KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_loops_bb2_in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_loops_bb1_in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_loops_bb5_in

Found invariant 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_loops_bb3_in

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_loops_bb4_in

Problem after Preprocessing

Start: eval_loops_start
Program_Vars: X₀, X₁, X₂
Temp_Vars:
Locations: eval_loops_bb0_in, eval_loops_bb1_in, eval_loops_bb2_in, eval_loops_bb3_in, eval_loops_bb4_in, eval_loops_bb5_in, eval_loops_bb6_in, eval_loops_start, eval_loops_stop
Transitions:
t₁: eval_loops_bb0_in(X₀, X₁, X₂) → eval_loops_bb1_in(X₀, X₀, X₂) :|: 0 ≤ X₀
t₂: eval_loops_bb0_in(X₀, X₁, X₂) → eval_loops_bb6_in(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₃: eval_loops_bb1_in(X₀, X₁, X₂) → eval_loops_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀
t₄: eval_loops_bb1_in(X₀, X₁, X₂) → eval_loops_bb6_in(X₀, X₁, X₂) :|: 1+X₁ ≤ 0 ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀
t₅: eval_loops_bb2_in(X₀, X₁, X₂) → eval_loops_bb3_in(X₀, X₁, 1) :|: 2 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₆: eval_loops_bb2_in(X₀, X₁, X₂) → eval_loops_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₇: eval_loops_bb3_in(X₀, X₁, X₂) → eval_loops_bb4_in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₈: eval_loops_bb3_in(X₀, X₁, X₂) → eval_loops_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₉: eval_loops_bb4_in(X₀, X₁, X₂) → eval_loops_bb3_in(X₀, X₁, 2⋅X₂) :|: 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ X₁ ≤ X₀
t₁₀: eval_loops_bb5_in(X₀, X₁, X₂) → eval_loops_bb1_in(X₀, X₁-1, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁
t₁₁: eval_loops_bb6_in(X₀, X₁, X₂) → eval_loops_stop(X₀, X₁, X₂)
t₀: eval_loops_start(X₀, X₁, X₂) → eval_loops_bb0_in(X₀, X₁, X₂)

MPRF for transition t₃: eval_loops_bb1_in(X₀, X₁, X₂) → eval_loops_bb2_in(X₀, X₁, X₂) :|: 0 ≤ X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ X₀ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀+2 {O(n)}

MPRF:

• eval_loops_bb1_in: [2+X₁]
• eval_loops_bb2_in: [1+X₁]
• eval_loops_bb3_in: [1+X₁]
• eval_loops_bb4_in: [1+X₁]
• eval_loops_bb5_in: [1+X₁]

MPRF for transition t₅: eval_loops_bb2_in(X₀, X₁, X₂) → eval_loops_bb3_in(X₀, X₁, 1) :|: 2 ≤ X₁ ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_loops_bb1_in: [1+X₁]
• eval_loops_bb2_in: [1+X₁]
• eval_loops_bb3_in: [X₁]
• eval_loops_bb4_in: [X₁]
• eval_loops_bb5_in: [X₁]

MPRF for transition t₆: eval_loops_bb2_in(X₀, X₁, X₂) → eval_loops_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ 1 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_loops_bb1_in: [1+X₁]
• eval_loops_bb2_in: [1+X₁]
• eval_loops_bb3_in: [X₁]
• eval_loops_bb4_in: [X₁]
• eval_loops_bb5_in: [X₁]

MPRF for transition t₈: eval_loops_bb3_in(X₀, X₁, X₂) → eval_loops_bb5_in(X₀, X₁, X₂) :|: X₁ ≤ X₂ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ X₁ ≤ X₀ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_loops_bb1_in: [X₁-1]
• eval_loops_bb2_in: [X₁-1]
• eval_loops_bb3_in: [X₁-1]
• eval_loops_bb4_in: [X₁-1]
• eval_loops_bb5_in: [X₁-2]

MPRF for transition t₁₀: eval_loops_bb5_in(X₀, X₁, X₂) → eval_loops_bb1_in(X₀, X₁-1, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₁ ∧ X₁ ≤ X₀ ∧ 0 ≤ X₁ of depth 1:

new bound:

X₀+1 {O(n)}

MPRF:

• eval_loops_bb1_in: [1+X₁]
• eval_loops_bb2_in: [1+X₁]
• eval_loops_bb3_in: [1+X₁]
• eval_loops_bb4_in: [1+X₁]
• eval_loops_bb5_in: [1+X₁]

MPRF for transition t₇: eval_loops_bb3_in(X₀, X₁, X₂) → eval_loops_bb4_in(X₀, X₁, X₂) :|: 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ X₁ ≤ X₀ of depth 1:

new bound:

