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+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(X₁-X₀, 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+X₃ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location eval_foo_bb2_in

Found invariant X₃ ≤ X₁ for location eval_foo_bb1_in

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

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

new bound:

16⋅X₃+8⋅X₂+15 {O(n)}

MPRF:

• eval_foo_bb1_in: [-2-X₁; X₀-X₁]
• eval_foo_bb2_in: [-3-X₁; X₀-3-2⋅X₁]

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

new bound:

16⋅X₃+8⋅X₂+23 {O(n)}

MPRF:

• eval_foo_bb1_in: [-2-X₁; X₀-1-X₁]
• eval_foo_bb2_in: [-2-X₁; 0]

All Bounds

Timebounds

Overall timebound:16⋅X₂+32⋅X₃+42 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₃+8⋅X₂+15 {O(n)}
t₃: 1 {O(1)}
t₄: 16⋅X₃+8⋅X₂+23 {O(n)}
t₅: 1 {O(1)}

Costbounds

Overall costbound: 16⋅X₂+32⋅X₃+42 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 16⋅X₃+8⋅X₂+15 {O(n)}
t₃: 1 {O(1)}
t₄: 16⋅X₃+8⋅X₂+23 {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: 18⋅X₃+9⋅X₂+11 {O(n)}
t₂, X₁: 17⋅X₃+8⋅X₂+9 {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₂+3⋅X₃+3 {O(n)}
t₃, X₁: 3⋅X₃+1 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 3⋅X₃+X₂+3 {O(n)}
t₄, X₁: 2⋅X₃+1 {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 2⋅X₂+3⋅X₃+3 {O(n)}
t₅, X₁: 3⋅X₃+1 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}