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_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, 0, X₂, X₃) :|: X₀ ≤ 4
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: 5 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 9 ∧ 3 ≤ X₀
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 2
t₆: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: 10 ≤ X₁
t₇: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, 1+X₁, X₂, X₃)
t₈: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃)
t₉: eval_foo_bb5_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

Eliminate variables [X₃] that do not contribute to the problem

Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 for location eval_foo_bb2_in

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₂ ≤ X₀ ∧ 5 ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ 4 ∧ X₂ ≤ 4+X₁ ∧ X₁+X₂ ≤ 13 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 9 ∧ X₁ ≤ 6+X₀ ∧ X₀+X₁ ≤ 13 ∧ 0 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ X₀ ≤ 4+X₁ ∧ X₀ ≤ 4 ∧ 3 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₂ ≤ 4 ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ 0 ≤ X₁ ∧ X₀ ≤ 2+X₁ ∧ X₀ ≤ 4 for location eval_foo_bb4_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₉: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂)
t₂₀: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 0, X₂) :|: X₀ ≤ 4 ∧ X₂ ≤ X₀
t₂₁: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: 5 ≤ X₀ ∧ X₂ ≤ X₀
t₂₂: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 9 ∧ 3 ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₃: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 10 ≤ X₁ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₅: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 1+X₁, X₂) :|: X₀+X₁ ≤ 13 ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 6+X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₆: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₀ ≤ 2+X₁ ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁
t₂₇: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: 5 ≤ X₀ ∧ X₂ ≤ X₀
t₂₈: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂₀: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 0, X₂) :|: X₀ ≤ 4 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_foo_bb1_in: [5-X₀]
• eval_foo_bb2_in: [4-X₀]
• eval_foo_bb3_in: [4-X₀]
• eval_foo_bb4_in: [4-X₀]

MPRF for transition t₂₃: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: X₀ ≤ 2 ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_foo_bb1_in: [5-X₀]
• eval_foo_bb2_in: [5-X₀]
• eval_foo_bb3_in: [5-X₀]
• eval_foo_bb4_in: [4-X₀]

MPRF for transition t₂₄: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 10 ≤ X₁ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_foo_bb1_in: [5-X₀]
• eval_foo_bb2_in: [5-X₀]
• eval_foo_bb3_in: [5-X₀]
• eval_foo_bb4_in: [4-X₀]

MPRF for transition t₂₆: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₂ ≤ 4 ∧ X₀ ≤ 2+X₁ ∧ X₂ ≤ 2+X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂+5 {O(n)}

MPRF:

• eval_foo_bb1_in: [5-X₀]
• eval_foo_bb2_in: [5-X₀]
• eval_foo_bb3_in: [5-X₀]
• eval_foo_bb4_in: [5-X₀]

MPRF for transition t₂₂: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₁ ≤ 9 ∧ 3 ≤ X₀ ∧ X₀+X₂ ≤ 8 ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

10⋅X₂+60 {O(n)}

MPRF:

• eval_foo_bb1_in: [10]
• eval_foo_bb2_in: [10-X₁]
• eval_foo_bb3_in: [9-X₁]
• eval_foo_bb4_in: [-X₁]

MPRF for transition t₂₅: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 1+X₁, X₂) :|: X₀+X₁ ≤ 13 ∧ X₁+X₂ ≤ 13 ∧ X₁ ≤ 9 ∧ X₀+X₂ ≤ 8 ∧ X₁ ≤ 6+X₀ ∧ X₀ ≤ 4 ∧ X₀ ≤ 4+X₁ ∧ X₂ ≤ 4+X₁ ∧ X₂ ≤ 4 ∧ 3 ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ X₂ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

10⋅X₂+60 {O(n)}

MPRF:

• eval_foo_bb1_in: [10]
• eval_foo_bb2_in: [10-X₁]
• eval_foo_bb3_in: [10-X₁]
• eval_foo_bb4_in: [10-X₁]

All Bounds

Timebounds

Overall timebound:24⋅X₂+144 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+5 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 10⋅X₂+60 {O(n)}
t₂₃: X₂+5 {O(n)}
t₂₄: X₂+5 {O(n)}
t₂₅: 10⋅X₂+60 {O(n)}
t₂₆: X₂+5 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Costbounds

Overall costbound: 24⋅X₂+144 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: X₂+5 {O(n)}
t₂₁: 1 {O(1)}
t₂₂: 10⋅X₂+60 {O(n)}
t₂₃: X₂+5 {O(n)}
t₂₄: X₂+5 {O(n)}
t₂₅: 10⋅X₂+60 {O(n)}
t₂₆: X₂+5 {O(n)}
t₂₇: 1 {O(1)}
t₂₈: 1 {O(1)}

Sizebounds

t₁₉, X₀: X₂ {O(n)}
t₁₉, X₁: X₁ {O(n)}
t₁₉, X₂: X₂ {O(n)}
t₂₀, X₀: 2⋅X₂+9 {O(n)}
t₂₀, X₁: 0 {O(1)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: 3⋅X₂+9 {O(n)}
t₂₁, X₁: X₁+10 {O(n)}
t₂₁, X₂: 2⋅X₂ {O(n)}
t₂₂, X₀: 4 {O(1)}
t₂₂, X₁: 9 {O(1)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: 2⋅X₂+9 {O(n)}
t₂₃, X₁: 0 {O(1)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 4 {O(1)}
t₂₄, X₁: 10 {O(1)}
t₂₄, X₂: X₂ {O(n)}
t₂₅, X₀: 4 {O(1)}
t₂₅, X₁: 10 {O(1)}
t₂₅, X₂: X₂ {O(n)}
t₂₆, X₀: 2⋅X₂+9 {O(n)}
t₂₆, X₁: 10 {O(1)}
t₂₆, X₂: X₂ {O(n)}
t₂₇, X₀: 3⋅X₂+9 {O(n)}
t₂₇, X₁: X₁+10 {O(n)}
t₂₇, X₂: 2⋅X₂ {O(n)}
t₂₈, X₀: X₀ {O(n)}
t₂₈, X₁: X₁ {O(n)}
t₂₈, X₂: X₂ {O(n)}