KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈
Temp_Vars: nondef.0, nondef.1
Locations: eval_foo_2, eval_foo_3, eval_foo_6, eval_foo_7, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₆: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_3(X₀, X₁, nondef.0, X₃, X₄, X₅, X₆, X₇, X₈)
t₇: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_6(X₀, X₁, X₂, X₀+X₂, X₄, X₅, X₆, X₇, X₈) :|: 1+X₂ ≤ 0
t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_6(X₀, X₁, X₂, X₀-X₂, X₄, X₅, X₆, X₇, X₈) :|: 0 ≤ X₂
t₁₀: eval_foo_6(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_7(X₀, X₁, X₂, X₃, nondef.1, X₅, X₆, X₇, X₈)
t₁₁: eval_foo_7(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb1_in(X₃, 1+X₁-X₄, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 1+X₄ ≤ 0
t₁₂: eval_foo_7(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb1_in(X₃, 1+X₁+X₄, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 0 ≤ X₄
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb1_in(X₅, X₆, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: 1+X₁ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) :|: X₀ ≤ X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₁₃: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆, X₇, X₈)

Preprocessing

Eliminate variables [X₇; X₈] that do not contribute to the problem

Found invariant 1+X₆ ≤ X₅ ∧ X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ 1+X₁ ≤ X₀ for location eval_foo_2

Found invariant 1+X₆ ≤ X₅ ∧ X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ X₃ ≤ X₅ ∧ 1+X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location eval_foo_6

Found invariant 1+X₆ ≤ X₅ ∧ X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ X₃ ≤ X₅ ∧ 1+X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location eval_foo_7

Found invariant X₆ ≤ X₁ ∧ X₀ ≤ X₅ ∧ X₀ ≤ X₁ for location eval_foo_stop

Found invariant X₆ ≤ X₁ ∧ X₀ ≤ X₅ ∧ X₀ ≤ X₁ for location eval_foo_bb3_in

Found invariant 1+X₆ ≤ X₅ ∧ X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ 1+X₁ ≤ X₀ for location eval_foo_3

Found invariant 1+X₆ ≤ X₅ ∧ X₆ ≤ X₁ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ X₀ ≤ X₅ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₆ ≤ X₁ ∧ X₀ ≤ X₅ for location eval_foo_bb1_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅, X₆
Temp_Vars: nondef.0, nondef.1
Locations: eval_foo_2, eval_foo_3, eval_foo_6, eval_foo_7, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₂₅: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_3(X₀, X₁, nondef.0, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₂₆: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_6(X₀, X₁, X₂, X₀+X₂, X₄, X₅, X₆) :|: 1+X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₂₇: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_6(X₀, X₁, X₂, X₀-X₂, X₄, X₅, X₆) :|: 0 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₂₈: eval_foo_6(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_7(X₀, X₁, X₂, X₃, nondef.1, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ ∧ X₃ ≤ X₅
t₂₉: eval_foo_7(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₃, 1+X₁-X₄, X₂, X₃, X₄, X₅, X₆) :|: 1+X₄ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ ∧ X₃ ≤ X₅
t₃₀: eval_foo_7(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₃, 1+X₁+X₄, X₂, X₃, X₄, X₅, X₆) :|: 0 ≤ X₄ ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ ∧ X₃ ≤ X₅
t₃₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₅, X₆, X₂, X₃, X₄, X₅, X₆)
t₃₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₃₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ X₁ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₃₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₃₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: X₀ ≤ X₁ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁
t₃₆: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆)

MPRF for transition t₂₅: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_3(X₀, X₁, nondef.0, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_2: [X₀-X₁]
• eval_foo_3: [X₀-1-X₁]
• eval_foo_6: [X₀-1-X₁]
• eval_foo_7: [X₃-1-X₁]
• eval_foo_bb1_in: [X₀-X₁]
• eval_foo_bb2_in: [X₀-X₁]

MPRF for transition t₂₆: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_6(X₀, X₁, X₂, X₀+X₂, X₄, X₅, X₆) :|: 1+X₂ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_2: [X₀-X₆]
• eval_foo_3: [X₀-X₆]
• eval_foo_6: [X₃-X₆]
• eval_foo_7: [X₃-X₆]
• eval_foo_bb1_in: [X₀-X₆]
• eval_foo_bb2_in: [X₀-X₆]

MPRF for transition t₂₇: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_6(X₀, X₁, X₂, X₀-X₂, X₄, X₅, X₆) :|: 0 ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_2: [X₅-X₁]
• eval_foo_3: [X₅-X₁]
• eval_foo_6: [X₅-1-X₁]
• eval_foo_7: [X₅-1-X₁]
• eval_foo_bb1_in: [X₅-X₁]
• eval_foo_bb2_in: [X₅-X₁]

MPRF for transition t₂₈: eval_foo_6(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_7(X₀, X₁, X₂, X₃, nondef.1, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_2: [X₀-X₁]
• eval_foo_3: [X₀-X₁]
• eval_foo_6: [X₀-X₁]
• eval_foo_7: [X₃-1-X₁]
• eval_foo_bb1_in: [X₀-X₁]
• eval_foo_bb2_in: [X₀-X₁]

