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(0, X₁, X₂, X₃)
t₂: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb2_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 99
t₃: eval_foo_bb1_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 5, X₂, X₃) :|: 100 ≤ X₀
t₄: eval_foo_bb2_in(X₀, X₁, X₂, X₃) → eval_foo_bb1_in(1+X₀, X₁, X₂, X₃)
t₅: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb4_in(X₀, X₁, X₂, X₃) :|: X₁ ≤ 20
t₆: eval_foo_bb3_in(X₀, X₁, X₂, X₃) → eval_foo_bb5_in(X₀, X₁, X₂, X₃) :|: 21 ≤ X₁
t₇: eval_foo_bb4_in(X₀, X₁, X₂, X₃) → eval_foo_bb3_in(X₀, 3+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 [X₂; X₃] that do not contribute to the problem

Found invariant 21 ≤ X₁ ∧ 121 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location eval_foo_bb5_in

Found invariant X₀ ≤ 99 ∧ 0 ≤ X₀ for location eval_foo_bb2_in

Found invariant 0 ≤ X₀ for location eval_foo_bb1_in

Found invariant 21 ≤ X₁ ∧ 121 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location eval_foo_stop

Found invariant 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location eval_foo_bb3_in

Found invariant X₁ ≤ 20 ∧ 80+X₁ ≤ X₀ ∧ 5 ≤ X₁ ∧ 105 ≤ X₀+X₁ ∧ 100 ≤ X₀ for location eval_foo_bb4_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_bb4_in, eval_foo_bb5_in, eval_foo_start, eval_foo_stop
Transitions:
t₁₇: eval_foo_bb0_in(X₀, X₁) → eval_foo_bb1_in(0, X₁)
t₁₈: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: X₀ ≤ 99 ∧ 0 ≤ X₀
t₁₉: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb3_in(X₀, 5) :|: 100 ≤ X₀ ∧ 0 ≤ X₀
t₂₀: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 99 ∧ 0 ≤ X₀
t₂₁: eval_foo_bb3_in(X₀, X₁) → eval_foo_bb4_in(X₀, X₁) :|: X₁ ≤ 20 ∧ 5 ≤ X₁ ∧ 100 ≤ X₀ ∧ 105 ≤ X₀+X₁
t₂₂: eval_foo_bb3_in(X₀, X₁) → eval_foo_bb5_in(X₀, X₁) :|: 21 ≤ X₁ ∧ 5 ≤ X₁ ∧ 100 ≤ X₀ ∧ 105 ≤ X₀+X₁
t₂₃: eval_foo_bb4_in(X₀, X₁) → eval_foo_bb3_in(X₀, 3+X₁) :|: X₁ ≤ 20 ∧ 5 ≤ X₁ ∧ 80+X₁ ≤ X₀ ∧ 100 ≤ X₀ ∧ 105 ≤ X₀+X₁
t₂₄: eval_foo_bb5_in(X₀, X₁) → eval_foo_stop(X₀, X₁) :|: 21 ≤ X₁ ∧ 100 ≤ X₀ ∧ 121 ≤ X₀+X₁
t₂₅: eval_foo_start(X₀, X₁) → eval_foo_bb0_in(X₀, X₁)

MPRF for transition t₁₈: eval_foo_bb1_in(X₀, X₁) → eval_foo_bb2_in(X₀, X₁) :|: X₀ ≤ 99 ∧ 0 ≤ X₀ of depth 1:

new bound:

100 {O(1)}

MPRF:

• eval_foo_bb1_in: [100-X₀]
• eval_foo_bb2_in: [99-X₀]

MPRF for transition t₂₀: eval_foo_bb2_in(X₀, X₁) → eval_foo_bb1_in(1+X₀, X₁) :|: X₀ ≤ 99 ∧ 0 ≤ X₀ of depth 1:

new bound:

100 {O(1)}

MPRF:

• eval_foo_bb1_in: [100-X₀]
• eval_foo_bb2_in: [100-X₀]

MPRF for transition t₂₁: eval_foo_bb3_in(X₀, X₁) → eval_foo_bb4_in(X₀, X₁) :|: X₁ ≤ 20 ∧ 5 ≤ X₁ ∧ 100 ≤ X₀ ∧ 105 ≤ X₀+X₁ of depth 1:

new bound:

26 {O(1)}

MPRF:

• eval_foo_bb3_in: [21-X₁]
• eval_foo_bb4_in: [18-X₁]

MPRF for transition t₂₃: eval_foo_bb4_in(X₀, X₁) → eval_foo_bb3_in(X₀, 3+X₁) :|: X₁ ≤ 20 ∧ 5 ≤ X₁ ∧ 80+X₁ ≤ X₀ ∧ 100 ≤ X₀ ∧ 105 ≤ X₀+X₁ of depth 1:

new bound:

28 {O(1)}

MPRF:

• eval_foo_bb3_in: [23-X₁]
• eval_foo_bb4_in: [21-X₁]

All Bounds

Timebounds

Overall timebound:259 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 100 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 100 {O(1)}
t₂₁: 26 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 28 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Costbounds

Overall costbound: 259 {O(1)}
t₁₇: 1 {O(1)}
t₁₈: 100 {O(1)}
t₁₉: 1 {O(1)}
t₂₀: 100 {O(1)}
t₂₁: 26 {O(1)}
t₂₂: 1 {O(1)}
t₂₃: 28 {O(1)}
t₂₄: 1 {O(1)}
t₂₅: 1 {O(1)}

Sizebounds

t₁₇, X₀: 0 {O(1)}
t₁₇, X₁: X₁ {O(n)}
t₁₈, X₀: 99 {O(1)}
t₁₈, X₁: X₁ {O(n)}
t₁₉, X₀: 100 {O(1)}
t₁₉, X₁: 5 {O(1)}
t₂₀, X₀: 100 {O(1)}
t₂₀, X₁: X₁ {O(n)}
t₂₁, X₀: 100 {O(1)}
t₂₁, X₁: 20 {O(1)}
t₂₂, X₀: 100 {O(1)}
t₂₂, X₁: 23 {O(1)}
t₂₃, X₀: 100 {O(1)}
t₂₃, X₁: 23 {O(1)}
t₂₄, X₀: 100 {O(1)}
t₂₄, X₁: 23 {O(1)}
t₂₅, X₀: X₀ {O(n)}
t₂₅, X₁: X₁ {O(n)}