KoAT2 Proof Output

Initial Problem

Start: eval_rsd_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: nondef.0
Locations: eval_rsd_3, eval_rsd_4, eval_rsd_bb0_in, eval_rsd_bb1_in, eval_rsd_bb2_in, eval_rsd_bb3_in, eval_rsd_bb4_in, eval_rsd_start, eval_rsd_stop
Transitions:
t₈: eval_rsd_3(X₀, X₁, X₂, X₃) → eval_rsd_4(nondef.0, X₁, X₂, X₃)
t₉: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₀
t₁₀: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₂-1, X₂, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ 0
t₁₁: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₁-1, X₁, X₃) :|: 1 ≤ X₀ ∧ X₀ ≤ 0
t₁₂: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₂-1, X₁, X₃) :|: X₀ ≤ 0
t₁: eval_rsd_bb0_in(X₀, X₁, X₂, X₃) → eval_rsd_bb1_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃
t₂: eval_rsd_bb0_in(X₀, X₁, X₂, X₃) → eval_rsd_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0
t₃: eval_rsd_bb1_in(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, 2⋅X₃, 2⋅X₃, X₃)
t₄: eval_rsd_bb2_in(X₀, X₁, X₂, X₃) → eval_rsd_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₁
t₅: eval_rsd_bb2_in(X₀, X₁, X₂, X₃) → eval_rsd_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₃
t₆: eval_rsd_bb3_in(X₀, X₁, X₂, X₃) → eval_rsd_3(X₀, X₁, X₂, X₃)
t₁₃: eval_rsd_bb4_in(X₀, X₁, X₂, X₃) → eval_rsd_stop(X₀, X₁, X₂, X₃)
t₀: eval_rsd_start(X₀, X₁, X₂, X₃) → eval_rsd_bb0_in(X₀, X₁, X₂, X₃)

Preprocessing

Cut unsatisfiable transition [t₁₀: eval_rsd_4→eval_rsd_bb2_in; t₁₁: eval_rsd_4→eval_rsd_bb2_in]

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ 1+X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ 1+X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ 1+X₁ for location eval_rsd_bb2_in

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_rsd_4

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_rsd_3

Found invariant X₃ ≤ X₂ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₃ ∧ 0 ≤ X₂+X₃ ∧ 0 ≤ X₁+X₃ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁ for location eval_rsd_bb3_in

Found invariant 0 ≤ X₃ for location eval_rsd_bb1_in

Problem after Preprocessing

Start: eval_rsd_start
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: nondef.0
Locations: eval_rsd_3, eval_rsd_4, eval_rsd_bb0_in, eval_rsd_bb1_in, eval_rsd_bb2_in, eval_rsd_bb3_in, eval_rsd_bb4_in, eval_rsd_start, eval_rsd_stop
Transitions:
t₈: eval_rsd_3(X₀, X₁, X₂, X₃) → eval_rsd_4(nondef.0, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₉: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₁₂: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₂-1, X₁, X₃) :|: X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₁: eval_rsd_bb0_in(X₀, X₁, X₂, X₃) → eval_rsd_bb1_in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃
t₂: eval_rsd_bb0_in(X₀, X₁, X₂, X₃) → eval_rsd_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0
t₃: eval_rsd_bb1_in(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, 2⋅X₃, 2⋅X₃, X₃) :|: 0 ≤ X₃
t₄: eval_rsd_bb2_in(X₀, X₁, X₂, X₃) → eval_rsd_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ 1+X₁+X₃ ∧ X₃ ≤ 1+X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₅: eval_rsd_bb2_in(X₀, X₁, X₂, X₃) → eval_rsd_bb4_in(X₀, X₁, X₂, X₃) :|: 1+X₁ ≤ X₃ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ 1+X₁+X₃ ∧ X₃ ≤ 1+X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₆: eval_rsd_bb3_in(X₀, X₁, X₂, X₃) → eval_rsd_3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃
t₁₃: eval_rsd_bb4_in(X₀, X₁, X₂, X₃) → eval_rsd_stop(X₀, X₁, X₂, X₃)
t₀: eval_rsd_start(X₀, X₁, X₂, X₃) → eval_rsd_bb0_in(X₀, X₁, X₂, X₃)

MPRF for transition t₄: eval_rsd_bb2_in(X₀, X₁, X₂, X₃) → eval_rsd_bb3_in(X₀, X₁, X₂, X₃) :|: X₃ ≤ X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ 1+X₁+X₃ ∧ X₃ ≤ 1+X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

