• eval_foo_bb5_in: [X₂]

MPRF for transition t₂₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀-1, X₁, X₄, X₅) :|: 1 ≤ X₀ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀]
• eval_foo_bb5_in: [X₂]

MPRF for transition t₃₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀, X₁-1, X₄, X₅) :|: 1 ≤ X₁ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]
• eval_foo_bb5_in: [X₃]

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

new bound:

X₄+X₅+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1]
• eval_foo_bb2_in: [1-X₁]
• eval_foo_bb3_in: [1]
• eval_foo_bb4_in: [1-X₁]
• eval_foo_bb5_in: [1]

MPRF for transition t₂₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

X₄⋅X₅+X₅⋅X₅+X₅ {O(n^2)}

MPRF:

• eval_foo_bb1_in: [X₅]
• eval_foo_bb2_in: [1+X₅-X₁]
• eval_foo_bb3_in: [X₅]
• eval_foo_bb4_in: [X₅-X₁]
• eval_foo_bb5_in: [X₅]

MPRF for transition t₃₁: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₀, X₁, X₄, X₅) :|: X₁ ≤ 0 ∧ X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅ of depth 1:

new bound:

0 {O(1)}

MPRF:

• eval_foo_bb1_in: [0]
• eval_foo_bb2_in: [1-X₁]
• eval_foo_bb3_in: [0]
• eval_foo_bb4_in: [1-X₁]
• eval_foo_bb5_in: [0]

MPRF for transition t₃₂: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ X₁ ≤ 1+X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₅ ∧ X₂ ≤ X₄ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₄+X₅ {O(n)}

MPRF:

• eval_foo_bb1_in: [0]
• eval_foo_bb2_in: [1-X₁]
• eval_foo_bb3_in: [1]
• eval_foo_bb4_in: [1-X₁]
• eval_foo_bb5_in: [1]

knowledge_propagation leads to new time bound X₄+X₅+1 {O(n)} for transition t₂₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅

knowledge_propagation leads to new time bound X₄+X₅+1 {O(n)} for transition t₂₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₅

All Bounds

Timebounds

Overall timebound:5⋅X₅+6⋅X₄+7 {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₄+X₅+1 {O(n)}
t₂₅: X₄+X₅+1 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₄ {O(n)}
t₂₈: X₄+X₅+1 {O(n)}
t₂₉: X₄ {O(n)}
t₃₀: X₅ {O(n)}
t₃₁: 0 {O(1)}
t₃₂: X₄+X₅ {O(n)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₅+6⋅X₄+7 {O(n)}
t₂₃: 1 {O(1)}
t₂₄: X₄+X₅+1 {O(n)}
t₂₅: X₄+X₅+1 {O(n)}
t₂₆: 1 {O(1)}
t₂₇: X₄ {O(n)}
t₂₈: X₄+X₅+1 {O(n)}
t₂₉: X₄ {O(n)}
t₃₀: X₅ {O(n)}
t₃₁: 0 {O(1)}
t₃₂: X₄+X₅ {O(n)}
t₃₃: 1 {O(1)}
t₃₄: 1 {O(1)}

