KoAT2 Proof Output

Initial Problem

Start: eval_random1d_start
Program_Vars: X₀, X₁, X₂
Temp_Vars: nondef.0
Locations: eval_random1d_0, eval_random1d_1, eval_random1d_bb0_in, eval_random1d_bb1_in, eval_random1d_bb2_in, eval_random1d_bb3_in, eval_random1d_start, eval_random1d_stop
Transitions:
t₇: eval_random1d_0(X₀, X₁, X₂) → eval_random1d_1(nondef.0, X₁, X₂)
t₈: eval_random1d_1(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀
t₉: eval_random1d_1(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1+X₂) :|: X₀ ≤ 0
t₁: eval_random1d_bb0_in(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1) :|: 1 ≤ X₁
t₂: eval_random1d_bb0_in(X₀, X₁, X₂) → eval_random1d_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 0
t₃: eval_random1d_bb1_in(X₀, X₁, X₂) → eval_random1d_bb2_in(X₀, X₁, X₂) :|: X₂ ≤ X₁
t₄: eval_random1d_bb1_in(X₀, X₁, X₂) → eval_random1d_bb3_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂
t₅: eval_random1d_bb2_in(X₀, X₁, X₂) → eval_random1d_0(X₀, X₁, X₂)
t₁₀: eval_random1d_bb3_in(X₀, X₁, X₂) → eval_random1d_stop(X₀, X₁, X₂)
t₀: eval_random1d_start(X₀, X₁, X₂) → eval_random1d_bb0_in(X₀, X₁, X₂)

Preprocessing

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_random1d_1

Found invariant X₂ ≤ 1+X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_random1d_bb1_in

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_random1d_0

Found invariant X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 1 ≤ X₁ for location eval_random1d_bb2_in

Problem after Preprocessing

Start: eval_random1d_start
Program_Vars: X₀, X₁, X₂
Temp_Vars: nondef.0
Locations: eval_random1d_0, eval_random1d_1, eval_random1d_bb0_in, eval_random1d_bb1_in, eval_random1d_bb2_in, eval_random1d_bb3_in, eval_random1d_start, eval_random1d_stop
Transitions:
t₇: eval_random1d_0(X₀, X₁, X₂) → eval_random1d_1(nondef.0, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁
t₈: eval_random1d_1(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁
t₉: eval_random1d_1(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1+X₂) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁
t₁: eval_random1d_bb0_in(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1) :|: 1 ≤ X₁
t₂: eval_random1d_bb0_in(X₀, X₁, X₂) → eval_random1d_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 0
t₃: eval_random1d_bb1_in(X₀, X₁, X₂) → eval_random1d_bb2_in(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂
t₄: eval_random1d_bb1_in(X₀, X₁, X₂) → eval_random1d_bb3_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₂ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂
t₅: eval_random1d_bb2_in(X₀, X₁, X₂) → eval_random1d_0(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁
t₁₀: eval_random1d_bb3_in(X₀, X₁, X₂) → eval_random1d_stop(X₀, X₁, X₂)
t₀: eval_random1d_start(X₀, X₁, X₂) → eval_random1d_bb0_in(X₀, X₁, X₂)

MPRF for transition t₃: eval_random1d_bb1_in(X₀, X₁, X₂) → eval_random1d_bb2_in(X₀, X₁, X₂) :|: X₂ ≤ X₁ ∧ X₂ ≤ 1+X₁ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

• eval_random1d_0: [X₁-X₂]
• eval_random1d_1: [X₁-X₂]
• eval_random1d_bb1_in: [1+X₁-X₂]
• eval_random1d_bb2_in: [X₁-X₂]

MPRF for transition t₅: eval_random1d_bb2_in(X₀, X₁, X₂) → eval_random1d_0(X₀, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

• eval_random1d_0: [X₁-X₂]
• eval_random1d_1: [X₁-X₂]
• eval_random1d_bb1_in: [1+X₁-X₂]
• eval_random1d_bb2_in: [1+X₁-X₂]

MPRF for transition t₇: eval_random1d_0(X₀, X₁, X₂) → eval_random1d_1(nondef.0, X₁, X₂) :|: 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• eval_random1d_0: [2⋅X₁-X₂]
• eval_random1d_1: [2⋅X₁-1-X₂]
• eval_random1d_bb1_in: [2⋅X₁-X₂]
• eval_random1d_bb2_in: [2⋅X₁-X₂]

MPRF for transition t₈: eval_random1d_1(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1+X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₁+2 {O(n)}

MPRF:

• eval_random1d_0: [1+X₁-X₂]
• eval_random1d_1: [1+X₁-X₂]
• eval_random1d_bb1_in: [1+X₁-X₂]
• eval_random1d_bb2_in: [1+X₁-X₂]

MPRF for transition t₉: eval_random1d_1(X₀, X₁, X₂) → eval_random1d_bb1_in(X₀, X₁, 1+X₂) :|: X₀ ≤ 0 ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₂ ≤ X₁ of depth 1:

new bound:

2⋅X₁+1 {O(n)}

MPRF:

• eval_random1d_0: [2⋅X₁-X₂]
• eval_random1d_1: [2⋅X₁-X₂]
• eval_random1d_bb1_in: [2⋅X₁-X₂]
• eval_random1d_bb2_in: [2⋅X₁-X₂]

All Bounds

Timebounds

Overall timebound:7⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₁+2 {O(n)}
t₄: 1 {O(1)}
t₅: X₁+2 {O(n)}
t₇: 2⋅X₁+1 {O(n)}
t₈: X₁+2 {O(n)}
t₉: 2⋅X₁+1 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 7⋅X₁+13 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₁+2 {O(n)}
t₄: 1 {O(1)}
t₅: X₁+2 {O(n)}
t₇: 2⋅X₁+1 {O(n)}
t₈: X₁+2 {O(n)}
t₉: 2⋅X₁+1 {O(n)}
t₁₀: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: 1 {O(1)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₁: X₁ {O(n)}
t₃, X₂: 3⋅X₁+4 {O(n)}
t₄, X₁: 2⋅X₁ {O(n)}
t₄, X₂: 6⋅X₁+8 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₂: 3⋅X₁+4 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₂: 3⋅X₁+4 {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₂: 3⋅X₁+4 {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₂: 3⋅X₁+4 {O(n)}
t₁₀, X₁: 3⋅X₁ {O(n)}
t₁₀, X₂: 6⋅X₁+X₂+8 {O(n)}