KoAT2 Proof Output

Initial Problem

Start: eval_foo_start
Program_Vars: X₀, X₁, X₂, X₃, X₄
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, X₄) → eval_foo_bb1_in(0, 3, X₂, X₃, X₄)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, X₁, X₃, X₄) :|: X₀ ≤ 9
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) :|: 10 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) :|: X₂ ≤ 11
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) :|: 12 ≤ X₂
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb2_in(X₀, X₁, 1+X₂, X₃, X₄)
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb1_in(1+X₀, X₂, X₂, X₃, X₄)
t₈: eval_foo_bb5_in(X₀, X₁, X₂, X₃, X₄) → eval_foo_stop(X₀, X₁, X₂, X₃, X₄)
t₀: eval_foo_start(X₀, X₁, X₂, X₃, X₄) → eval_foo_bb0_in(X₀, X₁, X₂, X₃, X₄)

Preprocessing

Eliminate variables [X₃; X₄] that do not contribute to the problem

Found invariant 3 ≤ X₁ ∧ 13 ≤ X₀+X₁ ∧ 10 ≤ X₀ for location eval_foo_bb5_in

Found invariant 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in

Found invariant 3 ≤ X₁ ∧ 13 ≤ X₀+X₁ ∧ 10 ≤ X₀ for location eval_foo_stop

Found invariant X₂ ≤ 11 ∧ X₂ ≤ 8+X₁ ∧ X₁+X₂ ≤ 22 ∧ X₂ ≤ 11+X₀ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ 11 ∧ X₁ ≤ 11+X₀ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb3_in

Found invariant 12 ≤ X₂ ∧ 15 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 12 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀ 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(0, 3, X₂)
t₁₈: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, X₁) :|: X₀ ≤ 9 ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁ ∧ 0 ≤ X₀
t₁₉: eval_foo_bb1_in(X₀, X₁, X₂) → eval_foo_bb5_in(X₀, X₁, X₂) :|: 10 ≤ X₀ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁ ∧ 0 ≤ X₀
t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ 11 ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₂₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 12 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₂₂: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, 1+X₂) :|: X₁+X₂ ≤ 22 ∧ X₁ ≤ 11+X₀ ∧ X₂ ≤ 11+X₀ ∧ X₁ ≤ 11 ∧ X₂ ≤ 11 ∧ X₂ ≤ 8+X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₂₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₂, X₂) :|: 3 ≤ X₀+X₁ ∧ 3 ≤ X₁ ∧ 12 ≤ X₀+X₂ ∧ 12 ≤ X₂ ∧ 15 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂
t₂₄: eval_foo_bb5_in(X₀, X₁, X₂) → eval_foo_stop(X₀, X₁, X₂) :|: 3 ≤ X₁ ∧ 10 ≤ X₀ ∧ 13 ≤ X₀+X₁
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, X₁, X₁) :|: X₀ ≤ 9 ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₁ ∧ 0 ≤ X₀ of depth 1:

new bound:

10 {O(1)}

MPRF:

• eval_foo_bb1_in: [10-X₀]
• eval_foo_bb2_in: [9-X₀]
• eval_foo_bb3_in: [9-X₀]
• eval_foo_bb4_in: [9-X₀]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb3_in(X₀, X₁, X₂) :|: X₂ ≤ 11 ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

15 {O(1)}

MPRF:

• eval_foo_bb1_in: [12-X₁]
• eval_foo_bb2_in: [12-X₂]
• eval_foo_bb3_in: [11-X₂]
• eval_foo_bb4_in: [12-X₂]

MPRF for transition t₂₂: eval_foo_bb3_in(X₀, X₁, X₂) → eval_foo_bb2_in(X₀, X₁, 1+X₂) :|: X₁+X₂ ≤ 22 ∧ X₁ ≤ 11+X₀ ∧ X₂ ≤ 11+X₀ ∧ X₁ ≤ 11 ∧ X₂ ≤ 11 ∧ X₂ ≤ 8+X₁ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂ of depth 1:

new bound:

23 {O(1)}

MPRF:

• eval_foo_bb1_in: [20-X₁]
• eval_foo_bb2_in: [20-X₂]
• eval_foo_bb3_in: [20-X₂]
• eval_foo_bb4_in: [20-X₂]

knowledge_propagation leads to new time bound 33 {O(1)} for transition t₂₁: eval_foo_bb2_in(X₀, X₁, X₂) → eval_foo_bb4_in(X₀, X₁, X₂) :|: 12 ≤ X₂ ∧ 3 ≤ X₀+X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁ ∧ 3 ≤ X₂ ∧ 6 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂

knowledge_propagation leads to new time bound 33 {O(1)} for transition t₂₃: eval_foo_bb4_in(X₀, X₁, X₂) → eval_foo_bb1_in(1+X₀, X₂, X₂) :|: 3 ≤ X₀+X₁ ∧ 3 ≤ X₁ ∧ 12 ≤ X₀+X₂ ∧ 12 ≤ X₂ ∧ 15 ≤ X₁+X₂ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₂

All Bounds

Timebounds

Overall timebound:118 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 10 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 15 {O(1)}
t₂₁: 33 {O(1)}
t₂₂: 23 {O(1)}
t₂₃: 33 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 118 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 10 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 15 {O(1)}
t₂₁: 33 {O(1)}
t₂₂: 23 {O(1)}
t₂₃: 33 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: 0 {O(1)}
t₁₇, X₁: 3 {O(1)}
t₁₇, X₂: X₂ {O(n)}
t₁₈, X₀: 9 {O(1)}
t₁₈, X₁: 18 {O(1)}
t₁₈, X₂: 15 {O(1)}
t₁₉, X₀: 19 {O(1)}
t₁₉, X₁: 15 {O(1)}
t₁₉, X₂: 15 {O(1)}
t₂₀, X₀: 9 {O(1)}
t₂₀, X₁: 11 {O(1)}
t₂₀, X₂: 11 {O(1)}
t₂₁, X₀: 18 {O(1)}
t₂₁, X₁: 29 {O(1)}
t₂₁, X₂: 15 {O(1)}
t₂₂, X₀: 9 {O(1)}
t₂₂, X₁: 11 {O(1)}
t₂₂, X₂: 12 {O(1)}
t₂₃, X₀: 19 {O(1)}
t₂₃, X₁: 15 {O(1)}
t₂₃, X₂: 15 {O(1)}
t₂₄, X₀: 19 {O(1)}
t₂₄, X₁: 15 {O(1)}
t₂₄, X₂: 15 {O(1)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}
t₂₅, X₂: X₂ {O(n)}