KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location eval_speed_pldi10_ex4_stop

Found invariant 1 ≤ X₄ ∧ 1 ≤ X₃+X₄ ∧ X₃ ≤ X₄ ∧ 2 ≤ X₂+X₄ ∧ X₂ ≤ X₄ ∧ 2 ≤ X₁+X₄ ∧ X₁ ≤ X₄ ∧ 2 ≤ X₀+X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₂ ≤ 1 ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speed_pldi10_ex4_bb3_in

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 1 ∧ 0 ≤ X₂ for location eval_speed_pldi10_ex4_bb1_in

Found invariant X₁ ≤ X₄ ∧ X₀ ≤ X₄ ∧ X₃ ≤ 1 ∧ X₃ ≤ X₂ ∧ X₂+X₃ ≤ 2 ∧ 0 ≤ X₃ ∧ 1 ≤ X₂+X₃ ∧ X₂ ≤ 1+X₃ ∧ X₂ ≤ 1 ∧ 1 ≤ X₂ ∧ X₁ ≤ X₀ for location eval_speed_pldi10_ex4_bb2_in

Found invariant X₀ ≤ X₄ ∧ X₂ ≤ 0 ∧ 0 ≤ X₂ for location eval_speed_pldi10_ex4_bb4_in

Problem after Preprocessing

Start: eval_speed_pldi10_ex4_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars:
Locations: eval_speed_pldi10_ex4_bb0_in, eval_speed_pldi10_ex4_bb1_in, eval_speed_pldi10_ex4_bb2_in, eval_speed_pldi10_ex4_bb3_in, eval_speed_pldi10_ex4_bb4_in, eval_speed_pldi10_ex4_start, eval_speed_pldi10_ex4_stop
Transitions:
t₁: eval_speed_pldi10_ex4_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb1_in(X₄, X₁, 1, X₃, X₄)
t₂: eval_speed_pldi10_ex4_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb2_in(X₀, X₀, X₂, 0, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂
t₃: eval_speed_pldi10_ex4_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 0 ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂
t₅: eval_speed_pldi10_ex4_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb1_in(X₁, X₁, X₃, X₃, X₄) :|: X₁ ≤ 0 ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₄: eval_speed_pldi10_ex4_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₆: eval_speed_pldi10_ex4_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb2_in(X₀, X₁-1, X₂, 1, X₄) :|: X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄
t₇: eval_speed_pldi10_ex4_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_stop(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ X₄ ∧ 0 ≤ X₂ ∧ X₂ ≤ 0
t₀: eval_speed_pldi10_ex4_start(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₄: eval_speed_pldi10_ex4_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₁ ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_speed_pldi10_ex4_bb1_in: [X₀]
• eval_speed_pldi10_ex4_bb2_in: [X₁]
• eval_speed_pldi10_ex4_bb3_in: [X₁-1]

MPRF for transition t₆: eval_speed_pldi10_ex4_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb2_in(X₀, X₁-1, X₂, 1, X₄) :|: X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1 ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ 1 ≤ X₃+X₄ ∧ 1 ≤ X₄ ∧ 2 ≤ X₀+X₁ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₀+X₄ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₁+X₄ ∧ 2 ≤ X₂+X₄ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ X₂ ≤ X₄ ∧ 0 ≤ X₃ ∧ X₃ ≤ X₄ of depth 1:

new bound:

X₄ {O(n)}

MPRF:

• eval_speed_pldi10_ex4_bb1_in: [X₀]
• eval_speed_pldi10_ex4_bb2_in: [X₁]
• eval_speed_pldi10_ex4_bb3_in: [X₁]

MPRF for transition t₂: eval_speed_pldi10_ex4_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb2_in(X₀, X₀, X₂, 0, X₄) :|: 1 ≤ X₂ ∧ X₂ ≤ 1 ∧ X₀ ≤ X₄ ∧ 0 ≤ X₂ of depth 1:

new bound:

2⋅X₄+2 {O(n)}

MPRF:

• eval_speed_pldi10_ex4_bb1_in: [1+X₂]
• eval_speed_pldi10_ex4_bb2_in: [X₂+X₃]
• eval_speed_pldi10_ex4_bb3_in: [X₂+X₃]

MPRF for transition t₅: eval_speed_pldi10_ex4_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_speed_pldi10_ex4_bb1_in(X₁, X₁, X₃, X₃, X₄) :|: X₁ ≤ 0 ∧ X₂+X₃ ≤ 2 ∧ X₂ ≤ 1 ∧ X₂ ≤ 1+X₃ ∧ X₃ ≤ 1 ∧ 1 ≤ X₂ ∧ 1 ≤ X₂+X₃ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₄ ∧ X₁ ≤ X₄ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

2⋅X₄+1 {O(n)}

MPRF:

• eval_speed_pldi10_ex4_bb1_in: [X₂]
• eval_speed_pldi10_ex4_bb2_in: [1+X₃]
• eval_speed_pldi10_ex4_bb3_in: [X₂+X₃]

All Bounds

Timebounds

Overall timebound:6⋅X₄+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₄+2 {O(n)}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: 2⋅X₄+1 {O(n)}
t₆: X₄ {O(n)}
t₇: 1 {O(1)}

Costbounds

Overall costbound: 6⋅X₄+7 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 2⋅X₄+2 {O(n)}
t₃: 1 {O(1)}
t₄: X₄ {O(n)}
t₅: 2⋅X₄+1 {O(n)}
t₆: X₄ {O(n)}
t₇: 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: X₄ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 1 {O(1)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₀: 2⋅X₄ {O(n)}
t₂, X₁: X₄ {O(n)}
t₂, X₂: 1 {O(1)}
t₂, X₃: 0 {O(1)}
t₂, X₄: X₄ {O(n)}
t₃, X₀: X₄ {O(n)}
t₃, X₁: 2⋅X₄ {O(n)}
t₃, X₂: 0 {O(1)}
t₃, X₃: 1 {O(1)}
t₃, X₄: X₄ {O(n)}
t₄, X₀: 2⋅X₄ {O(n)}
t₄, X₁: X₄ {O(n)}
t₄, X₂: 1 {O(1)}
t₄, X₃: 1 {O(1)}
t₄, X₄: X₄ {O(n)}
t₅, X₀: X₄ {O(n)}
t₅, X₁: 2⋅X₄ {O(n)}
t₅, X₂: 1 {O(1)}
t₅, X₃: 1 {O(1)}
t₅, X₄: X₄ {O(n)}
t₆, X₀: 2⋅X₄ {O(n)}
t₆, X₁: X₄ {O(n)}
t₆, X₂: 1 {O(1)}
t₆, X₃: 1 {O(1)}
t₆, X₄: X₄ {O(n)}
t₇, X₀: X₄ {O(n)}
t₇, X₁: 2⋅X₄ {O(n)}
t₇, X₂: 0 {O(1)}
t₇, X₃: 1 {O(1)}
t₇, X₄: X₄ {O(n)}