KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 2 ≤ X₅ ∧ 3 ≤ X₁+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀ ≤ X₅ for location eval_foo_bb1_in

Found invariant X₀ ≤ X₅ for location eval_foo_stop

Found invariant X₀ ≤ X₅ for location eval_foo_.critedge_in

Found invariant 2 ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 3 ≤ X₁+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_3

Found invariant 2 ≤ X₅ ∧ 3 ≤ X₂+X₅ ∧ 1+X₂ ≤ X₅ ∧ 3 ≤ X₁+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_4

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄, X₅
Temp_Vars: nondef.0
Locations: eval_foo_.critedge_in, eval_foo_3, eval_foo_4, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₀: eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ X₅
t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_4(X₀, X₁, X₂, nondef.0, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₅ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₅ ∧ 3 ≤ X₂+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅
t₉: eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₅ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₅ ∧ 3 ≤ X₂+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₅, X₄, X₂, X₃, X₄, X₅)
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: X₀ ≤ 0 ∧ X₀ ≤ X₅
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+X₀ ≤ 2⋅X₁ ∧ X₀ ≤ X₅
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1+2⋅X₁ ≤ X₀ ∧ X₀ ≤ X₅
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 2⋅X₁ ≤ X₀ ∧ X₀ ≤ 2⋅X₁ ∧ X₀ ≤ X₅
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_3(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₅ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄, X₅)

MPRF for transition t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) :|: 1 ≤ X₀ ∧ 2⋅X₁ ≤ X₀ ∧ X₀ ≤ 2⋅X₁ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₅ {O(n)}

MPRF:

• eval_foo_3: [X₀-1]
• eval_foo_4: [X₂]
• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀-1]

MPRF for transition t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_3(X₀, X₁, X₀-1, X₃, X₄, X₅) :|: 1 ≤ X₁ ∧ 2 ≤ X₀ ∧ 2 ≤ X₅ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_3: [X₂-1]
• eval_foo_4: [X₂-1]
• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-1]

MPRF for transition t₈: eval_foo_3(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_4(X₀, X₁, X₂, nondef.0, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₅ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₅ ∧ 3 ≤ X₂+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_3: [X₂]
• eval_foo_4: [X₀-2]
• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-1]

MPRF for transition t₉: eval_foo_4(X₀, X₁, X₂, X₃, X₄, X₅) → eval_foo_bb1_in(X₂, X₃, X₂, X₃, X₄, X₅) :|: X₀ ≤ 1+X₂ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1+X₂ ≤ X₅ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₂ ∧ 2 ≤ X₅ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₅ ∧ 3 ≤ X₂+X₅ ∧ 4 ≤ X₀+X₅ ∧ X₀ ≤ X₅ of depth 1:

new bound:

X₅+1 {O(n)}

MPRF:

• eval_foo_3: [X₂]
• eval_foo_4: [X₂]
• eval_foo_bb1_in: [X₀-1]
• eval_foo_bb2_in: [X₀-1]

All Bounds

Timebounds

Overall timebound:4⋅X₅+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₅ {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: X₅+1 {O(n)}
t₈: X₅+1 {O(n)}
t₉: X₅+1 {O(n)}
t₁₀: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₅+9 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₅ {O(n)}
t₃: 1 {O(1)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: X₅+1 {O(n)}
t₈: X₅+1 {O(n)}
t₉: X₅+1 {O(n)}
t₁₀: 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)}
t₀, X₅: X₅ {O(n)}
t₁, X₀: X₅ {O(n)}
t₁, X₁: X₄ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₁, X₄: X₄ {O(n)}
t₁, X₅: X₅ {O(n)}
t₂, X₀: X₅ {O(n)}
t₂, X₂: X₂+X₅ {O(n)}
t₂, X₄: X₄ {O(n)}
t₂, X₅: X₅ {O(n)}
t₃, X₀: X₅ {O(n)}
t₃, X₁: X₄ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₃, X₄: X₄ {O(n)}
t₃, X₅: X₅ {O(n)}
t₄, X₀: 2⋅X₅ {O(n)}
t₄, X₂: X₂+X₅ {O(n)}
t₄, X₄: 2⋅X₄ {O(n)}
t₄, X₅: 2⋅X₅ {O(n)}
t₅, X₀: 2⋅X₅ {O(n)}
t₅, X₂: X₂+X₅ {O(n)}
t₅, X₄: 2⋅X₄ {O(n)}
t₅, X₅: 2⋅X₅ {O(n)}
t₆, X₀: X₅ {O(n)}
t₆, X₂: X₅ {O(n)}
t₆, X₄: X₄ {O(n)}
t₆, X₅: X₅ {O(n)}
t₈, X₀: X₅ {O(n)}
t₈, X₂: X₅ {O(n)}
t₈, X₄: X₄ {O(n)}
t₈, X₅: X₅ {O(n)}
t₉, X₀: X₅ {O(n)}
t₉, X₂: X₅ {O(n)}
t₉, X₄: X₄ {O(n)}
t₉, X₅: X₅ {O(n)}
t₁₀, X₀: 5⋅X₅ {O(n)}
t₁₀, X₂: 2⋅X₅+3⋅X₂ {O(n)}
t₁₀, X₄: 5⋅X₄ {O(n)}
t₁₀, X₅: 5⋅X₅ {O(n)}