KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 2 ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_2

Found invariant 2 ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₃ for location eval_foo_stop

Found invariant X₁ ≤ X₃ for location eval_foo_.critedge_in

Found invariant 2 ≤ X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_3

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
Temp_Vars: nondef.0
Locations: eval_foo_.critedge_in, eval_foo_2, eval_foo_3, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_start, eval_foo_stop
Transitions:
t₉: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄) :|: X₁ ≤ X₃
t₇: eval_foo_2(X₀, X₁, X₂, X₃, X₄) → eval_foo_3(X₀, X₁, nondef.0, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃
t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₀, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₄, X₃, X₂, X₃, X₄)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: X₀ ≤ 0 ∧ X₁ ≤ X₃
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄) :|: 1+X₁ ≤ 2⋅X₀ ∧ X₁ ≤ X₃
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 2⋅X₀ ≤ X₁ ∧ X₁ ≤ X₃
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_2(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 2⋅X₀ ≤ X₁ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_2: [X₁-2]
• eval_foo_3: [X₀-1]
• eval_foo_bb1_in: [X₁-1]
• eval_foo_bb2_in: [X₁-2]

MPRF for transition t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_2(X₀, X₁, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_2: [X₀-1]
• eval_foo_3: [X₀-1]
• eval_foo_bb1_in: [X₁-1]
• eval_foo_bb2_in: [X₀]

MPRF for transition t₇: eval_foo_2(X₀, X₁, X₂, X₃, X₄) → eval_foo_3(X₀, X₁, nondef.0, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_2: [X₁-1]
• eval_foo_3: [X₀-1]
• eval_foo_bb1_in: [X₁-1]
• eval_foo_bb2_in: [X₁-1]

MPRF for transition t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(X₂, X₀, X₂, X₃, X₄) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1+X₀ ≤ X₃ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃+1 {O(n)}

MPRF:

• eval_foo_2: [X₁-1]
• eval_foo_3: [X₁-1]
• eval_foo_bb1_in: [X₁-1]
• eval_foo_bb2_in: [X₁-1]

All Bounds

Timebounds

Overall timebound:4⋅X₃+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₃+1 {O(n)}
t₇: X₃+1 {O(n)}
t₈: X₃+1 {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₃+1 {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: X₃+1 {O(n)}
t₇: X₃+1 {O(n)}
t₈: X₃+1 {O(n)}
t₉: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₀, X₄: X₄ {O(n)}
t₁, X₀: X₄ {O(n)}
t₁, X₁: X₃ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₂, X₃: X₃ {O(n)}
t₂, X₄: X₄ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₃, X₄: 2⋅X₄ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₅, X₃: X₃ {O(n)}
t₅, X₄: X₄ {O(n)}
t₇, X₃: X₃ {O(n)}
t₇, X₄: X₄ {O(n)}
t₈, X₃: X₃ {O(n)}
t₈, X₄: X₄ {O(n)}
t₉, X₃: 4⋅X₃ {O(n)}
t₉, X₄: 4⋅X₄ {O(n)}