KoAT2 Proof Output

Initial Problem

Start: eval_t20_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_t20_bb0_in, eval_t20_bb1_in, eval_t20_bb2_in, eval_t20_bb3_in, eval_t20_bb4_in, eval_t20_bb5_in, eval_t20_start, eval_t20_stop
Transitions:
t₁: eval_t20_bb0_in(X₀, X₁, X₂, X₃) → eval_t20_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_t20_bb1_in(X₀, X₁, X₂, X₃) → eval_t20_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃
t₃: eval_t20_bb1_in(X₀, X₁, X₂, X₃) → eval_t20_bb3_in(X₀, X₃, X₂, X₃) :|: X₃ ≤ X₀
t₄: eval_t20_bb2_in(X₀, X₁, X₂, X₃) → eval_t20_bb1_in(1+X₀, X₁, X₂, X₃)
t₅: eval_t20_bb3_in(X₀, X₁, X₂, X₃) → eval_t20_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀
t₆: eval_t20_bb3_in(X₀, X₁, X₂, X₃) → eval_t20_bb5_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁
t₇: eval_t20_bb4_in(X₀, X₁, X₂, X₃) → eval_t20_bb3_in(X₀, 1+X₁, X₂, X₃)
t₈: eval_t20_bb5_in(X₀, X₁, X₂, X₃) → eval_t20_stop(X₀, X₁, X₂, X₃)
t₀: eval_t20_start(X₀, X₁, X₂, X₃) → eval_t20_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Found invariant X₂ ≤ X₀ for location eval_t20_bb1_in

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ for location eval_t20_bb3_in

Found invariant X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ 1+X₁ ≤ X₀ for location eval_t20_bb4_in

Found invariant 1+X₂ ≤ X₃ ∧ 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ for location eval_t20_bb2_in

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_t20_bb5_in

Found invariant X₃ ≤ X₁ ∧ X₃ ≤ X₀ ∧ X₂ ≤ X₁ ∧ X₂ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₀ ≤ X₁ for location eval_t20_stop

Problem after Preprocessing

Start: eval_t20_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars:
Locations: eval_t20_bb0_in, eval_t20_bb1_in, eval_t20_bb2_in, eval_t20_bb3_in, eval_t20_bb4_in, eval_t20_bb5_in, eval_t20_start, eval_t20_stop
Transitions:
t₁: eval_t20_bb0_in(X₀, X₁, X₂, X₃) → eval_t20_bb1_in(X₂, X₁, X₂, X₃)
t₂: eval_t20_bb1_in(X₀, X₁, X₂, X₃) → eval_t20_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀
t₃: eval_t20_bb1_in(X₀, X₁, X₂, X₃) → eval_t20_bb3_in(X₀, X₃, X₂, X₃) :|: X₃ ≤ X₀ ∧ X₂ ≤ X₀
t₄: eval_t20_bb2_in(X₀, X₁, X₂, X₃) → eval_t20_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀
t₅: eval_t20_bb3_in(X₀, X₁, X₂, X₃) → eval_t20_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₃ ≤ X₁
t₆: eval_t20_bb3_in(X₀, X₁, X₂, X₃) → eval_t20_bb5_in(X₀, X₁, X₂, X₃) :|: X₀ ≤ X₁ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₃ ≤ X₁
t₇: eval_t20_bb4_in(X₀, X₁, X₂, X₃) → eval_t20_bb3_in(X₀, 1+X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁
t₈: eval_t20_bb5_in(X₀, X₁, X₂, X₃) → eval_t20_stop(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₀ ≤ X₁ ∧ X₂ ≤ X₁ ∧ X₃ ≤ X₁
t₀: eval_t20_start(X₀, X₁, X₂, X₃) → eval_t20_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₂: eval_t20_bb1_in(X₀, X₁, X₂, X₃) → eval_t20_bb2_in(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_t20_bb1_in: [X₃-X₀]
• eval_t20_bb2_in: [X₃-1-X₀]

MPRF for transition t₄: eval_t20_bb2_in(X₀, X₁, X₂, X₃) → eval_t20_bb1_in(1+X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₃ ∧ 1+X₂ ≤ X₃ ∧ X₂ ≤ X₀ of depth 1:

new bound:

X₂+X₃ {O(n)}

MPRF:

• eval_t20_bb1_in: [X₃-X₀]
• eval_t20_bb2_in: [X₃-X₀]

MPRF for transition t₅: eval_t20_bb3_in(X₀, X₁, X₂, X₃) → eval_t20_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ X₁ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₀ ∧ X₃ ≤ X₁ of depth 1:

new bound:

3⋅X₂+3⋅X₃+1 {O(n)}

MPRF:

• eval_t20_bb3_in: [1+X₀-X₁]
• eval_t20_bb4_in: [X₀-X₁]

MPRF for transition t₇: eval_t20_bb4_in(X₀, X₁, X₂, X₃) → eval_t20_bb3_in(X₀, 1+X₁, X₂, X₃) :|: 1+X₁ ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ X₂ ≤ X₀ ∧ X₃ ≤ X₁ of depth 1:

new bound:

3⋅X₂+3⋅X₃ {O(n)}

MPRF:

• eval_t20_bb3_in: [X₀-X₁]
• eval_t20_bb4_in: [X₀-X₁]

All Bounds

Timebounds

Overall timebound:8⋅X₂+8⋅X₃+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₂+X₃ {O(n)}
t₅: 3⋅X₂+3⋅X₃+1 {O(n)}
t₆: 1 {O(1)}
t₇: 3⋅X₂+3⋅X₃ {O(n)}
t₈: 1 {O(1)}

Costbounds

Overall costbound: 8⋅X₂+8⋅X₃+6 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+X₃ {O(n)}
t₃: 1 {O(1)}
t₄: X₂+X₃ {O(n)}
t₅: 3⋅X₂+3⋅X₃+1 {O(n)}
t₆: 1 {O(1)}
t₇: 3⋅X₂+3⋅X₃ {O(n)}
t₈: 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: 2⋅X₂+X₃ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: 3⋅X₂+X₃ {O(n)}
t₃, X₁: 2⋅X₃ {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₃, X₃: 2⋅X₃ {O(n)}
t₄, X₀: 2⋅X₂+X₃ {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₂: X₂ {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₀: 3⋅X₂+X₃ {O(n)}
t₅, X₁: 3⋅X₂+5⋅X₃ {O(n)}
t₅, X₂: 2⋅X₂ {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₆, X₀: 2⋅X₃+6⋅X₂ {O(n)}
t₆, X₁: 3⋅X₂+7⋅X₃ {O(n)}
t₆, X₂: 4⋅X₂ {O(n)}
t₆, X₃: 4⋅X₃ {O(n)}
t₇, X₀: 3⋅X₂+X₃ {O(n)}
t₇, X₁: 3⋅X₂+5⋅X₃ {O(n)}
t₇, X₂: 2⋅X₂ {O(n)}
t₇, X₃: 2⋅X₃ {O(n)}
t₈, X₀: 2⋅X₃+6⋅X₂ {O(n)}
t₈, X₁: 3⋅X₂+7⋅X₃ {O(n)}
t₈, X₂: 4⋅X₂ {O(n)}
t₈, X₃: 4⋅X₃ {O(n)}