KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₄ for location eval_foo_stop

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

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

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

Found invariant X₀ ≤ X₄ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₄ 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_3, eval_foo_4, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_start, eval_foo_stop
Transitions:
t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_4(X₀, X₁, X₂, nondef.0, X₄, X₅) :|: 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₂+X₄ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₄
t₁₀: eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₂+X₄ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₄
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 ≤ X₀ ∧ 0 ≤ X₁ ∧ X₀ ≤ X₄
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₄
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₁ ≤ 0 ∧ X₀ ≤ X₄
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃, X₄, X₅) :|: 1 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ 0 ≤ X₄
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ 0 ≤ X₄
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_3(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₄
t₁₁: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: 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 t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₁ ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_foo_3: [1+X₂]
• eval_foo_4: [1+X₂]
• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₀]
• eval_foo_bb3_in: [X₀]

MPRF for transition t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_3(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_foo_3: [1+X₂]
• eval_foo_4: [1+X₂]
• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₀]
• eval_foo_bb3_in: [1+X₀]

MPRF for transition t₉: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_4(X₀, X₁, X₂, nondef.0, X₄, X₅) :|: 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₂+X₄ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₄+1 {O(n)}

MPRF:

• eval_foo_3: [1+X₀]
• eval_foo_4: [X₀]
• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₀]
• eval_foo_bb3_in: [1+X₀]

MPRF for transition t₁₀: eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ X₁ ≤ 1+X₂ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₂+X₄ ∧ 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₄ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₄ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

2⋅X₄+2 {O(n)}

MPRF:

• eval_foo_3: [1+2⋅X₀]
• eval_foo_4: [1+2⋅X₀]
• eval_foo_bb1_in: [2+2⋅X₀]
• eval_foo_bb2_in: [2+2⋅X₀]
• eval_foo_bb3_in: [2+2⋅X₀]

Cut unreachable locations [eval_foo_bb2_in] from the program graph

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

Found invariant X₀ ≤ X₄ for location eval_foo_stop

Found invariant X₀ ≤ X₄ for location eval_foo_bb1_in

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

Found invariant X₀ ≤ X₄ for location eval_foo_bb4_in

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₄+1 {O(n)}
t₆: inf {Infinity}
t₇: X₄+1 {O(n)}
t₉: X₄+1 {O(n)}
t₁₀: 2⋅X₄+2 {O(n)}
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: 1 {O(1)}
t₅: X₄+1 {O(n)}
t₆: inf {Infinity}
t₇: X₄+1 {O(n)}
t₉: X₄+1 {O(n)}
t₁₀: 2⋅X₄+2 {O(n)}
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, X₀: X₄+1 {O(n)}
t₂, X₂: X₂+X₄+2 {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: 2⋅X₄+1 {O(n)}
t₃, X₂: X₂+X₄+2 {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₃, X₅: 2⋅X₅ {O(n)}
t₄, X₀: 2⋅X₄+1 {O(n)}
t₄, X₂: X₂+X₄+2 {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₅, X₀: X₄+1 {O(n)}
t₅, X₁: 0 {O(1)}
t₅, X₂: X₂+X₄+2 {O(n)}
t₅, X₄: X₄ {O(n)}
t₅, X₅: X₅ {O(n)}
t₆, X₀: X₄+1 {O(n)}
t₆, X₂: X₂+X₄+2 {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₇, X₀: X₄+1 {O(n)}
t₇, X₁: 0 {O(1)}
t₇, X₂: X₄+2 {O(n)}
t₇, X₄: X₄ {O(n)}
t₇, X₅: X₅ {O(n)}
t₉, X₀: X₄+1 {O(n)}
t₉, X₁: 0 {O(1)}
t₉, X₂: X₄+2 {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₀: X₄+1 {O(n)}
t₁₀, X₂: X₄+2 {O(n)}
t₁₀, X₄: X₄ {O(n)}
t₁₀, X₅: X₅ {O(n)}
t₁₁, X₀: 4⋅X₄+2 {O(n)}
t₁₁, X₂: 2⋅X₂+2⋅X₄+4 {O(n)}
t₁₁, X₄: 4⋅X₄ {O(n)}
t₁₁, X₅: 4⋅X₅ {O(n)}