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(10000, 1, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(X₀-1, 1+X₁, X₂, X₃)
t₅: 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 ≤ 5000 ∧ 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 10001 ∧ 1 ≤ X₁ ∧ 10001 ≤ X₀+X₁ ∧ X₀ ≤ 9999+X₁ ∧ X₀ ≤ 10000 ∧ 5001 ≤ X₀ for location eval_foo_bb2_in

Found invariant X₀+X₁ ≤ 10001 ∧ 1 ≤ X₁ ∧ 10001 ≤ X₀+X₁ ∧ X₀ ≤ 9999+X₁ ∧ X₀ ≤ 10000 for location eval_foo_bb1_in

Found invariant X₀+X₁ ≤ 10001 ∧ 5001 ≤ X₁ ∧ 10001 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 5000 for location eval_foo_stop

Found invariant X₀+X₁ ≤ 10001 ∧ 5001 ≤ X₁ ∧ 10001 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 5000 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:
t₁₁: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(10000, 1)
t₁₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 10001 ∧ X₀ ≤ 10000 ∧ X₀ ≤ 9999+X₁ ∧ 1 ≤ X₁ ∧ 10001 ≤ X₀+X₁
t₁₃: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, X₁) :|: X₀ ≤ X₁ ∧ X₀+X₁ ≤ 10001 ∧ X₀ ≤ 10000 ∧ X₀ ≤ 9999+X₁ ∧ 1 ≤ X₁ ∧ 10001 ≤ X₀+X₁
t₁₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀-1, 1+X₁) :|: X₀+X₁ ≤ 10001 ∧ X₀ ≤ 10000 ∧ X₀ ≤ 9999+X₁ ∧ X₁ ≤ 5000 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 5001 ≤ X₀ ∧ 10001 ≤ X₀+X₁
t₁₅: eval_foo_bb3_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: X₀+X₁ ≤ 10001 ∧ X₀ ≤ 5000 ∧ 1+X₀ ≤ X₁ ∧ 5001 ≤ X₁ ∧ 10001 ≤ X₀+X₁
t₁₆: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

MPRF for transition t₁₂: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: 1+X₁ ≤ X₀ ∧ X₀+X₁ ≤ 10001 ∧ X₀ ≤ 10000 ∧ X₀ ≤ 9999+X₁ ∧ 1 ≤ X₁ ∧ 10001 ≤ X₀+X₁ of depth 1:

new bound:

10001 {O(1)}

MPRF:

• eval_foo_bb1_in: [10000-X₁]
• eval_foo_bb2_in: [9999-X₁]

MPRF for transition t₁₄: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(X₀-1, 1+X₁) :|: X₀+X₁ ≤ 10001 ∧ X₀ ≤ 10000 ∧ X₀ ≤ 9999+X₁ ∧ X₁ ≤ 5000 ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 5001 ≤ X₀ ∧ 10001 ≤ X₀+X₁ of depth 1:

new bound:

15000 {O(1)}

MPRF:

• eval_foo_bb1_in: [X₀-5000]
• eval_foo_bb2_in: [X₀-5000]

All Bounds

Timebounds

Overall timebound:25005 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 10001 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 15000 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Costbounds

Overall costbound: 25005 {O(1)}
t₁₁: 1 {O(1)}
t₁₂: 10001 {O(1)}
t₁₃: 1 {O(1)}
t₁₄: 15000 {O(1)}
t₁₅: 1 {O(1)}
t₁₆: 1 {O(1)}

Sizebounds

t₁₁, X₀: 10000 {O(1)}
t₁₁, X₁: 1 {O(1)}
t₁₂, X₀: 10000 {O(1)}
t₁₂, X₁: 10002 {O(1)}
t₁₃, X₀: 10000 {O(1)}
t₁₃, X₁: 5004 {O(1)}
t₁₄, X₀: 10000 {O(1)}
t₁₄, X₁: 5004 {O(1)}
t₁₅, X₀: 10000 {O(1)}
t₁₅, X₁: 5004 {O(1)}
t₁₆, X₀: X₀ {O(n)}
t₁₆, X₁: X₁ {O(n)}