KoAT2 Proof Output

Initial Problem

Start: eval_speed_popl10_simple_single_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_speed_popl10_simple_single_0, eval_speed_popl10_simple_single_1, eval_speed_popl10_simple_single_bb0_in, eval_speed_popl10_simple_single_bb1_in, eval_speed_popl10_simple_single_bb2_in, eval_speed_popl10_simple_single_bb3_in, eval_speed_popl10_simple_single_start, eval_speed_popl10_simple_single_stop
Transitions:
t₆: eval_speed_popl10_simple_single_0(X₀, X₁) → eval_speed_popl10_simple_single_1(X₀, X₁)
t₇: eval_speed_popl10_simple_single_1(X₀, X₁) → eval_speed_popl10_simple_single_bb1_in(X₀, 1+X₁)
t₁: eval_speed_popl10_simple_single_bb0_in(X₀, X₁) → eval_speed_popl10_simple_single_bb1_in(X₀, 0)
t₂: eval_speed_popl10_simple_single_bb1_in(X₀, X₁) → eval_speed_popl10_simple_single_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀
t₃: eval_speed_popl10_simple_single_bb1_in(X₀, X₁) → eval_speed_popl10_simple_single_bb3_in(X₀, X₁) :|: X₀ ≤ X₁
t₄: eval_speed_popl10_simple_single_bb2_in(X₀, X₁) → eval_speed_popl10_simple_single_0(X₀, X₁)
t₈: eval_speed_popl10_simple_single_bb3_in(X₀, X₁) → eval_speed_popl10_simple_single_stop(X₀, X₁)
t₀: eval_speed_popl10_simple_single_start(X₀, X₁) → eval_speed_popl10_simple_single_bb0_in(X₀, X₁)

Preprocessing

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speed_popl10_simple_single_0

Found invariant 0 ≤ X₁ for location eval_speed_popl10_simple_single_bb1_in

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speed_popl10_simple_single_1

Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_popl10_simple_single_bb3_in

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_speed_popl10_simple_single_bb2_in

Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_speed_popl10_simple_single_stop

Problem after Preprocessing

Start: eval_speed_popl10_simple_single_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_speed_popl10_simple_single_0, eval_speed_popl10_simple_single_1, eval_speed_popl10_simple_single_bb0_in, eval_speed_popl10_simple_single_bb1_in, eval_speed_popl10_simple_single_bb2_in, eval_speed_popl10_simple_single_bb3_in, eval_speed_popl10_simple_single_start, eval_speed_popl10_simple_single_stop
Transitions:
t₆: eval_speed_popl10_simple_single_0(X₀, X₁) → eval_speed_popl10_simple_single_1(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₇: eval_speed_popl10_simple_single_1(X₀, X₁) → eval_speed_popl10_simple_single_bb1_in(X₀, 1+X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁: eval_speed_popl10_simple_single_bb0_in(X₀, X₁) → eval_speed_popl10_simple_single_bb1_in(X₀, 0)
t₂: eval_speed_popl10_simple_single_bb1_in(X₀, X₁) → eval_speed_popl10_simple_single_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₃: eval_speed_popl10_simple_single_bb1_in(X₀, X₁) → eval_speed_popl10_simple_single_bb3_in(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₄: eval_speed_popl10_simple_single_bb2_in(X₀, X₁) → eval_speed_popl10_simple_single_0(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₈: eval_speed_popl10_simple_single_bb3_in(X₀, X₁) → eval_speed_popl10_simple_single_stop(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₀: eval_speed_popl10_simple_single_start(X₀, X₁) → eval_speed_popl10_simple_single_bb0_in(X₀, X₁)

MPRF for transition t₂: eval_speed_popl10_simple_single_bb1_in(X₀, X₁) → eval_speed_popl10_simple_single_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_speed_popl10_simple_single_0: [X₀-1-X₁]
• eval_speed_popl10_simple_single_1: [X₀-1-X₁]
• eval_speed_popl10_simple_single_bb1_in: [X₀-X₁]
• eval_speed_popl10_simple_single_bb2_in: [X₀-1-X₁]

MPRF for transition t₄: eval_speed_popl10_simple_single_bb2_in(X₀, X₁) → eval_speed_popl10_simple_single_0(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

MPRF for transition t₆: eval_speed_popl10_simple_single_0(X₀, X₁) → eval_speed_popl10_simple_single_1(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_speed_popl10_simple_single_0: [X₀-X₁]
• eval_speed_popl10_simple_single_1: [X₀-1-X₁]
• eval_speed_popl10_simple_single_bb1_in: [X₀-X₁]
• eval_speed_popl10_simple_single_bb2_in: [X₀-X₁]

MPRF for transition t₇: eval_speed_popl10_simple_single_1(X₀, X₁) → eval_speed_popl10_simple_single_bb1_in(X₀, 1+X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_speed_popl10_simple_single_0: [X₀-X₁]
• eval_speed_popl10_simple_single_1: [X₀-X₁]
• eval_speed_popl10_simple_single_bb1_in: [X₀-X₁]
• eval_speed_popl10_simple_single_bb2_in: [X₀-X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₀+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: X₀ {O(n)}
t₆: X₀ {O(n)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+4 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₀ {O(n)}
t₃: 1 {O(1)}
t₄: X₀ {O(n)}
t₆: X₀ {O(n)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: 0 {O(1)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₀ {O(n)}
t₃, X₀: 2⋅X₀ {O(n)}
t₃, X₁: X₀ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₀ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: X₀ {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀ {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: X₀ {O(n)}