KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(X₁, X₁)
t₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: X₀ ≤ 9
t₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 10 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 2
t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 4 ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀, X₁) :|: X₀ ≤ 3 ∧ 3 ≤ X₀
t₇: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁)
t₀: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

Preprocessing

Found invariant X₁ ≤ 9 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ 9 for location eval_foo_bb2_in

Found invariant X₁ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₀ ∧ 10 ≤ X₀ for location eval_foo_stop

Found invariant X₁ ≤ X₀ ∧ 10 ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(X₁, X₁)
t₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: X₀ ≤ 9 ∧ X₁ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 10 ≤ X₀ ∧ X₁ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 2 ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ 9 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 4 ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ 9 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀, X₁) :|: X₀ ≤ 3 ∧ 3 ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ 9 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀
t₇: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 10 ≤ X₀ ∧ X₁ ≤ X₀
t₀: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 2 ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ 9 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+3 {O(n)}

MPRF:

• eval_foo_bb1_in: [3-X₀]
• eval_foo_bb2_in: [3-X₀]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 4 ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ 9 ∧ X₁ ≤ 9 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+10 {O(n)}

MPRF:

• eval_foo_bb1_in: [10-X₀]
• eval_foo_bb2_in: [10-X₀]

Found invariant X₁ ≤ 3 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 6 ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ for location eval_foo_bb1_in_v1

Found invariant X₁ ≤ 2 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 5 ∧ X₀ ≤ 3 for location eval_foo_bb1_in_v3

Found invariant X₁ ≤ 3 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 6 ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ for location eval_foo_bb2_in_v5

Found invariant X₁ ≤ 9 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 19 ∧ 4 ≤ X₁ ∧ 9 ≤ X₀+X₁ ∧ X₀ ≤ 6+X₁ ∧ X₀ ≤ 10 ∧ 5 ≤ X₀ for location eval_foo_bb1_in_v2

Found invariant X₁ ≤ X₀ ∧ 4 ≤ X₁ ∧ 14 ≤ X₀+X₁ ∧ X₀ ≤ 6+X₁ ∧ 10 ≤ X₀ for location eval_foo_stop

Found invariant X₁ ≤ X₀ ∧ 4 ≤ X₁ ∧ 14 ≤ X₀+X₁ ∧ X₀ ≤ 6+X₁ ∧ 10 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ 8 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 17 ∧ 4 ≤ X₁ ∧ 9 ≤ X₀+X₁ ∧ X₀ ≤ 5+X₁ ∧ X₀ ≤ 9 ∧ 5 ≤ X₀ for location eval_foo_bb2_in_v4

Found invariant X₁ ≤ 2 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 5 ∧ X₀ ≤ 3 for location eval_foo_bb2_in_v2

Found invariant X₁ ≤ 2 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 5 ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ for location eval_foo_bb1_in_v4

Found invariant X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_bb1_in

Found invariant X₁ ≤ 9 ∧ X₁ ≤ X₀ ∧ X₀+X₁ ≤ 18 ∧ X₀ ≤ X₁ ∧ X₀ ≤ 9 for location eval_foo_bb2_in_v1

Found invariant X₁ ≤ 2 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 5 ∧ X₀ ≤ 3 ∧ 3 ≤ X₀ for location eval_foo_bb2_in_v3

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: X₁+3 {O(n)}
t₅: X₁+10 {O(n)}
t₆: inf {Infinity}
t₇: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: 1 {O(1)}
t₄: X₁+3 {O(n)}
t₅: X₁+10 {O(n)}
t₆: inf {Infinity}
t₇: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₁, X₀: X₁ {O(n)}
t₁, X₁: X₁ {O(n)}
t₂, X₀: 2⋅X₁+16 {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: X₁+10 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₄, X₀: 2⋅X₁+30 {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: 10 {O(1)}
t₅, X₁: X₁ {O(n)}
t₆, X₀: 3 {O(1)}
t₆, X₁: X₁ {O(n)}
t₇, X₀: X₁+10 {O(n)}
t₇, X₁: 2⋅X₁ {O(n)}