KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_bb1_in

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_stop

Found invariant X₁ ≤ X₃ ∧ X₀ ≤ X₂ for location eval_foo_.critedge_in

Found invariant 1 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 2 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_foo_bb3_in

Found invariant 2 ≤ X₃ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₁ ≤ X₃ ∧ 3 ≤ X₀+X₃ ∧ X₀ ≤ X₃ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 1 ≤ X₀ for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_foo_.critedge_in, eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₂: eval_foo_.critedge_in(X₀, X₁, X₂, X₃) → eval_foo_stop(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₃, X₂, X₃)
t₄: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₅: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 0 ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₆: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_.critedge_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃
t₁₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃
t₁₁: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃
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 ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁-1]
• eval_foo_bb2_in: [X₀+X₁-2]
• eval_foo_bb3_in: [X₀+X₁-2]
• eval_foo_bb4_in: [X₀+X₁-2]

MPRF for transition t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂+X₃+1 {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀+X₁-1]
• eval_foo_bb2_in: [X₀+X₁-2]
• eval_foo_bb3_in: [X₀+X₁-2]
• eval_foo_bb4_in: [X₀+X₁-2]

MPRF for transition t₇: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁-1]
• eval_foo_bb4_in: [X₁]

MPRF for transition t₈: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₁ ∧ 1 ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁+X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀-1]

MPRF for transition t₉: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁-1, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₃ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁+X₃ ∧ 2 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₃ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₁]
• eval_foo_bb2_in: [X₁]
• eval_foo_bb3_in: [X₁]
• eval_foo_bb4_in: [X₁]

MPRF for transition t₁₀: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃ of depth 1:

new bound:

X₂ {O(n)}

MPRF:

• eval_foo_bb1_in: [X₀]
• eval_foo_bb2_in: [X₀]
• eval_foo_bb3_in: [X₀]
• eval_foo_bb4_in: [X₀]

knowledge_propagation leads to new time bound X₂ {O(n)} for transition t₁₁: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 1 ≤ X₀ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ 2 ≤ X₁ ∧ 2 ≤ X₃ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₃ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₂+X₃ ∧ 4 ≤ X₁+X₃ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ X₀ ≤ X₃ ∧ X₁ ≤ X₃

All Bounds

Timebounds

Overall timebound:4⋅X₃+5⋅X₂+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃+1 {O(n)}
t₃: X₂+X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₃ {O(n)}
t₈: X₂ {O(n)}
t₉: X₃ {O(n)}
t₁₀: X₂ {O(n)}
t₁₁: X₂ {O(n)}
t₁₂: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+5⋅X₂+8 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃+1 {O(n)}
t₃: X₂+X₃+1 {O(n)}
t₄: 1 {O(1)}
t₅: 1 {O(1)}
t₆: 1 {O(1)}
t₇: X₃ {O(n)}
t₈: X₂ {O(n)}
t₉: X₃ {O(n)}
t₁₀: X₂ {O(n)}
t₁₁: X₂ {O(n)}
t₁₂: 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: X₂ {O(n)}
t₂, X₁: X₃ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₂ {O(n)}
t₃, X₁: X₃ {O(n)}
t₃, X₂: X₂ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₀: 2⋅X₂ {O(n)}
t₄, X₁: 2⋅X₃ {O(n)}
t₄, X₂: 2⋅X₂ {O(n)}
t₄, X₃: 2⋅X₃ {O(n)}
t₅, X₀: 2⋅X₂ {O(n)}
t₅, X₁: 2⋅X₃ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₆, X₀: 2⋅X₂ {O(n)}
t₆, X₁: 2⋅X₃ {O(n)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}
t₇, X₀: X₂ {O(n)}
t₇, X₁: X₃ {O(n)}
t₇, X₂: X₂ {O(n)}
t₇, X₃: X₃ {O(n)}
t₈, X₀: X₂ {O(n)}
t₈, X₁: X₃ {O(n)}
t₈, X₂: X₂ {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₀: X₂ {O(n)}
t₉, X₁: X₃ {O(n)}
t₉, X₂: X₂ {O(n)}
t₉, X₃: X₃ {O(n)}
t₁₀, X₀: X₂ {O(n)}
t₁₀, X₁: X₃ {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₀, X₃: X₃ {O(n)}
t₁₁, X₀: X₂ {O(n)}
t₁₁, X₁: X₃ {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₁, X₃: X₃ {O(n)}
t₁₂, X₀: 6⋅X₂ {O(n)}
t₁₂, X₁: 6⋅X₃ {O(n)}
t₁₂, X₂: 6⋅X₂ {O(n)}
t₁₂, X₃: 6⋅X₃ {O(n)}