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
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 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

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

Found invariant X₃ ≤ X₁ for location eval_foo_bb1_in

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ 0 for location eval_foo_stop

Found invariant X₃ ≤ X₁ ∧ X₀ ≤ 0 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 ∧ X₃ ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₃ ≤ X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₃ ≤ X₁
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ 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 ≤ X₀ ∧ X₃ ≤ X₁ of depth 2:

new bound:

8⋅X₂+8⋅X₃+9 {O(n)}

MPRF:

• eval_foo_bb1_in: [1-X₁; X₀]
• eval_foo_bb2_in: [-X₁; X₀-X₁]

MPRF for transition t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-X₁, 1+X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₃ ≤ X₁ of depth 2:

new bound:

8⋅X₂+8⋅X₃+9 {O(n)}

MPRF:

• eval_foo_bb1_in: [1-X₁; X₀]
• eval_foo_bb2_in: [1-X₁; X₀]

All Bounds

Timebounds

Overall timebound:16⋅X₂+16⋅X₃+22 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 8⋅X₂+8⋅X₃+9 {O(n)}
t₃: 1 {O(1)}
t₄: 8⋅X₂+8⋅X₃+9 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₂+16⋅X₃+22 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 8⋅X₂+8⋅X₃+9 {O(n)}
t₃: 1 {O(1)}
t₄: 8⋅X₂+8⋅X₃+9 {O(n)}
t₅: 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, X₀: 136⋅X₂⋅X₃+64⋅X₂⋅X₂+72⋅X₃⋅X₃+153⋅X₂+161⋅X₃+90 {O(n^2)}
t₂, X₁: 8⋅X₂+9⋅X₃+9 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 136⋅X₂⋅X₃+64⋅X₂⋅X₂+72⋅X₃⋅X₃+154⋅X₂+161⋅X₃+90 {O(n^2)}
t₃, X₁: 10⋅X₃+8⋅X₂+9 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 136⋅X₂⋅X₃+64⋅X₂⋅X₂+72⋅X₃⋅X₃+153⋅X₂+161⋅X₃+90 {O(n^2)}
t₄, X₁: 8⋅X₂+9⋅X₃+9 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 136⋅X₂⋅X₃+64⋅X₂⋅X₂+72⋅X₃⋅X₃+154⋅X₂+161⋅X₃+90 {O(n^2)}
t₅, X₁: 10⋅X₃+8⋅X₂+9 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}