KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₄ ∧ 1+X₁ ≤ 0 for location eval_foo_bb5_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_2

Found invariant X₀ ≤ X₄ ∧ 1+X₁ ≤ 0 for location eval_foo_stop

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ X₀ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 1 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_3

Found invariant X₀ ≤ X₄ ∧ 0 ≤ X₁ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₄ for location eval_foo_bb1_in

Found invariant 1+X₂ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀+X₂ ≤ 0 ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ 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: nondef.0
Locations: eval_foo_2, eval_foo_3, 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:
t₈: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_3(X₀, X₁, X₂, nondef.0, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂
t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₄, X₅, X₂, X₃, X₄, X₅)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 0 ≤ X₁ ∧ X₀ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ 0 ∧ X₀ ≤ X₄
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂
t₁₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₁-1, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀+X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ 0 ∧ 1+X₂ ≤ X₄ ∧ X₀ ≤ 0 ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁
t₁₁: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ 0 ∧ X₀ ≤ X₄
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_2: [X₂]
• eval_foo_3: [X₀-1]
• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀-1]
• eval_foo_bb4_in: [X₀]

MPRF for transition t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_2: [X₂]
• eval_foo_3: [X₂]
• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [1+X₂]
• eval_foo_bb4_in: [X₀]

MPRF for transition t₈: eval_foo_2(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_3(X₀, X₁, X₂, nondef.0, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_2: [1+X₂]
• eval_foo_3: [X₀-1]
• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀]

MPRF for transition t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1+X₂ ≤ X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_foo_2: [1+X₂]
• eval_foo_3: [1+X₂]
• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀]

Cut unreachable locations [eval_foo_bb2_in] from the program graph

Found invariant X₀ ≤ X₄ ∧ 1+X₁ ≤ 0 for location eval_foo_bb5_in

Found invariant 2+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 2+X₂ ≤ 0 ∧ 2+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 3+X₀+X₂ ≤ 0 ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb4_in_v2

Found invariant 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ 0 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2+X₀+X₂ ≤ 0 ∧ X₀ ≤ X₂ ∧ 0 ≤ 1+X₁ ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb1_in_v1

Found invariant X₀ ≤ X₄ ∧ 1+X₁ ≤ 0 for location eval_foo_stop

Found invariant 1+X₂ ≤ X₄ ∧ 1+X₀ ≤ X₄ ∧ 1+X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 2+X₀+X₂ ≤ 0 ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb2_in_v2

Found invariant X₀ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₄ ∧ 0 ≤ X₁ for location eval_foo_bb2_in_v1

Found invariant 1+X₂ ≤ X₄ ∧ X₀ ≤ X₄ ∧ 1+X₂ ≤ 0 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₀+X₂ ≤ 0 ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 0 for location eval_foo_bb4_in_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: inf {Infinity}
t₆: X₄ {O(n)}
t₈: X₄ {O(n)}
t₉: X₄ {O(n)}
t₁₀: inf {Infinity}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: inf {Infinity}
t₆: X₄ {O(n)}
t₈: X₄ {O(n)}
t₉: X₄ {O(n)}
t₁₀: inf {Infinity}
t₁₁: 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: X₄ {O(n)}
t₁, X₁: X₅ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₄: 3⋅X₄ {O(n)}
t₃, X₅: 3⋅X₅ {O(n)}
t₄, X₄: X₄ {O(n)}
t₄, X₅: X₅ {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₀, X₅: X₅ {O(n)}
t₁₁, X₄: 3⋅X₄ {O(n)}
t₁₁, X₅: 3⋅X₅ {O(n)}