KoAT2 Proof Output

Initial Problem

Preprocessing

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb1in

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb2in

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb3in

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb4in

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalinsertsortstop

Found invariant 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbbin

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalinsertsortreturnin

Found invariant 1 ≤ X₀ for location evalinsertsortbb5in

Problem after Preprocessing

Start: evalinsertsortstart
Program_Vars: X₀, X₁, X₂, X₃
Temp_Vars: E
Locations: evalinsertsortbb1in, evalinsertsortbb2in, evalinsertsortbb3in, evalinsertsortbb4in, evalinsertsortbb5in, evalinsertsortbbin, evalinsertsortentryin, evalinsertsortreturnin, evalinsertsortstart, evalinsertsortstop
Transitions:
t₉: evalinsertsortbb1in(X₀, X₁, X₂, X₃) → evalinsertsortbb2in(X₀, X₁, X₂, X₃-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₃
t₆: evalinsertsortbb2in(X₀, X₁, X₂, X₃) → evalinsertsortbb3in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₁ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃
t₅: evalinsertsortbb2in(X₀, X₁, X₂, X₃) → evalinsertsortbb4in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0 ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₁ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃
t₇: evalinsertsortbb3in(X₀, X₁, X₂, X₃) → evalinsertsortbb1in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ E ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₃
t₈: evalinsertsortbb3in(X₀, X₁, X₂, X₃) → evalinsertsortbb4in(X₀, X₁, X₂, X₃) :|: E ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₃
t₁₀: evalinsertsortbb4in(X₀, X₁, X₂, X₃) → evalinsertsortbb5in(1+X₀, X₁, X₂, X₃) :|: 0 ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₁ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃
t₂: evalinsertsortbb5in(X₀, X₁, X₂, X₃) → evalinsertsortbbin(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀
t₃: evalinsertsortbb5in(X₀, X₁, X₂, X₃) → evalinsertsortreturnin(X₀, X₁, X₂, X₃) :|: X₁ ≤ X₀ ∧ 1 ≤ X₀
t₄: evalinsertsortbbin(X₀, X₁, X₂, X₃) → evalinsertsortbb2in(X₀, X₁, E, X₀-1) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁
t₁: evalinsertsortentryin(X₀, X₁, X₂, X₃) → evalinsertsortbb5in(1, X₁, X₂, X₃)
t₁₁: evalinsertsortreturnin(X₀, X₁, X₂, X₃) → evalinsertsortstop(X₀, X₁, X₂, X₃) :|: 1 ≤ X₀ ∧ X₁ ≤ X₀
t₀: evalinsertsortstart(X₀, X₁, X₂, X₃) → evalinsertsortentryin(X₀, X₁, X₂, X₃)

MPRF for transition t₂: evalinsertsortbb5in(X₀, X₁, X₂, X₃) → evalinsertsortbbin(X₀, X₁, X₂, X₃) :|: 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalinsertsortbb1in: [X₁-1-X₀]
• evalinsertsortbb2in: [X₁-1-X₀]
• evalinsertsortbb3in: [X₁-1-X₀]
• evalinsertsortbb4in: [X₁-1-X₀]
• evalinsertsortbb5in: [X₁-X₀]
• evalinsertsortbbin: [X₁-1-X₀]

MPRF for transition t₄: evalinsertsortbbin(X₀, X₁, X₂, X₃) → evalinsertsortbb2in(X₀, X₁, E, X₀-1) :|: 1 ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

MPRF for transition t₅: evalinsertsortbb2in(X₀, X₁, X₂, X₃) → evalinsertsortbb4in(X₀, X₁, X₂, X₃) :|: 1+X₃ ≤ 0 ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₁ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalinsertsortbb1in: [X₁-X₀]
• evalinsertsortbb2in: [X₁-X₀]
• evalinsertsortbb3in: [X₁-X₀]
• evalinsertsortbb4in: [X₁-1-X₀]
• evalinsertsortbb5in: [X₁-X₀]
• evalinsertsortbbin: [X₁-X₀]

MPRF for transition t₈: evalinsertsortbb3in(X₀, X₁, X₂, X₃) → evalinsertsortbb4in(X₀, X₁, X₂, X₃) :|: E ≤ X₂ ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

MPRF for transition t₁₀: evalinsertsortbb4in(X₀, X₁, X₂, X₃) → evalinsertsortbb5in(1+X₀, X₁, X₂, X₃) :|: 0 ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₁ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃ of depth 1:

new bound:

X₁+1 {O(n)}

MPRF:

• evalinsertsortbb1in: [X₁-X₀]
• evalinsertsortbb2in: [X₁-X₀]
• evalinsertsortbb3in: [X₁-X₀]
• evalinsertsortbb4in: [X₁-X₀]
• evalinsertsortbb5in: [X₁-X₀]
• evalinsertsortbbin: [X₁-X₀]