Sizebounds

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: 2⋅X₄+X₅ {O(n)}
t₂₄, X₁: 2⋅X₅+X₄ {O(n)}
t₂₄, X₂: 3⋅X₅+6⋅X₄+X₂ {O(n)}
t₂₄, X₃: 3⋅X₄+6⋅X₅+X₃ {O(n)}
t₂₄, X₄: X₄ {O(n)}
t₂₄, X₅: X₅ {O(n)}
t₂₅, X₀: 2⋅X₄+X₅ {O(n)}
t₂₅, X₁: 2⋅X₅+X₄ {O(n)}
t₂₅, X₂: 3⋅X₅+6⋅X₄+X₂ {O(n)}
t₂₅, X₃: 3⋅X₄+6⋅X₅+X₃ {O(n)}
t₂₅, X₄: X₄ {O(n)}
t₂₅, X₅: X₅ {O(n)}
t₂₆, X₀: 3⋅X₄+X₅ {O(n)}
t₂₆, X₁: 3⋅X₅+X₄ {O(n)}
t₂₆, X₂: 3⋅X₅+6⋅X₄+X₂ {O(n)}
t₂₆, X₃: 3⋅X₄+6⋅X₅+X₃ {O(n)}
t₂₆, X₄: 2⋅X₄ {O(n)}
t₂₆, X₅: 2⋅X₅ {O(n)}
t₂₇, X₀: 2⋅X₄+X₅ {O(n)}
t₂₇, X₁: 2⋅X₅+X₄ {O(n)}
t₂₇, X₂: 12⋅X₄+2⋅X₂+6⋅X₅ {O(n)}
t₂₇, X₃: 12⋅X₅+2⋅X₃+6⋅X₄ {O(n)}
t₂₇, X₄: X₄ {O(n)}
t₂₇, X₅: X₅ {O(n)}
t₂₈, X₀: 2⋅X₄+X₅ {O(n)}
t₂₈, X₁: 2⋅X₅+X₄ {O(n)}
t₂₈, X₂: 3⋅X₅+6⋅X₄+X₂ {O(n)}
t₂₈, X₃: 3⋅X₄+6⋅X₅+X₃ {O(n)}
t₂₈, X₄: X₄ {O(n)}
t₂₈, X₅: X₅ {O(n)}
t₂₉, X₀: 2⋅X₄+X₅ {O(n)}
t₂₉, X₁: 2⋅X₅+X₄ {O(n)}
t₂₉, X₂: 2⋅X₄+X₅ {O(n)}
t₂₉, X₃: 2⋅X₅+X₄ {O(n)}
t₂₉, X₄: X₄ {O(n)}
t₂₉, X₅: X₅ {O(n)}
t₃₀, X₀: 2⋅X₄+X₅ {O(n)}
t₃₀, X₁: 2⋅X₅+X₄ {O(n)}
t₃₀, X₂: 2⋅X₄+X₅ {O(n)}
t₃₀, X₃: 2⋅X₅+X₄ {O(n)}
t₃₀, X₄: X₄ {O(n)}
t₃₀, X₅: X₅ {O(n)}
t₃₁, X₀: 2⋅X₄+X₅ {O(n)}
t₃₁, X₁: 2⋅X₅+X₄ {O(n)}
t₃₁, X₂: 2⋅X₄+X₅ {O(n)}
t₃₁, X₃: 2⋅X₅+X₄ {O(n)}
t₃₁, X₄: X₄ {O(n)}
t₃₁, X₅: X₅ {O(n)}
t₃₂, X₀: 2⋅X₄+X₅ {O(n)}
t₃₂, X₁: 2⋅X₅+X₄ {O(n)}
t₃₂, X₂: 3⋅X₅+6⋅X₄ {O(n)}
t₃₂, X₃: 3⋅X₄+6⋅X₅ {O(n)}
t₃₂, X₄: X₄ {O(n)}
t₃₂, X₅: X₅ {O(n)}
t₃₃, X₀: 3⋅X₄+X₅ {O(n)}
t₃₃, X₁: 3⋅X₅+X₄ {O(n)}
t₃₃, X₂: 3⋅X₅+6⋅X₄+X₂ {O(n)}
t₃₃, X₃: 3⋅X₄+6⋅X₅+X₃ {O(n)}
t₃₃, X₄: 2⋅X₄ {O(n)}
t₃₃, X₅: 2⋅X₅ {O(n)}
t₃₄, X₀: X₀ {O(n)}
t₃₄, X₁: X₁ {O(n)}
t₃₄, X₂: X₂ {O(n)}
t₃₄, X₃: X₃ {O(n)}
t₃₄, X₄: X₄ {O(n)}
t₃₄, X₅: X₅ {O(n)}