KoAT2 Proof Output

Initial Problem

Start: eval_easy1_start
Program_Vars: X₀, X₁
Temp_Vars: nondef_0
Locations: eval_easy1_1, eval_easy1_2, eval_easy1_3, eval_easy1_4, eval_easy1_5, eval_easy1_6, eval_easy1_bb0_in, eval_easy1_bb1_in, eval_easy1_bb2_in, eval_easy1_bb3_in, eval_easy1_start, eval_easy1_stop
Transitions:
t₂: eval_easy1_1(X₀, X₁) → eval_easy1_2(X₀, X₁)
t₃: eval_easy1_2(X₀, X₁) → eval_easy1_3(X₀, X₁)
t₄: eval_easy1_3(X₀, X₁) → eval_easy1_4(X₀, X₁)
t₅: eval_easy1_4(X₀, X₁) → eval_easy1_5(X₀, X₁)
t₆: eval_easy1_5(X₀, X₁) → eval_easy1_6(X₀, X₁)
t₇: eval_easy1_6(X₀, X₁) → eval_easy1_bb1_in(X₀, 0)
t₁: eval_easy1_bb0_in(X₀, X₁) → eval_easy1_1(nondef_0, X₁)
t₈: eval_easy1_bb1_in(X₀, X₁) → eval_easy1_bb2_in(X₀, X₁) :|: X₁ ≤ 39
t₉: eval_easy1_bb1_in(X₀, X₁) → eval_easy1_bb3_in(X₀, X₁) :|: 40 ≤ X₁
t₁₀: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 1+X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0
t₁₁: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 2+X₁) :|: 1+X₀ ≤ 0
t₁₂: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 2+X₁) :|: 1 ≤ X₀
t₁₃: eval_easy1_bb3_in(X₀, X₁) → eval_easy1_stop(X₀, X₁)
t₀: eval_easy1_start(X₀, X₁) → eval_easy1_bb0_in(X₀, X₁)

Preprocessing

Found invariant 40 ≤ X₁ for location eval_easy1_stop

Found invariant 0 ≤ X₁ for location eval_easy1_bb1_in

Found invariant X₁ ≤ 39 ∧ 0 ≤ X₁ for location eval_easy1_bb2_in

Found invariant 40 ≤ X₁ for location eval_easy1_bb3_in

Problem after Preprocessing

Start: eval_easy1_start
Program_Vars: X₀, X₁
Temp_Vars: nondef_0
Locations: eval_easy1_1, eval_easy1_2, eval_easy1_3, eval_easy1_4, eval_easy1_5, eval_easy1_6, eval_easy1_bb0_in, eval_easy1_bb1_in, eval_easy1_bb2_in, eval_easy1_bb3_in, eval_easy1_start, eval_easy1_stop
Transitions:
t₂: eval_easy1_1(X₀, X₁) → eval_easy1_2(X₀, X₁)
t₃: eval_easy1_2(X₀, X₁) → eval_easy1_3(X₀, X₁)
t₄: eval_easy1_3(X₀, X₁) → eval_easy1_4(X₀, X₁)
t₅: eval_easy1_4(X₀, X₁) → eval_easy1_5(X₀, X₁)
t₆: eval_easy1_5(X₀, X₁) → eval_easy1_6(X₀, X₁)
t₇: eval_easy1_6(X₀, X₁) → eval_easy1_bb1_in(X₀, 0)
t₁: eval_easy1_bb0_in(X₀, X₁) → eval_easy1_1(nondef_0, X₁)
t₈: eval_easy1_bb1_in(X₀, X₁) → eval_easy1_bb2_in(X₀, X₁) :|: X₁ ≤ 39 ∧ 0 ≤ X₁
t₉: eval_easy1_bb1_in(X₀, X₁) → eval_easy1_bb3_in(X₀, X₁) :|: 40 ≤ X₁ ∧ 0 ≤ X₁
t₁₀: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 1+X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ 39 ∧ 0 ≤ X₁
t₁₁: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 2+X₁) :|: 1+X₀ ≤ 0 ∧ X₁ ≤ 39 ∧ 0 ≤ X₁
t₁₂: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 2+X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 39 ∧ 0 ≤ X₁
t₁₃: eval_easy1_bb3_in(X₀, X₁) → eval_easy1_stop(X₀, X₁) :|: 40 ≤ X₁
t₀: eval_easy1_start(X₀, X₁) → eval_easy1_bb0_in(X₀, X₁)

MPRF for transition t₈: eval_easy1_bb1_in(X₀, X₁) → eval_easy1_bb2_in(X₀, X₁) :|: X₁ ≤ 39 ∧ 0 ≤ X₁ of depth 1:

new bound:

40 {O(1)}

MPRF:

• eval_easy1_bb1_in: [40-X₁]
• eval_easy1_bb2_in: [39-X₁]

MPRF for transition t₁₀: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 1+X₁) :|: 0 ≤ X₀ ∧ X₀ ≤ 0 ∧ X₁ ≤ 39 ∧ 0 ≤ X₁ of depth 1:

new bound:

40 {O(1)}

MPRF:

• eval_easy1_bb1_in: [40-X₁]
• eval_easy1_bb2_in: [40-X₁]

MPRF for transition t₁₁: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 2+X₁) :|: 1+X₀ ≤ 0 ∧ X₁ ≤ 39 ∧ 0 ≤ X₁ of depth 1:

new bound:

40 {O(1)}

MPRF:

• eval_easy1_bb1_in: [40-X₁]
• eval_easy1_bb2_in: [40-X₁]

MPRF for transition t₁₂: eval_easy1_bb2_in(X₀, X₁) → eval_easy1_bb1_in(X₀, 2+X₁) :|: 1 ≤ X₀ ∧ X₁ ≤ 39 ∧ 0 ≤ X₁ of depth 1:

new bound:

40 {O(1)}

MPRF:

• eval_easy1_bb1_in: [40-X₁]
• eval_easy1_bb2_in: [40-X₁]

All Bounds

Timebounds

Overall timebound:170 {O(1)}
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: 1 {O(1)}
t₈: 40 {O(1)}
t₉: 1 {O(1)}
t₁₀: 40 {O(1)}
t₁₁: 40 {O(1)}
t₁₂: 40 {O(1)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 170 {O(1)}
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: 1 {O(1)}
t₈: 40 {O(1)}
t₉: 1 {O(1)}
t₁₀: 40 {O(1)}
t₁₁: 40 {O(1)}
t₁₂: 40 {O(1)}
t₁₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₁: X₁ {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₁: X₁ {O(n)}
t₆, X₁: X₁ {O(n)}
t₇, X₁: 0 {O(1)}
t₈, X₁: 39 {O(1)}
t₉, X₁: 122 {O(1)}
t₁₀, X₀: 0 {O(1)}
t₁₀, X₁: 40 {O(1)}
t₁₁, X₁: 41 {O(1)}
t₁₂, X₁: 41 {O(1)}
t₁₃, X₁: 122 {O(1)}