KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_3

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location eval_sipmabubble_bb4_in

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_sipmabubble_stop

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_2

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_sipmabubble_bb2_in

Found invariant X₀ ≤ X₂ for location eval_sipmabubble_bb1_in

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_sipmabubble_bb5_in

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_4

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_bb3_in

Problem after Preprocessing

Start: eval_sipmabubble_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_sipmabubble_2, eval_sipmabubble_3, eval_sipmabubble_4, eval_sipmabubble_bb0_in, eval_sipmabubble_bb1_in, eval_sipmabubble_bb2_in, eval_sipmabubble_bb3_in, eval_sipmabubble_bb4_in, eval_sipmabubble_bb5_in, eval_sipmabubble_start, eval_sipmabubble_stop
Transitions:
t₈: eval_sipmabubble_2(X₀, X₁, X₂) → eval_sipmabubble_3(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₁₀: eval_sipmabubble_3(X₀, X₁, X₂) → eval_sipmabubble_4(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₁₁: eval_sipmabubble_4(X₀, X₁, X₂) → eval_sipmabubble_bb2_in(X₀, 1+X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₁: eval_sipmabubble_bb0_in(X₀, X₁, X₂) → eval_sipmabubble_bb1_in(X₂, X₁, X₂)
t₂: eval_sipmabubble_bb1_in(X₀, X₁, X₂) → eval_sipmabubble_bb2_in(X₀, 0, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ X₂
t₃: eval_sipmabubble_bb1_in(X₀, X₁, X₂) → eval_sipmabubble_bb5_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₂
t₄: eval_sipmabubble_bb2_in(X₀, X₁, X₂) → eval_sipmabubble_bb3_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₅: eval_sipmabubble_bb2_in(X₀, X₁, X₂) → eval_sipmabubble_bb4_in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₆: eval_sipmabubble_bb3_in(X₀, X₁, X₂) → eval_sipmabubble_2(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁
t₁₂: eval_sipmabubble_bb4_in(X₀, X₁, X₂) → eval_sipmabubble_bb1_in(X₀-1, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂
t₁₃: eval_sipmabubble_bb5_in(X₀, X₁, X₂) → eval_sipmabubble_stop(X₀, X₁, X₂) :|: 1+X₀ ≤ 0 ∧ X₀ ≤ X₂
t₀: eval_sipmabubble_start(X₀, X₁, X₂) → eval_sipmabubble_bb0_in(X₀, X₁, X₂)

MPRF for transition t₂: eval_sipmabubble_bb1_in(X₀, X₁, X₂) → eval_sipmabubble_bb2_in(X₀, 0, X₂) :|: 0 ≤ X₀ ∧ X₀ ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_sipmabubble_2: [X₀]
• eval_sipmabubble_3: [X₀]
• eval_sipmabubble_4: [X₀]
• eval_sipmabubble_bb1_in: [1+X₀]
• eval_sipmabubble_bb2_in: [X₀]
• eval_sipmabubble_bb3_in: [X₀]
• eval_sipmabubble_bb4_in: [X₀]

MPRF for transition t₅: eval_sipmabubble_bb2_in(X₀, X₁, X₂) → eval_sipmabubble_bb4_in(X₀, X₁, X₂) :|: X₀ ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

• eval_sipmabubble_2: [1+X₀]
• eval_sipmabubble_3: [1+X₀]
• eval_sipmabubble_4: [1+X₀]
• eval_sipmabubble_bb1_in: [1+X₀]
• eval_sipmabubble_bb2_in: [1+X₀]
• eval_sipmabubble_bb3_in: [1+X₀]
• eval_sipmabubble_bb4_in: [X₀]

MPRF for transition t₁₂: eval_sipmabubble_bb4_in(X₀, X₁, X₂) → eval_sipmabubble_bb1_in(X₀-1, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₁ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂+1 {O(n)}

MPRF:

MPRF for transition t₄: eval_sipmabubble_bb2_in(X₀, X₁, X₂) → eval_sipmabubble_bb3_in(X₀, X₁, X₂) :|: 1+X₁ ≤ X₀ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

X₂⋅X₂+4⋅X₂+3 {O(n^2)}

MPRF:

• eval_sipmabubble_2: [X₀-X₁]
• eval_sipmabubble_3: [X₀-X₁]
• eval_sipmabubble_4: [X₀-X₁]
• eval_sipmabubble_bb1_in: [1+X₀]
• eval_sipmabubble_bb2_in: [1+X₀-X₁]
• eval_sipmabubble_bb3_in: [X₀-X₁]
• eval_sipmabubble_bb4_in: [1+X₀-X₁]

MPRF for transition t₆: eval_sipmabubble_bb3_in(X₀, X₁, X₂) → eval_sipmabubble_2(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+4⋅X₂+3 {O(n^2)}

MPRF:

MPRF for transition t₈: eval_sipmabubble_2(X₀, X₁, X₂) → eval_sipmabubble_3(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+5⋅X₂+5 {O(n^2)}

MPRF:

