KoAT2 Proof Output

Initial Problem

Start: eval_nd_loop_start
Program_Vars: X₀, X₁
Temp_Vars: nondef.0
Locations: eval_nd_loop_0, eval_nd_loop_1, eval_nd_loop_bb0_in, eval_nd_loop_bb1_in, eval_nd_loop_bb2_in, eval_nd_loop_start, eval_nd_loop_stop
Transitions:
t₄: eval_nd_loop_0(X₀, X₁) → eval_nd_loop_1(nondef.0, X₁)
t₅: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb1_in(X₀, X₀) :|: X₀ ≤ 9 ∧ X₀ ≤ 2+X₁ ∧ 1+X₁ ≤ X₀
t₆: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb2_in(X₀, X₁) :|: 3+X₁ ≤ X₀
t₇: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb2_in(X₀, X₁) :|: X₀ ≤ X₁
t₈: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb2_in(X₀, X₁) :|: 10 ≤ X₀
t₁: eval_nd_loop_bb0_in(X₀, X₁) → eval_nd_loop_bb1_in(X₀, 0)
t₂: eval_nd_loop_bb1_in(X₀, X₁) → eval_nd_loop_0(X₀, X₁)
t₉: eval_nd_loop_bb2_in(X₀, X₁) → eval_nd_loop_stop(X₀, X₁)
t₀: eval_nd_loop_start(X₀, X₁) → eval_nd_loop_bb0_in(X₀, X₁)

Preprocessing

Found invariant 0 ≤ X₁ for location eval_nd_loop_1

Found invariant 0 ≤ X₁ for location eval_nd_loop_0

Found invariant 0 ≤ X₁ for location eval_nd_loop_bb2_in

Found invariant 0 ≤ X₁ for location eval_nd_loop_stop

Found invariant X₁ ≤ 9 ∧ 0 ≤ X₁ for location eval_nd_loop_bb1_in

Problem after Preprocessing

Start: eval_nd_loop_start
Program_Vars: X₀, X₁
Temp_Vars: nondef.0
Locations: eval_nd_loop_0, eval_nd_loop_1, eval_nd_loop_bb0_in, eval_nd_loop_bb1_in, eval_nd_loop_bb2_in, eval_nd_loop_start, eval_nd_loop_stop
Transitions:
t₄: eval_nd_loop_0(X₀, X₁) → eval_nd_loop_1(nondef.0, X₁) :|: 0 ≤ X₁
t₅: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb1_in(X₀, X₀) :|: X₀ ≤ 9 ∧ X₀ ≤ 2+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₆: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb2_in(X₀, X₁) :|: 3+X₁ ≤ X₀ ∧ 0 ≤ X₁
t₇: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb2_in(X₀, X₁) :|: X₀ ≤ X₁ ∧ 0 ≤ X₁
t₈: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb2_in(X₀, X₁) :|: 10 ≤ X₀ ∧ 0 ≤ X₁
t₁: eval_nd_loop_bb0_in(X₀, X₁) → eval_nd_loop_bb1_in(X₀, 0)
t₂: eval_nd_loop_bb1_in(X₀, X₁) → eval_nd_loop_0(X₀, X₁) :|: X₁ ≤ 9 ∧ 0 ≤ X₁
t₉: eval_nd_loop_bb2_in(X₀, X₁) → eval_nd_loop_stop(X₀, X₁) :|: 0 ≤ X₁
t₀: eval_nd_loop_start(X₀, X₁) → eval_nd_loop_bb0_in(X₀, X₁)

MPRF for transition t₂: eval_nd_loop_bb1_in(X₀, X₁) → eval_nd_loop_0(X₀, X₁) :|: X₁ ≤ 9 ∧ 0 ≤ X₁ of depth 1:

new bound:

10 {O(1)}

MPRF:

• eval_nd_loop_0: [9-X₁]
• eval_nd_loop_1: [9-X₁]
• eval_nd_loop_bb1_in: [10-X₁]

MPRF for transition t₅: eval_nd_loop_1(X₀, X₁) → eval_nd_loop_bb1_in(X₀, X₀) :|: X₀ ≤ 9 ∧ X₀ ≤ 2+X₁ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

9 {O(1)}

MPRF:

• eval_nd_loop_0: [9-X₁]
• eval_nd_loop_1: [9-X₁]
• eval_nd_loop_bb1_in: [9-X₁]

knowledge_propagation leads to new time bound 10 {O(1)} for transition t₄: eval_nd_loop_0(X₀, X₁) → eval_nd_loop_1(nondef.0, X₁) :|: 0 ≤ X₁

All Bounds

Timebounds

Overall timebound:35 {O(1)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 10 {O(1)}
t₄: 10 {O(1)}
t₅: 9 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 35 {O(1)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 10 {O(1)}
t₄: 10 {O(1)}
t₅: 9 {O(1)}
t₆: 1 {O(1)}
t₇: 1 {O(1)}
t₈: 1 {O(1)}
t₉: 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+9 {O(n)}
t₂, X₁: 9 {O(1)}
t₄, X₁: 9 {O(1)}
t₅, X₀: 9 {O(1)}
t₅, X₁: 9 {O(1)}
t₆, X₁: 9 {O(1)}
t₇, X₁: 9 {O(1)}
t₈, X₁: 9 {O(1)}
t₉, X₁: 27 {O(1)}