KoAT2 Proof Output

Initial Problem

Start: eval_t19_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_t19_bb0_in, eval_t19_bb1_in, eval_t19_bb2_in, eval_t19_bb3_in, eval_t19_bb4_in, eval_t19_bb5_in, eval_t19_bb6_in, eval_t19_start, eval_t19_stop
Transitions:
t₁: eval_t19_bb0_in(X₀, X₁, X₂, X₃) → eval_t19_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_t19_bb1_in(X₀, X₁, X₂, X₃) → eval_t19_bb2_in(X₀, X₁, X₂, X₃) :|: 101 ≤ X₀
t₃: eval_t19_bb1_in(X₀, X₁, X₂, X₃) → eval_t19_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 100
t₄: eval_t19_bb2_in(X₀, X₁, X₂, X₃) → eval_t19_bb1_in(X₀-1, X₁, X₂, X₃)
t₅: eval_t19_bb3_in(X₀, X₁, X₂, X₃) → eval_t19_bb4_in(X₀, 50+X₀+X₃, X₂, X₃)
t₆: eval_t19_bb4_in(X₀, X₁, X₂, X₃) → eval_t19_bb5_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁
t₇: eval_t19_bb4_in(X₀, X₁, X₂, X₃) → eval_t19_bb6_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0
t₈: eval_t19_bb5_in(X₀, X₁, X₂, X₃) → eval_t19_bb4_in(X₀, X₁-1, X₂, X₃)
t₉: eval_t19_bb6_in(X₀, X₁, X₂, X₃) → eval_t19_stop(X₀, X₁, X₂, X₃)
t₀: eval_t19_start(X₀, X₁, X₂, X₃) → eval_t19_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₁ ≤ 150+X₃ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ 0 ∧ X₀+X₁ ≤ 99 ∧ X₀ ≤ 100 for location eval_t19_bb6_in

Found invariant X₀ ≤ X₂ ∧ X₀ ≤ 100 for location eval_t19_bb3_in

Found invariant X₀ ≤ X₂ for location eval_t19_bb1_in

Found invariant 0 ≤ 150+X₃ ∧ 0 ≤ 150+X₁+X₃ ∧ X₁ ≤ 150+X₃ ∧ X₀ ≤ 250+X₃ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ X₀ ≤ 100+X₁ ∧ X₀ ≤ 100 for location eval_t19_bb5_in

Found invariant 101 ≤ X₂ ∧ 202 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 101 ≤ X₀ for location eval_t19_bb2_in

Found invariant X₁ ≤ 150+X₃ ∧ X₀ ≤ X₂ ∧ X₀ ≤ 100 for location eval_t19_bb4_in

Found invariant X₁ ≤ 150+X₃ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ 0 ∧ X₀+X₁ ≤ 99 ∧ X₀ ≤ 100 for location eval_t19_stop

Problem after Preprocessing

Start: eval_t19_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_t19_bb0_in, eval_t19_bb1_in, eval_t19_bb2_in, eval_t19_bb3_in, eval_t19_bb4_in, eval_t19_bb5_in, eval_t19_bb6_in, eval_t19_start, eval_t19_stop
Transitions:
t₁: eval_t19_bb0_in(X₀, X₁, X₂, X₃) → eval_t19_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_t19_bb1_in(X₀, X₁, X₂, X₃) → eval_t19_bb2_in(X₀, X₁, X₂, X₃) :|: 101 ≤ X₀ ∧ X₀ ≤ X₂
t₃: eval_t19_bb1_in(X₀, X₁, X₂, X₃) → eval_t19_bb3_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ 100 ∧ X₀ ≤ X₂
t₄: eval_t19_bb2_in(X₀, X₁, X₂, X₃) → eval_t19_bb1_in(X₀-1, X₁, X₂, X₃) :|: 101 ≤ X₀ ∧ 101 ≤ X₂ ∧ 202 ≤ X₀+X₂ ∧ X₀ ≤ X₂
t₅: eval_t19_bb3_in(X₀, X₁, X₂, X₃) → eval_t19_bb4_in(X₀, 50+X₀+X₃, X₂, X₃) :|: X₀ ≤ 100 ∧ X₀ ≤ X₂
t₆: eval_t19_bb4_in(X₀, X₁, X₂, X₃) → eval_t19_bb5_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ X₁ ≤ 150+X₃ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₂
t₇: eval_t19_bb4_in(X₀, X₁, X₂, X₃) → eval_t19_bb6_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ 0 ∧ X₁ ≤ 150+X₃ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₂
t₈: eval_t19_bb5_in(X₀, X₁, X₂, X₃) → eval_t19_bb4_in(X₀, X₁-1, X₂, X₃) :|: X₀ ≤ 250+X₃ ∧ 0 ≤ 150+X₁+X₃ ∧ X₁ ≤ 150+X₃ ∧ 0 ≤ 150+X₃ ∧ X₀ ≤ 100 ∧ X₀ ≤ 100+X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₉: eval_t19_bb6_in(X₀, X₁, X₂, X₃) → eval_t19_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ 150+X₃ ∧ X₀ ≤ 100 ∧ X₀+X₁ ≤ 99 ∧ 1+X₁ ≤ 0 ∧ X₀ ≤ X₂
t₀: eval_t19_start(X₀, X₁, X₂, X₃) → eval_t19_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_t19_bb1_in(X₀, X₁, X₂, X₃) → eval_t19_bb2_in(X₀, X₁, X₂, X₃) :|: 101 ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+100 {O(n)}