4⋅X₃+2 {O(n)}

MPRF:

• eval_rsd_3: [1+X₁+X₂]
• eval_rsd_4: [1+X₁+X₂]
• eval_rsd_bb2_in: [2+X₁+X₂]
• eval_rsd_bb3_in: [1+X₁+X₂]

MPRF for transition t₆: eval_rsd_bb3_in(X₀, X₁, X₂, X₃) → eval_rsd_3(X₀, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

4⋅X₃+1 {O(n)}

MPRF:

• eval_rsd_3: [X₁+X₂]
• eval_rsd_4: [X₁+X₂]
• eval_rsd_bb2_in: [1+X₁+X₂]
• eval_rsd_bb3_in: [1+X₁+X₂]

MPRF for transition t₈: eval_rsd_3(X₀, X₁, X₂, X₃) → eval_rsd_4(nondef.0, X₁, X₂, X₃) :|: 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

4⋅X₃+1 {O(n)}

MPRF:

• eval_rsd_3: [1+X₁+X₂]
• eval_rsd_4: [X₁+X₂]
• eval_rsd_bb2_in: [1+X₁+X₂]
• eval_rsd_bb3_in: [1+X₁+X₂]

MPRF for transition t₉: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₁-1, X₂, X₃) :|: 1 ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

4⋅X₃+1 {O(n)}

MPRF:

• eval_rsd_3: [1+X₁+X₂]
• eval_rsd_4: [1+X₁+X₂]
• eval_rsd_bb2_in: [1+X₁+X₂]
• eval_rsd_bb3_in: [1+X₁+X₂]

MPRF for transition t₁₂: eval_rsd_4(X₀, X₁, X₂, X₃) → eval_rsd_bb2_in(X₀, X₂-1, X₁, X₃) :|: X₀ ≤ 0 ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₁+X₃ ∧ X₃ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₂+X₃ ∧ X₃ ≤ X₂ ∧ 0 ≤ X₃ of depth 1:

new bound:

4⋅X₃+1 {O(n)}

MPRF:

• eval_rsd_3: [1+X₁+X₂]
• eval_rsd_4: [1+X₁+X₂]
• eval_rsd_bb2_in: [1+X₁+X₂]
• eval_rsd_bb3_in: [1+X₁+X₂]

All Bounds

Timebounds

Overall timebound:20⋅X₃+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 4⋅X₃+2 {O(n)}
t₅: 1 {O(1)}
t₆: 4⋅X₃+1 {O(n)}
t₈: 4⋅X₃+1 {O(n)}
t₉: 4⋅X₃+1 {O(n)}
t₁₂: 4⋅X₃+1 {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 20⋅X₃+12 {O(n)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 4⋅X₃+2 {O(n)}
t₅: 1 {O(1)}
t₆: 4⋅X₃+1 {O(n)}
t₈: 4⋅X₃+1 {O(n)}
t₉: 4⋅X₃+1 {O(n)}
t₁₂: 4⋅X₃+1 {O(n)}
t₁₃: 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: X₀ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₂, X₃: X₃ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: 2⋅X₃ {O(n)}
t₃, X₂: 2⋅X₃ {O(n)}
t₃, X₃: X₃ {O(n)}
t₄, X₁: 4⋅X₃+1 {O(n)}
t₄, X₂: 4⋅X₃+1 {O(n)}
t₄, X₃: X₃ {O(n)}
t₅, X₁: 8⋅X₃+2 {O(n)}
t₅, X₂: 8⋅X₃+2 {O(n)}
t₅, X₃: 2⋅X₃ {O(n)}
t₆, X₁: 4⋅X₃+1 {O(n)}
t₆, X₂: 4⋅X₃+1 {O(n)}
t₆, X₃: X₃ {O(n)}
t₈, X₁: 4⋅X₃+1 {O(n)}
t₈, X₂: 4⋅X₃+1 {O(n)}
t₈, X₃: X₃ {O(n)}
t₉, X₁: 4⋅X₃+1 {O(n)}
t₉, X₂: 4⋅X₃+1 {O(n)}
t₉, X₃: X₃ {O(n)}
t₁₂, X₁: 4⋅X₃+1 {O(n)}
t₁₂, X₂: 4⋅X₃+1 {O(n)}
t₁₂, X₃: X₃ {O(n)}
t₁₃, X₁: 8⋅X₃+X₁+2 {O(n)}
t₁₃, X₂: 8⋅X₃+X₂+2 {O(n)}
t₁₃, X₃: 3⋅X₃ {O(n)}