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
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₂, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, 4+X₁, X₂, X₃)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, 2+X₁, X₂, X₃)
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb2_in

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

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ 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
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₃, X₂, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, 4+X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, 2+X₁, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁
t₀: eval_foo_start(X₀, X₁, X₂, X₃) → eval_foo_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-X₁]
• eval_foo_bb2_in: [X₀-4-X₁]
• eval_foo_bb3_in: [X₀-X₁]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, 4+X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀-X₁]
• eval_foo_bb2_in: [X₀-X₁]
• eval_foo_bb3_in: [X₀-X₁]

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₀ ≤ 1+X₃ ∧ 2+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₁ for location eval_foo_bb3_in_v2

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₀ ≤ 1+X₃ ∧ 2+X₂ ≤ X₁ ∧ 1+X₀ ≤ X₁ for location eval_foo_bb1_in_v1

Found invariant 4+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 4+X₂ ≤ X₁ ∧ 2+X₀ ≤ X₁ for location eval_foo_bb1_in_v2

Found invariant X₃ ≤ X₀ ∧ 1+X₂ ≤ X₃ ∧ 1+X₁ ≤ X₃ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb2_in

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

Found invariant 4+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 4+X₂ ≤ X₁ ∧ 2+X₀ ≤ X₁ for location eval_foo_bb3_in_v3

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₃ ∧ X₂ ≤ X₁ ∧ X₀ ≤ X₁ for location eval_foo_bb3_in_v1

All Bounds

Timebounds

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

Costbounds

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

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₃ {O(n)}
t₁, X₁: X₂ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₃ {O(n)}
t₂, X₁: 4⋅X₃+5⋅X₂ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₃ {O(n)}
t₄, X₁: 4⋅X₃+5⋅X₂ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}