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_start, eval_foo_stop
Transitions:
t₁: eval_foo_bb0_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(0, 0, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, 1+X₁, X₂, X₃) :|: X₀ ≤ 50
t₅: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁-1, X₂, X₃) :|: 51 ≤ X₀
t₆: eval_foo_bb3_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 [X₂; X₃] that do not contribute to the problem

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_foo_bb1_in

Found invariant 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_stop

Found invariant 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_foo_bb3_in

Problem after Preprocessing

Start: eval_foo_start
Program_Vars: X₀, X₁
Temp_Vars:
Locations: eval_foo_bb0_in, eval_foo_bb1_in, eval_foo_bb2_in, eval_foo_bb3_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₃: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(0, 0)
t₁₄: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 0 ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₁₅: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₁₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, 1+X₁) :|: X₀ ≤ 50 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₇: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁-1) :|: 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₈: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 0 ≤ 1+X₁ ∧ 1 ≤ X₀ ∧ 1+X₁ ≤ 0 ∧ 2+X₁ ≤ X₀ ∧ 0 ≤ X₀+X₁
t₁₉: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

MPRF for transition t₁₆: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, 1+X₁) :|: X₀ ≤ 50 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

51 {O(1)}

MPRF:

• eval_foo_bb1_in: [51-X₀]
• eval_foo_bb2_in: [51-X₀]

MPRF for transition t₁₄: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 0 ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ of depth 1:

new bound:

2653 {O(1)}

MPRF:

• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [X₁]

MPRF for transition t₁₇: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁-1) :|: 51 ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

2653 {O(1)}

MPRF:

• eval_foo_bb1_in: [1+X₁]
• eval_foo_bb2_in: [1+X₁]

All Bounds

Timebounds

Overall timebound:5361 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 2653 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 51 {O(1)}
t₁₇: 2653 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Costbounds

Overall costbound: 5361 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 2653 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 51 {O(1)}
t₁₇: 2653 {O(1)}
t₁₈: 1 {O(1)}
t₁₉: 1 {O(1)}

Sizebounds

t₁₃, X₀: 0 {O(1)}
t₁₃, X₁: 0 {O(1)}
t₁₄, X₀: 106 {O(1)}
t₁₄, X₁: 52 {O(1)}
t₁₅, X₀: 108 {O(1)}
t₁₅, X₁: 1 {O(1)}
t₁₆, X₀: 51 {O(1)}
t₁₆, X₁: 51 {O(1)}
t₁₇, X₀: 108 {O(1)}
t₁₇, X₁: 52 {O(1)}
t₁₈, X₀: 108 {O(1)}
t₁₈, X₁: 1 {O(1)}
t₁₉, X₀: X₀ {O(n)}
t₁₉, X₁: X₁ {O(n)}