KoAT2 Proof Output

Initial Problem

Start: eval_start_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁) → eval_start_1(X₀, X₁)
t₃: eval_start_1(X₀, X₁) → eval_start_2(X₀, X₁)
t₄: eval_start_2(X₀, X₁) → eval_start_3(X₀, X₁)
t₅: eval_start_3(X₀, X₁) → eval_start_4(X₀, X₁)
t₆: eval_start_4(X₀, X₁) → eval_start_bb1_in(X₀, 0)
t₁₀: eval_start_5(X₀, X₁) → eval_start_6(X₀, X₁)
t₁₁: eval_start_6(X₀, X₁) → eval_start_bb1_in(X₀, 1+X₁)
t₁: eval_start_bb0_in(X₀, X₁) → eval_start_0(X₀, X₁)
t₇: eval_start_bb1_in(X₀, X₁) → eval_start_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀
t₈: eval_start_bb1_in(X₀, X₁) → eval_start_bb3_in(X₀, X₁) :|: X₀ ≤ X₁
t₉: eval_start_bb2_in(X₀, X₁) → eval_start_5(X₀, X₁)
t₁₂: eval_start_bb3_in(X₀, X₁) → eval_start_stop(X₀, X₁)
t₀: eval_start_start(X₀, X₁) → eval_start_bb0_in(X₀, X₁)

Preprocessing

Found invariant 0 ≤ X₁ for location eval_start_bb1_in

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_5

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

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_6

Found invariant 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_start_bb2_in

Found invariant 0 ≤ X₁ ∧ X₀ ≤ X₁ for location eval_start_bb3_in

Problem after Preprocessing

Start: eval_start_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_start_0, eval_start_1, eval_start_2, eval_start_3, eval_start_4, eval_start_5, eval_start_6, eval_start_bb0_in, eval_start_bb1_in, eval_start_bb2_in, eval_start_bb3_in, eval_start_start, eval_start_stop
Transitions:
t₂: eval_start_0(X₀, X₁) → eval_start_1(X₀, X₁)
t₃: eval_start_1(X₀, X₁) → eval_start_2(X₀, X₁)
t₄: eval_start_2(X₀, X₁) → eval_start_3(X₀, X₁)
t₅: eval_start_3(X₀, X₁) → eval_start_4(X₀, X₁)
t₆: eval_start_4(X₀, X₁) → eval_start_bb1_in(X₀, 0)
t₁₀: eval_start_5(X₀, X₁) → eval_start_6(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₁: eval_start_6(X₀, X₁) → eval_start_bb1_in(X₀, 1+X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁: eval_start_bb0_in(X₀, X₁) → eval_start_0(X₀, X₁)
t₇: eval_start_bb1_in(X₀, X₁) → eval_start_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₈: eval_start_bb1_in(X₀, X₁) → eval_start_bb3_in(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₉: eval_start_bb2_in(X₀, X₁) → eval_start_5(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₂: eval_start_bb3_in(X₀, X₁) → eval_start_stop(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₀: eval_start_start(X₀, X₁) → eval_start_bb0_in(X₀, X₁)

MPRF for transition t₇: eval_start_bb1_in(X₀, X₁) → eval_start_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_start_5: [X₀-1-X₁]
• eval_start_6: [X₀-1-X₁]
• eval_start_bb1_in: [X₀-X₁]
• eval_start_bb2_in: [X₀-1-X₁]

MPRF for transition t₉: eval_start_bb2_in(X₀, X₁) → eval_start_5(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_start_5: [X₀-1-X₁]
• eval_start_6: [X₀-1-X₁]
• eval_start_bb1_in: [X₀-X₁]
• eval_start_bb2_in: [X₀-X₁]

MPRF for transition t₁₀: eval_start_5(X₀, X₁) → eval_start_6(X₀, X₁) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀ {O(n)}

MPRF:

• eval_start_5: [X₀-X₁]
• eval_start_6: [X₀-1-X₁]
• eval_start_bb1_in: [X₀-X₁]
• eval_start_bb2_in: [X₀-X₁]

MPRF for transition t₁₁: eval_start_6(X₀, X₁) → eval_start_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_start_5: [X₀-X₁]
• eval_start_6: [X₀-X₁]
• eval_start_bb1_in: [X₀-X₁]
• eval_start_bb2_in: [X₀-X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₀+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}
t₉: X₀ {O(n)}
t₁₀: X₀ {O(n)}
t₁₁: X₀ {O(n)}
t₁₂: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₀ {O(n)}
t₈: 1 {O(1)}
t₉: X₀ {O(n)}
t₁₀: X₀ {O(n)}
t₁₁: X₀ {O(n)}
t₁₂: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₁ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₁ {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: 0 {O(1)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀ {O(n)}
t₈, X₀: 2⋅X₀ {O(n)}
t₈, X₁: X₀ {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀ {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₀ {O(n)}
t₁₁, X₀: X₀ {O(n)}
t₁₁, X₁: X₀ {O(n)}
t₁₂, X₀: 2⋅X₀ {O(n)}
t₁₂, X₁: X₀ {O(n)}