MPRF:

• eval_t19_bb1_in: [X₀-100]
• eval_t19_bb2_in: [X₀-101]

MPRF for transition t₄: eval_t19_bb2_in(X₀, X₁, X₂, X₃) → eval_t19_bb1_in(X₀-1, X₁, X₂, X₃) :|: 101 ≤ X₀ ∧ 101 ≤ X₂ ∧ 202 ≤ X₀+X₂ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+100 {O(n)}

MPRF:

• eval_t19_bb1_in: [X₀-100]
• eval_t19_bb2_in: [X₀-100]

MPRF for transition t₆: eval_t19_bb4_in(X₀, X₁, X₂, X₃) → eval_t19_bb5_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ X₁ ≤ 150+X₃ ∧ X₀ ≤ 100 ∧ X₀ ≤ X₂ of depth 1:

new bound:

2⋅X₂+2⋅X₃+51 {O(n)}

MPRF:

• eval_t19_bb4_in: [1+X₁]
• eval_t19_bb5_in: [X₁]

MPRF for transition t₈: eval_t19_bb5_in(X₀, X₁, X₂, X₃) → eval_t19_bb4_in(X₀, X₁-1, X₂, X₃) :|: X₀ ≤ 250+X₃ ∧ 0 ≤ 150+X₁+X₃ ∧ X₁ ≤ 150+X₃ ∧ 0 ≤ 150+X₃ ∧ X₀ ≤ 100 ∧ X₀ ≤ 100+X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

2⋅X₂+2⋅X₃+51 {O(n)}

MPRF:

• eval_t19_bb4_in: [1+X₁]
• eval_t19_bb5_in: [1+X₁]

All Bounds

Timebounds

Overall timebound:4⋅X₃+6⋅X₂+308 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+100 {O(n)}
t₃: 1 {O(1)}
t₄: X₂+100 {O(n)}
t₅: 1 {O(1)}
t₆: 2⋅X₂+2⋅X₃+51 {O(n)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+2⋅X₃+51 {O(n)}
t₉: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₃+6⋅X₂+308 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+100 {O(n)}
t₃: 1 {O(1)}
t₄: X₂+100 {O(n)}
t₅: 1 {O(1)}
t₆: 2⋅X₂+2⋅X₃+51 {O(n)}
t₇: 1 {O(1)}
t₈: 2⋅X₂+2⋅X₃+51 {O(n)}
t₉: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₀, X₃: X₃ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₂ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 2⋅X₂ {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: X₂ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 2⋅X₂ {O(n)}
t₅, X₁: 2⋅X₂+2⋅X₃+50 {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₆, X₀: 2⋅X₂ {O(n)}
t₆, X₁: 2⋅X₂+2⋅X₃+51 {O(n)}
t₆, X₂: 2⋅X₂ {O(n)}
t₆, X₃: 2⋅X₃ {O(n)}
t₇, X₀: 4⋅X₂ {O(n)}
t₇, X₁: 4⋅X₂+4⋅X₃+101 {O(n)}
t₇, X₂: 4⋅X₂ {O(n)}
t₇, X₃: 4⋅X₃ {O(n)}
t₈, X₀: 2⋅X₂ {O(n)}
t₈, X₁: 2⋅X₂+2⋅X₃+51 {O(n)}
t₈, X₂: 2⋅X₂ {O(n)}
t₈, X₃: 2⋅X₃ {O(n)}
t₉, X₀: 4⋅X₂ {O(n)}
t₉, X₁: 4⋅X₂+4⋅X₃+101 {O(n)}
t₉, X₂: 4⋅X₂ {O(n)}
t₉, X₃: 4⋅X₃ {O(n)}