KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 0 ≤ X₃ for location eval_speedNestedMultipleDep_bb1_in

Found invariant 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₂ for location eval_speedNestedMultipleDep_bb2_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_speedNestedMultipleDep_stop

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_speedNestedMultipleDep_bb5_in

Found invariant 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_speedNestedMultipleDep_bb3_in

Found invariant 1+X₄ ≤ X₁ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedNestedMultipleDep_bb4_in

Problem after Preprocessing

Start: eval_speedNestedMultipleDep_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_speedNestedMultipleDep_bb0_in, eval_speedNestedMultipleDep_bb1_in, eval_speedNestedMultipleDep_bb2_in, eval_speedNestedMultipleDep_bb3_in, eval_speedNestedMultipleDep_bb4_in, eval_speedNestedMultipleDep_bb5_in, eval_speedNestedMultipleDep_start, eval_speedNestedMultipleDep_stop
Transitions:
t₁: eval_speedNestedMultipleDep_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb1_in(X₀, X₁, X₂, 0, X₄)
t₂: eval_speedNestedMultipleDep_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ X₂ ∧ 0 ≤ X₃
t₃: eval_speedNestedMultipleDep_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₃ ∧ 0 ≤ X₃
t₄: eval_speedNestedMultipleDep_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb3_in(1+X₃, X₁, X₂, X₃, 0) :|: 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₃
t₆: eval_speedNestedMultipleDep_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb1_in(X₀, X₁, X₂, X₀, X₄) :|: X₁ ≤ X₄ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄
t₅: eval_speedNestedMultipleDep_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₁ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄
t₇: eval_speedNestedMultipleDep_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb3_in(X₀, X₁, X₂, X₃, 1+X₄) :|: X₀ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁+X₄ ∧ 1+X₄ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄
t₈: eval_speedNestedMultipleDep_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_stop(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ X₃ ∧ 0 ≤ X₃
t₀: eval_speedNestedMultipleDep_start(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: eval_speedNestedMultipleDep_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_speedNestedMultipleDep_bb1_in: [X₂-X₃]
• eval_speedNestedMultipleDep_bb2_in: [X₂-1-X₃]
• eval_speedNestedMultipleDep_bb3_in: [X₂-X₀]
• eval_speedNestedMultipleDep_bb4_in: [X₂-X₀]

MPRF for transition t₄: eval_speedNestedMultipleDep_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb3_in(1+X₃, X₁, X₂, X₃, 0) :|: 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_speedNestedMultipleDep_bb1_in: [X₂-X₃]
• eval_speedNestedMultipleDep_bb2_in: [X₂-X₃]
• eval_speedNestedMultipleDep_bb3_in: [X₂-1-X₃]
• eval_speedNestedMultipleDep_bb4_in: [X₂-1-X₃]

MPRF for transition t₆: eval_speedNestedMultipleDep_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb1_in(X₀, X₁, X₂, X₀, X₄) :|: X₁ ≤ X₄ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_speedNestedMultipleDep_bb1_in: [X₂-X₃]
• eval_speedNestedMultipleDep_bb2_in: [X₂-X₃]
• eval_speedNestedMultipleDep_bb3_in: [X₂-X₃]
• eval_speedNestedMultipleDep_bb4_in: [X₂-X₃]

MPRF for transition t₅: eval_speedNestedMultipleDep_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₄ ≤ X₁ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₁⋅X₂+X₁ {O(n^2)}

MPRF:

• eval_speedNestedMultipleDep_bb1_in: [X₁]
• eval_speedNestedMultipleDep_bb2_in: [X₁]
• eval_speedNestedMultipleDep_bb3_in: [X₁-X₄]
• eval_speedNestedMultipleDep_bb4_in: [X₁-1-X₄]

MPRF for transition t₇: eval_speedNestedMultipleDep_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_speedNestedMultipleDep_bb3_in(X₀, X₁, X₂, X₃, 1+X₄) :|: X₀ ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1 ≤ X₀+X₄ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₁+X₄ ∧ 1+X₄ ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1+X₃ ≤ X₂ ∧ 1 ≤ X₂+X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₃ ∧ 0 ≤ X₃+X₄ ∧ 0 ≤ X₄ of depth 1:

new bound:

X₁⋅X₂+X₁ {O(n^2)}

MPRF:

• eval_speedNestedMultipleDep_bb1_in: [X₁]
• eval_speedNestedMultipleDep_bb2_in: [X₁]
• eval_speedNestedMultipleDep_bb3_in: [X₁-X₄]
• eval_speedNestedMultipleDep_bb4_in: [X₁-X₄]

Found invariant 0 ≤ X₃ for location eval_speedNestedMultipleDep_bb1_in

Found invariant 1+X₃ ≤ X₂ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₂ for location eval_speedNestedMultipleDep_bb2_in

Found invariant X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 2 ≤ X₂+X₄ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedNestedMultipleDep_bb3_in_v1

Found invariant 1+X₄ ≤ X₁ ∧ 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ 2 ≤ X₂+X₄ ∧ 3 ≤ X₁+X₄ ∧ 2 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedNestedMultipleDep_bb4_in_v2

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_speedNestedMultipleDep_bb5_in

Found invariant X₄ ≤ 0 ∧ X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₀ for location eval_speedNestedMultipleDep_bb3_in

Found invariant 0 ≤ X₃ ∧ X₂ ≤ X₃ for location eval_speedNestedMultipleDep_stop

Found invariant X₄ ≤ 0 ∧ X₄ ≤ X₃ ∧ 1+X₄ ≤ X₂ ∧ 1+X₄ ≤ X₁ ∧ 1+X₄ ≤ X₀ ∧ 0 ≤ X₄ ∧ 0 ≤ X₃+X₄ ∧ 1 ≤ X₂+X₄ ∧ 1 ≤ X₁+X₄ ∧ 1 ≤ X₀+X₄ ∧ 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speedNestedMultipleDep_bb4_in_v1

All Bounds

Timebounds

Overall timebound:2⋅X₁⋅X₂+2⋅X₁+3⋅X₂+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₄: X₂ {O(n)}
t₅: X₁⋅X₂+X₁ {O(n^2)}
t₆: X₂ {O(n)}
t₇: X₁⋅X₂+X₁ {O(n^2)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₁⋅X₂+2⋅X₁+3⋅X₂+4 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂ {O(n)}
t₃: 1 {O(1)}
t₄: X₂ {O(n)}
t₅: X₁⋅X₂+X₁ {O(n^2)}
t₆: X₂ {O(n)}
t₇: X₁⋅X₂+X₁ {O(n^2)}
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)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: 0 {O(1)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₂+X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₂ {O(n)}
t₂, X₄: X₁+X₄+1 {O(n)}
t₃, X₀: 2⋅X₂+X₀ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: X₂ {O(n)}
t₃, X₄: X₁+X₄+1 {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₂ {O(n)}
t₄, X₄: 0 {O(1)}
t₅, X₀: X₂ {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₂ {O(n)}
t₅, X₄: X₁+1 {O(n)}
t₆, X₀: 2⋅X₂ {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₂: X₂ {O(n)}
t₆, X₃: X₂ {O(n)}
t₆, X₄: X₁+1 {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₂ {O(n)}
t₇, X₄: X₁+1 {O(n)}
t₈, X₀: 2⋅X₂+X₀ {O(n)}
t₈, X₁: 2⋅X₁ {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: X₂ {O(n)}
t₈, X₄: X₁+X₄+1 {O(n)}