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) :|: 1+X₀ ≤ 0
t₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 6+X₀ ≤ 0
t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 0 ≤ 4+X₀
t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀, X₁) :|: 0 ≤ 5+X₀ ∧ 5+X₀ ≤ 0
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 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ 1+X₀ ≤ 0 for location eval_foo_bb2_in

Found invariant X₁ ≤ X₀ for location eval_foo_bb1_in

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

Found invariant X₁ ≤ X₀ ∧ 0 ≤ 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) :|: 1+X₀ ≤ 0 ∧ X₁ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 0 ≤ X₀ ∧ X₁ ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 6+X₀ ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 0 ≤ 4+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀
t₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀, X₁) :|: 0 ≤ 5+X₀ ∧ 5+X₀ ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀
t₇: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 0 ≤ 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) :|: 6+X₀ ≤ 0 ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁+5 {O(n)}

MPRF:

• eval_foo_bb1_in: [-5-X₀]
• eval_foo_bb2_in: [-5-X₀]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: 0 ≤ 4+X₀ ∧ 1+X₀ ≤ 0 ∧ 1+X₁ ≤ 0 ∧ 2+X₀+X₁ ≤ 0 ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₁ {O(n)}

MPRF:

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

Found invariant 5+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 10+X₀+X₁ ≤ 0 ∧ 0 ≤ 5+X₁ ∧ 0 ≤ 10+X₀+X₁ ∧ X₀ ≤ X₁ ∧ 5+X₀ ≤ 0 ∧ 0 ≤ 5+X₀ for location eval_foo_bb1_in_v1

Found invariant 6+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 11+X₀+X₁ ≤ 0 ∧ 5+X₀ ≤ 0 for location eval_foo_bb1_in_v3

Found invariant 5+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 10+X₀+X₁ ≤ 0 ∧ 0 ≤ 5+X₁ ∧ 0 ≤ 10+X₀+X₁ ∧ X₀ ≤ X₁ ∧ 5+X₀ ≤ 0 ∧ 0 ≤ 5+X₀ for location eval_foo_bb2_in_v5

Found invariant 1+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 1+X₀+X₁ ≤ 0 ∧ 0 ≤ 4+X₁ ∧ 0 ≤ 7+X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 0 ∧ 0 ≤ 3+X₀ for location eval_foo_bb1_in_v2

Found invariant X₁ ≤ X₀ ∧ 0 ≤ 4+X₁ ∧ 0 ≤ 4+X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ 0 ≤ X₀ for location eval_foo_stop

Found invariant X₁ ≤ X₀ ∧ 0 ≤ 4+X₁ ∧ 0 ≤ 4+X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb3_in

Found invariant 2+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 3+X₀+X₁ ≤ 0 ∧ 0 ≤ 4+X₁ ∧ 0 ≤ 7+X₀+X₁ ∧ X₀ ≤ 3+X₁ ∧ 1+X₀ ≤ 0 ∧ 0 ≤ 3+X₀ for location eval_foo_bb2_in_v4

Found invariant 6+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 11+X₀+X₁ ≤ 0 ∧ 5+X₀ ≤ 0 for location eval_foo_bb2_in_v2

Found invariant 6+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 11+X₀+X₁ ≤ 0 ∧ 5+X₀ ≤ 0 ∧ 0 ≤ 5+X₀ for location eval_foo_bb1_in_v4

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

Found invariant 1+X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 2+X₀+X₁ ≤ 0 ∧ X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb2_in_v1

Found invariant 6+X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 11+X₀+X₁ ≤ 0 ∧ 5+X₀ ≤ 0 ∧ 0 ≤ 5+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+5 {O(n)}
t₅: X₁ {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+5 {O(n)}
t₅: X₁ {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: X₁+8 {O(n)}
t₂, X₁: X₁ {O(n)}
t₃, X₀: X₁+3 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₄, X₀: X₁+8 {O(n)}
t₄, X₁: X₁ {O(n)}
t₅, X₀: 3 {O(1)}
t₅, X₁: X₁ {O(n)}
t₆, X₀: 5 {O(1)}
t₆, X₁: X₁ {O(n)}
t₇, X₀: X₁+3 {O(n)}
t₇, X₁: 2⋅X₁ {O(n)}