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, 23, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, 1+X₁, X₂, X₃)
t₅: 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

Eliminate variables [X₃] that do not contribute to the problem

Found invariant 0 ≤ X₂ ∧ 23 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 23 ≤ X₁ ∧ 23 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₂ ∧ 23 ≤ X₁ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₂ ∧ 23 ≤ X₁ ∧ 24+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_stop

Found invariant X₀ ≤ X₂ ∧ 23 ≤ X₁ ∧ 24+X₀ ≤ X₁ ∧ 1+X₀ ≤ 0 for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
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:
t₁₁: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, 23, X₂)
t₁₂: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ 23 ≤ X₁ ∧ X₀ ≤ X₂
t₁₃: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ 23 ≤ X₁ ∧ X₀ ≤ X₂
t₁₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-X₁, 1+X₁, X₂) :|: 23 ≤ X₀+X₁ ∧ 23 ≤ X₁ ∧ 23 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₁₅: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ 23 ≤ X₁ ∧ 24+X₀ ≤ X₁ ∧ X₀ ≤ X₂
t₁₆: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ 23 ≤ X₁ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₁₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₀-X₁, 1+X₁, X₂) :|: 23 ≤ X₀+X₁ ∧ 23 ≤ X₁ ∧ 23 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [1+X₀]
• eval_foo_bb2_in: [1+X₀]

All Bounds

Timebounds

Overall timebound:2⋅X₂+6 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₂+1 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 2⋅X₂+6 {O(n)}
t₁₁: 1 {O(1)}
t₁₂: X₂+1 {O(n)}
t₁₃: 1 {O(1)}
t₁₄: X₂+1 {O(n)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: X₂ {O(n)}
t₁₁, X₁: 23 {O(1)}
t₁₁, X₂: X₂ {O(n)}
t₁₂, X₀: 2⋅X₂+25 {O(n)}
t₁₂, X₁: X₂+25 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₃, X₀: 3⋅X₂+25 {O(n)}
t₁₃, X₁: X₂+48 {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}
t₁₄, X₀: 2⋅X₂+25 {O(n)}
t₁₄, X₁: X₂+25 {O(n)}
t₁₄, X₂: X₂ {O(n)}
t₁₅, X₀: 3⋅X₂+25 {O(n)}
t₁₅, X₁: X₂+48 {O(n)}
t₁₅, X₂: 2⋅X₂ {O(n)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}
t₁₆, X₂: X₂ {O(n)}