2⋅X₀⋅X₀+6⋅X₀+2 {O(n^2)}

MPRF:

• eval_loops_bb1_in: [X₀+X₁]
• eval_loops_bb2_in: [X₀+X₁]
• eval_loops_bb3_in: [X₁-X₂]
• eval_loops_bb4_in: [X₁-1-X₂]
• eval_loops_bb5_in: [X₀+X₁]

MPRF for transition t₉: eval_loops_bb4_in(X₀, X₁, X₂) → eval_loops_bb3_in(X₀, X₁, 2⋅X₂) :|: 1+X₂ ≤ X₀ ∧ 1+X₂ ≤ X₁ ∧ 1 ≤ X₂ ∧ 2 ≤ X₀ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₂ ∧ 3 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₁ ∧ X₁ ≤ X₀ of depth 1:

new bound:

2⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}

MPRF:

• eval_loops_bb1_in: [2⋅X₁]
• eval_loops_bb2_in: [2⋅X₁]
• eval_loops_bb3_in: [X₁-X₂]
• eval_loops_bb4_in: [X₁-X₂]
• eval_loops_bb5_in: [2⋅X₁]

Cut unsatisfiable transition [t₈: eval_loops_bb3_in→eval_loops_bb5_in; t₆₁: eval_loops_bb3_in→eval_loops_bb5_in]

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_loops_bb2_in

Found invariant X₁ ≤ X₀ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ 0 ≤ X₀ for location eval_loops_bb1_in

Found invariant 2 ≤ X₂ ∧ 4 ≤ X₁+X₂ ∧ 4 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_loops_bb3_in_v1

Found invariant 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 2 ≤ X₂ ∧ 5 ≤ X₁+X₂ ∧ 5 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 3 ≤ X₁ ∧ 6 ≤ X₀+X₁ ∧ 3 ≤ X₀ for location eval_loops_bb4_in_v2

Found invariant X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_loops_bb4_in_v1

Found invariant X₁ ≤ X₀ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ 0 ≤ X₀ for location eval_loops_bb5_in

Found invariant X₂ ≤ 1 ∧ 1+X₂ ≤ X₁ ∧ 1+X₂ ≤ X₀ ∧ 1 ≤ X₂ ∧ 3 ≤ X₁+X₂ ∧ 3 ≤ X₀+X₂ ∧ X₁ ≤ X₀ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ 2 ≤ X₀ for location eval_loops_bb3_in

All Bounds

Timebounds

Overall timebound:4⋅X₀⋅X₀+19⋅X₀+17 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+2 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₇: 2⋅X₀⋅X₀+6⋅X₀+2 {O(n^2)}
t₈: X₀+1 {O(n)}
t₉: 2⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
t₁₀: X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 4⋅X₀⋅X₀+19⋅X₀+17 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: X₀+2 {O(n)}
t₄: 1 {O(1)}
t₅: X₀+1 {O(n)}
t₆: X₀+1 {O(n)}
t₇: 2⋅X₀⋅X₀+6⋅X₀+2 {O(n^2)}
t₈: X₀+1 {O(n)}
t₉: 2⋅X₀⋅X₀+8⋅X₀+4 {O(n^2)}
t₁₀: X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Sizebounds

t₀, X₀: X₀ {O(n)}
t₀, X₁: X₁ {O(n)}
t₀, X₂: X₂ {O(n)}
t₁, X₀: X₀ {O(n)}
t₁, X₁: X₀ {O(n)}
t₁, X₂: X₂ {O(n)}
t₂, X₀: X₀ {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₂: X₂ {O(n)}
t₃, X₀: X₀ {O(n)}
t₃, X₁: X₀+2 {O(n)}
t₃, X₂: 16⋅2^(X₀)+X₂ {O(EXP)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: 1 {O(1)}
t₄, X₂: 16⋅2^(X₀)+X₂ {O(EXP)}
t₅, X₀: X₀ {O(n)}
t₅, X₁: X₀+2 {O(n)}
t₅, X₂: 1 {O(1)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: 1 {O(1)}
t₆, X₂: 16⋅2^(X₀)+X₂ {O(EXP)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: X₀+2 {O(n)}
t₇, X₂: 2^(X₀)⋅4 {O(EXP)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: X₀+2 {O(n)}
t₈, X₂: 16⋅2^(X₀) {O(EXP)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: X₀+2 {O(n)}
t₉, X₂: 16⋅2^(X₀) {O(EXP)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: X₀+2 {O(n)}
t₁₀, X₂: 16⋅2^(X₀)+X₂ {O(EXP)}
t₁₁, X₀: 2⋅X₀ {O(n)}
t₁₁, X₁: X₁+1 {O(n)}
t₁₁, X₂: 16⋅2^(X₀)+2⋅X₂ {O(EXP)}