KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1+X₁, X₂, X₃)
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ for location eval_foo_bb2_in

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_stop

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
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, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1+X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₃ ≤ X₁
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₁ ≤ X₂ ∧ X₃ ≤ X₂
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₀ ≤ X₂ ∧ X₃ ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-X₃]
• eval_foo_bb2_in: [X₀-1-X₃]

Found invariant 1+X₃ ≤ X₁ ∧ 1+X₀ ≤ X₂ ∧ 3+X₀ ≤ X₁ for location eval_foo_bb1_in_v1

Found invariant 1+X₃ ≤ X₂ ∧ 1+X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ 1+X₀ ≤ X₂ ∧ X₁ ≤ 1+X₀ for location eval_foo_bb1_in_v2

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_stop

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_foo_bb3_in

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 1+X₀ ≤ X₁ for location eval_foo_bb2_in_v2

Found invariant X₃ ≤ X₁ ∧ X₁ ≤ X₃ ∧ X₂ ≤ X₀ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant 1+X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 1+X₁ ≤ X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb2_in_v1

All Bounds

Timebounds

Overall timebound:inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: X₂+X₃ {O(n)}
t₄: 1 {O(1)}
t₅: inf {Infinity}
t₆: 1 {O(1)}

Costbounds

Overall costbound: inf {Infinity}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: inf {Infinity}
t₃: X₂+X₃ {O(n)}
t₄: 1 {O(1)}
t₅: inf {Infinity}
t₆: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₂: X₂ {O(n)}
t₅, X₃: X₃ {O(n)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}