• eval_sipmabubble_2: [2+X₀-X₁]
• eval_sipmabubble_3: [1+X₀-X₁]
• eval_sipmabubble_4: [1+X₀-X₁]
• eval_sipmabubble_bb1_in: [2+X₀]
• eval_sipmabubble_bb2_in: [2+X₀-X₁]
• eval_sipmabubble_bb3_in: [2+X₀-X₁]
• eval_sipmabubble_bb4_in: [1+X₀-X₁]

MPRF for transition t₁₀: eval_sipmabubble_3(X₀, X₁, X₂) → eval_sipmabubble_4(X₀, X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+4⋅X₂+3 {O(n^2)}

MPRF:

• eval_sipmabubble_2: [1+X₀-X₁]
• eval_sipmabubble_3: [1+X₀-X₁]
• eval_sipmabubble_4: [X₀-X₁]
• eval_sipmabubble_bb1_in: [1+X₀]
• eval_sipmabubble_bb2_in: [1+X₀-X₁]
• eval_sipmabubble_bb3_in: [1+X₀-X₁]
• eval_sipmabubble_bb4_in: [1+X₀-X₁]

MPRF for transition t₁₁: eval_sipmabubble_4(X₀, X₁, X₂) → eval_sipmabubble_bb2_in(X₀, 1+X₁, X₂) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₁ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₂⋅X₂+4⋅X₂+3 {O(n^2)}

MPRF:

• eval_sipmabubble_2: [1+X₀-X₁]
• eval_sipmabubble_3: [1+X₀-X₁]
• eval_sipmabubble_4: [1+X₀-X₁]
• eval_sipmabubble_bb1_in: [1+X₀]
• eval_sipmabubble_bb2_in: [1+X₀-X₁]
• eval_sipmabubble_bb3_in: [1+X₀-X₁]
• eval_sipmabubble_bb4_in: [1]

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_sipmabubble_4_v2

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_2_v1

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_3_v1

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_bb3_in_v1

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_sipmabubble_bb3_in_v2

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_sipmabubble_stop

Found invariant 1 ≤ X₂ ∧ 2 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 2 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_bb2_in_v1

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_sipmabubble_2_v2

Found invariant 2 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 4 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1+X₁ ≤ X₀ ∧ 1 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_sipmabubble_3_v2

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_sipmabubble_bb2_in

Found invariant 1 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 1+X₁ ≤ X₂ ∧ 2 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ X₁ ≤ 0 ∧ 1+X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1 ≤ X₀ for location eval_sipmabubble_4_v1

Found invariant X₀ ≤ X₂ for location eval_sipmabubble_bb1_in

Found invariant X₀ ≤ X₂ ∧ 1+X₀ ≤ 0 for location eval_sipmabubble_bb5_in

All Bounds

Timebounds

Overall timebound:5⋅X₂⋅X₂+24⋅X₂+24 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₅: X₂+1 {O(n)}
t₆: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₈: X₂⋅X₂+5⋅X₂+5 {O(n^2)}
t₁₀: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₁₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₁₂: X₂+1 {O(n)}
t₁₃: 1 {O(1)}

Costbounds

Overall costbound: 5⋅X₂⋅X₂+24⋅X₂+24 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₂+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₅: X₂+1 {O(n)}
t₆: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₈: X₂⋅X₂+5⋅X₂+5 {O(n^2)}
t₁₀: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₁₁: X₂⋅X₂+4⋅X₂+3 {O(n^2)}
t₁₂: X₂+1 {O(n)}
t₁₃: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₂ {O(n)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₂+1 {O(n)}
t₂, X₁: 0 {O(1)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: 2⋅X₂+1 {O(n)}
t₃, X₁: X₁+X₂+2 {O(n)}
t₃, X₂: 2⋅X₂ {O(n)}
t₄, X₀: X₂+1 {O(n)}
t₄, X₁: X₂+2 {O(n)}
t₄, X₂: X₂ {O(n)}
t₅, X₀: X₂+1 {O(n)}
t₅, X₁: X₂+2 {O(n)}
t₅, X₂: X₂ {O(n)}
t₆, X₀: X₂+1 {O(n)}
t₆, X₁: X₂+1 {O(n)}
t₆, X₂: X₂ {O(n)}
t₈, X₀: X₂+1 {O(n)}
t₈, X₁: X₂+2 {O(n)}
t₈, X₂: X₂ {O(n)}
t₁₀, X₀: X₂+1 {O(n)}
t₁₀, X₁: X₂+1 {O(n)}
t₁₀, X₂: X₂ {O(n)}
t₁₁, X₀: X₂+1 {O(n)}
t₁₁, X₁: X₂+2 {O(n)}
t₁₁, X₂: X₂ {O(n)}
t₁₂, X₀: X₂+1 {O(n)}
t₁₂, X₁: X₂+2 {O(n)}
t₁₂, X₂: X₂ {O(n)}
t₁₃, X₀: 2⋅X₂+1 {O(n)}
t₁₃, X₁: X₁+X₂+2 {O(n)}
t₁₃, X₂: 2⋅X₂ {O(n)}