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, 10, X₂, X₃) :|: X₀ ≤ 10
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: 11 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 2 ≤ X₁
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 1
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁-1, X₂, X₃)
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃)
t₈: 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 ∧ 11 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ X₀ ≤ 10 for location eval_foo_bb2_in

Found invariant X₂ ≤ X₀ for location eval_foo_bb1_in

Found invariant X₂ ≤ X₀ ∧ 11 ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ 10 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 20 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 10 ∧ X₀+X₁ ≤ 20 ∧ 2 ≤ X₁ ∧ X₀ ≤ 8+X₁ ∧ X₀ ≤ 10 for location eval_foo_bb3_in

Found invariant X₂ ≤ 10 ∧ X₁+X₂ ≤ 11 ∧ X₂ ≤ X₀ ∧ X₀+X₂ ≤ 20 ∧ X₁ ≤ 1 ∧ X₀+X₁ ≤ 11 ∧ X₀ ≤ 10 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:
t₁₇: eval_foo_bb0_in(X₀, X₁, X₂) → eval_foo_bb1_in(X₂, X₁, X₂)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: 11 ≤ X₀ ∧ X₂ ≤ X₀
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ X₀+X₁ ≤ 20 ∧ X₀+X₂ ≤ 20 ∧ X₁+X₂ ≤ 20 ∧ X₀ ≤ 10 ∧ X₁ ≤ 10 ∧ X₂ ≤ 10 ∧ X₂ ≤ X₀
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₀+X₁ ≤ 20 ∧ X₀+X₂ ≤ 20 ∧ X₁+X₂ ≤ 20 ∧ X₀ ≤ 10 ∧ X₁ ≤ 10 ∧ X₂ ≤ 10 ∧ X₂ ≤ X₀
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁-1, X₂) :|: X₀+X₁ ≤ 20 ∧ X₀+X₂ ≤ 20 ∧ X₁+X₂ ≤ 20 ∧ X₀ ≤ 10 ∧ X₁ ≤ 10 ∧ X₂ ≤ 10 ∧ X₀ ≤ 8+X₁ ∧ X₂ ≤ 8+X₁ ∧ 2 ≤ X₁ ∧ X₂ ≤ X₀
t₂₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: X₀+X₂ ≤ 20 ∧ X₀+X₁ ≤ 11 ∧ X₁+X₂ ≤ 11 ∧ X₀ ≤ 10 ∧ X₂ ≤ 10 ∧ X₁ ≤ 1 ∧ X₂ ≤ X₀
t₂₄: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: 11 ≤ X₀ ∧ X₂ ≤ X₀
t₂₅: eval_foo_start(X₀, X₁, X₂) → eval_foo_bb0_in(X₀, X₁, X₂)

MPRF for transition t₁₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, 10, X₂) :|: X₀ ≤ 10 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+11 {O(n)}

MPRF:

• eval_foo_bb1_in: [11-X₀]
• eval_foo_bb2_in: [10-X₀]
• eval_foo_bb3_in: [10-X₀]
• eval_foo_bb4_in: [10-X₀]

MPRF for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ X₀+X₁ ≤ 20 ∧ X₀+X₂ ≤ 20 ∧ X₁+X₂ ≤ 20 ∧ X₀ ≤ 10 ∧ X₁ ≤ 10 ∧ X₂ ≤ 10 ∧ X₂ ≤ X₀ of depth 1:

new bound:

2⋅X₂+21 {O(n)}

MPRF:

• eval_foo_bb1_in: [21-X₀-X₂]
• eval_foo_bb2_in: [21-X₀-X₂]
• eval_foo_bb3_in: [21-X₀-X₂]
• eval_foo_bb4_in: [20-X₀-X₂]

MPRF for transition t₂₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₁, X₂) :|: X₀+X₂ ≤ 20 ∧ X₀+X₁ ≤ 11 ∧ X₁+X₂ ≤ 11 ∧ X₀ ≤ 10 ∧ X₂ ≤ 10 ∧ X₁ ≤ 1 ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+11 {O(n)}

MPRF:

• eval_foo_bb1_in: [11-X₀]
• eval_foo_bb2_in: [11-X₀]
• eval_foo_bb3_in: [11-X₀]
• eval_foo_bb4_in: [11-X₀]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: 2 ≤ X₁ ∧ X₀+X₁ ≤ 20 ∧ X₀+X₂ ≤ 20 ∧ X₁+X₂ ≤ 20 ∧ X₀ ≤ 10 ∧ X₁ ≤ 10 ∧ X₂ ≤ 10 ∧ X₂ ≤ X₀ of depth 1:

new bound:

9⋅X₂+108 {O(n)}

MPRF:

• eval_foo_bb1_in: [9]
• eval_foo_bb2_in: [X₁-1]
• eval_foo_bb3_in: [X₁-2]
• eval_foo_bb4_in: [X₁-1]

MPRF for transition t₂₂: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁-1, X₂) :|: X₀+X₁ ≤ 20 ∧ X₀+X₂ ≤ 20 ∧ X₁+X₂ ≤ 20 ∧ X₀ ≤ 10 ∧ X₁ ≤ 10 ∧ X₂ ≤ 10 ∧ X₀ ≤ 8+X₁ ∧ X₂ ≤ 8+X₁ ∧ 2 ≤ X₁ ∧ X₂ ≤ X₀ of depth 1:

new bound:

17⋅X₂+204 {O(n)}

MPRF:

• eval_foo_bb1_in: [17]
• eval_foo_bb2_in: [7+X₁]
• eval_foo_bb3_in: [7+X₁]
• eval_foo_bb4_in: [7+X₁]

All Bounds

Timebounds

Overall timebound:30⋅X₂+359 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₂+11 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 9⋅X₂+108 {O(n)}
t₂₁: 2⋅X₂+21 {O(n)}
t₂₂: 17⋅X₂+204 {O(n)}
t₂₃: X₂+11 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 30⋅X₂+359 {O(n)}
t₁₇: 1 {O(1)}
t₁₈: X₂+11 {O(n)}
t₁₉: 1 {O(1)}
t₂₀: 9⋅X₂+108 {O(n)}
t₂₁: 2⋅X₂+21 {O(n)}
t₂₂: 17⋅X₂+204 {O(n)}
t₂₃: X₂+11 {O(n)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: X₂ {O(n)}
t₁₇, X₁: X₁ {O(n)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₀: 2⋅X₂+11 {O(n)}
t₁₈, X₁: 10 {O(1)}
t₁₈, X₂: X₂ {O(n)}
t₁₉, X₀: 3⋅X₂+11 {O(n)}
t₁₉, X₁: X₁+9 {O(n)}
t₁₉, X₂: 2⋅X₂ {O(n)}
t₂₀, X₀: 2⋅X₂+11 {O(n)}
t₂₀, X₁: 10 {O(1)}
t₂₀, X₂: X₂ {O(n)}
t₂₁, X₀: 2⋅X₂+11 {O(n)}
t₂₁, X₁: 9 {O(1)}
t₂₁, X₂: X₂ {O(n)}
t₂₂, X₀: 2⋅X₂+11 {O(n)}
t₂₂, X₁: 9 {O(1)}
t₂₂, X₂: X₂ {O(n)}
t₂₃, X₀: 2⋅X₂+11 {O(n)}
t₂₃, X₁: 9 {O(1)}
t₂₃, X₂: X₂ {O(n)}
t₂₄, X₀: 3⋅X₂+11 {O(n)}
t₂₄, X₁: X₁+9 {O(n)}
t₂₄, X₂: 2⋅X₂ {O(n)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}