MPRF for transition t₂₉: eval_foo_7(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₃, 1+X₁-X₄, X₂, X₃, X₄, X₅, X₆) :|: 1+X₄ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_2: [X₅-X₁]
• eval_foo_3: [X₅-X₁]
• eval_foo_6: [X₅-X₁]
• eval_foo_7: [X₅-X₁]
• eval_foo_bb1_in: [X₅-X₁]
• eval_foo_bb2_in: [X₅-X₁]

MPRF for transition t₃₀: eval_foo_7(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb1_in(X₃, 1+X₁+X₄, X₂, X₃, X₄, X₅, X₆) :|: 0 ≤ X₄ ∧ 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ ∧ X₃ ≤ X₅ of depth 1:

new bound:

X₅+X₆+1 {O(n)}

MPRF:

• eval_foo_2: [1+X₅-X₁]
• eval_foo_3: [1+X₅-X₁]
• eval_foo_6: [1+X₅-X₁]
• eval_foo_7: [1+X₅-X₁]
• eval_foo_bb1_in: [1+X₅-X₁]
• eval_foo_bb2_in: [1+X₅-X₁]

MPRF for transition t₃₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ of depth 1:

new bound:

X₅+X₆+2 {O(n)}

MPRF:

• eval_foo_2: [1+X₀-X₁]
• eval_foo_3: [1+X₀-X₁]
• eval_foo_6: [1+X₀-X₁]
• eval_foo_7: [1+X₀-X₁]
• eval_foo_bb1_in: [2+X₀-X₁]
• eval_foo_bb2_in: [1+X₀-X₁]

MPRF for transition t₃₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅, X₆) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅, X₆) :|: 1+X₁ ≤ X₀ ∧ 1+X₆ ≤ X₀ ∧ 1+X₁ ≤ X₅ ∧ 1+X₆ ≤ X₅ ∧ X₀ ≤ X₅ ∧ X₆ ≤ X₁ of depth 1:

new bound:

X₅+X₆ {O(n)}

MPRF:

• eval_foo_2: [X₀-1-X₁]
• eval_foo_3: [X₀-1-X₁]
• eval_foo_6: [X₀-1-X₁]
• eval_foo_7: [X₃-1-X₁]
• eval_foo_bb1_in: [X₀-X₁]
• eval_foo_bb2_in: [X₀-X₁]

All Bounds

Timebounds

Overall timebound:8⋅X₅+8⋅X₆+7 {O(n)}
t₂₅: X₅+X₆ {O(n)}
t₂₆: X₅+X₆ {O(n)}
t₂₇: X₅+X₆ {O(n)}
t₂₈: X₅+X₆ {O(n)}
t₂₉: X₅+X₆ {O(n)}
t₃₀: X₅+X₆+1 {O(n)}
t₃₁: 1 {O(1)}
t₃₂: X₅+X₆+2 {O(n)}
t₃₃: 1 {O(1)}
t₃₄: X₅+X₆ {O(n)}
t₃₅: 1 {O(1)}
t₃₆: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₅+8⋅X₆+7 {O(n)}
t₂₅: X₅+X₆ {O(n)}
t₂₆: X₅+X₆ {O(n)}
t₂₇: X₅+X₆ {O(n)}
t₂₈: X₅+X₆ {O(n)}
t₂₉: X₅+X₆ {O(n)}
t₃₀: X₅+X₆+1 {O(n)}
t₃₁: 1 {O(1)}
t₃₂: X₅+X₆+2 {O(n)}
t₃₃: 1 {O(1)}
t₃₄: X₅+X₆ {O(n)}
t₃₅: 1 {O(1)}
t₃₆: 1 {O(1)}

Sizebounds

t₂₅, X₅: X₅ {O(n)}
t₂₅, X₆: X₆ {O(n)}
t₂₆, X₅: X₅ {O(n)}
t₂₆, X₆: X₆ {O(n)}
t₂₇, X₅: X₅ {O(n)}
t₂₇, X₆: X₆ {O(n)}
t₂₈, X₅: X₅ {O(n)}
t₂₈, X₆: X₆ {O(n)}
t₂₉, X₅: X₅ {O(n)}
t₂₉, X₆: X₆ {O(n)}
t₃₀, X₅: X₅ {O(n)}
t₃₀, X₆: X₆ {O(n)}
t₃₁, X₀: X₅ {O(n)}
t₃₁, X₁: X₆ {O(n)}
t₃₁, X₂: X₂ {O(n)}
t₃₁, X₃: X₃ {O(n)}
t₃₁, X₄: X₄ {O(n)}
t₃₁, X₅: X₅ {O(n)}
t₃₁, X₆: X₆ {O(n)}
t₃₂, X₅: X₅ {O(n)}
t₃₂, X₆: X₆ {O(n)}
t₃₃, X₅: 3⋅X₅ {O(n)}
t₃₃, X₆: 3⋅X₆ {O(n)}
t₃₄, X₅: X₅ {O(n)}
t₃₄, X₆: X₆ {O(n)}
t₃₅, X₅: 3⋅X₅ {O(n)}
t₃₅, X₆: 3⋅X₆ {O(n)}
t₃₆, X₀: X₀ {O(n)}
t₃₆, X₁: X₁ {O(n)}
t₃₆, X₂: X₂ {O(n)}
t₃₆, X₃: X₃ {O(n)}
t₃₆, X₄: X₄ {O(n)}
t₃₆, X₅: X₅ {O(n)}
t₃₆, X₆: X₆ {O(n)}