MPRF for transition t₆: evalinsertsortbb2in(X₀, X₁, X₂, X₃) → evalinsertsortbb3in(X₀, X₁, X₂, X₃) :|: 0 ≤ X₃ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₀ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₁+X₃ ∧ 2 ≤ X₁ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₀+X₃ of depth 1:

new bound:

X₁⋅X₁+3⋅X₁+3 {O(n^2)}

MPRF:

• evalinsertsortbb1in: [X₃]
• evalinsertsortbb2in: [1+X₃]
• evalinsertsortbb3in: [X₃]
• evalinsertsortbb4in: [X₃]
• evalinsertsortbb5in: [X₀]
• evalinsertsortbbin: [X₀]

MPRF for transition t₇: evalinsertsortbb3in(X₀, X₁, X₂, X₃) → evalinsertsortbb1in(X₀, X₁, X₂, X₃) :|: 1+X₂ ≤ E ∧ 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁⋅X₁+4⋅X₁+5 {O(n^2)}

MPRF:

• evalinsertsortbb1in: [X₃]
• evalinsertsortbb2in: [1+X₃]
• evalinsertsortbb3in: [1+X₃]
• evalinsertsortbb4in: [X₃]
• evalinsertsortbb5in: [1+X₀]
• evalinsertsortbbin: [1+X₀]

MPRF for transition t₉: evalinsertsortbb1in(X₀, X₁, X₂, X₃) → evalinsertsortbb2in(X₀, X₁, X₂, X₃-1) :|: 1 ≤ X₀ ∧ 1 ≤ X₀+X₃ ∧ 1+X₃ ≤ X₀ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₁ ∧ 2 ≤ X₁+X₃ ∧ 2+X₃ ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 0 ≤ X₃ of depth 1:

new bound:

X₁⋅X₁+3⋅X₁+3 {O(n^2)}

MPRF:

• evalinsertsortbb1in: [1+X₃]
• evalinsertsortbb2in: [1+X₃]
• evalinsertsortbb3in: [1+X₃]
• evalinsertsortbb4in: [X₃]
• evalinsertsortbb5in: [X₀]
• evalinsertsortbbin: [X₀]

Cut unreachable locations [evalinsertsortbb2in] from the program graph

Found invariant 3+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb2in_v2

Found invariant 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ X₀ ≤ 1 ∧ 1 ≤ X₀ for location evalinsertsortbbin_v2

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb3in_v1

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalinsertsortreturnin

Found invariant 3+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₀ for location evalinsertsortbb1in_v2

Found invariant X₀ ≤ 1 ∧ 1 ≤ X₀ for location evalinsertsortbb5in

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb1in_v1

Found invariant 2+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 4 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 2 ≤ X₀ for location evalinsertsortbb5in_v1

Found invariant 3+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₀ for location evalinsertsortbb4in_v2

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb2in_v1

Found invariant 1+X₃ ≤ 0 ∧ 3+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ 1+X₃ ∧ 1 ≤ X₁+X₃ ∧ 0 ≤ X₀+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb4in

Found invariant X₁ ≤ X₀ ∧ 1 ≤ X₀ for location evalinsertsortstop

Found invariant 2+X₃ ≤ X₁ ∧ 1+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ X₀ ≤ 1+X₃ ∧ 2 ≤ X₁ ∧ 3 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 1 ≤ X₀ for location evalinsertsortbb4in_v1

Found invariant 3+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ X₃ ∧ 3 ≤ X₁+X₃ ∧ 2 ≤ X₀+X₃ ∧ 3 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₀ for location evalinsertsortbb3in_v2

Found invariant 3+X₃ ≤ X₁ ∧ 2+X₃ ≤ X₀ ∧ 0 ≤ 1+X₃ ∧ 2 ≤ X₁+X₃ ∧ 1 ≤ X₀+X₃ ∧ 3 ≤ X₁ ∧ 5 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 2 ≤ X₀ for location evalinsertsortbbin_v1

All Bounds

Timebounds

Overall timebound:3⋅X₁⋅X₁+15⋅X₁+20 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₁+1 {O(n)}
t₅: X₁+1 {O(n)}
t₆: X₁⋅X₁+3⋅X₁+3 {O(n^2)}
t₇: X₁⋅X₁+4⋅X₁+5 {O(n^2)}
t₈: X₁+1 {O(n)}
t₉: X₁⋅X₁+3⋅X₁+3 {O(n^2)}
t₁₀: X₁+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 3⋅X₁⋅X₁+15⋅X₁+20 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: X₁+1 {O(n)}
t₃: 1 {O(1)}
t₄: X₁+1 {O(n)}
t₅: X₁+1 {O(n)}
t₆: X₁⋅X₁+3⋅X₁+3 {O(n^2)}
t₇: X₁⋅X₁+4⋅X₁+5 {O(n^2)}
t₈: X₁+1 {O(n)}
t₉: X₁⋅X₁+3⋅X₁+3 {O(n^2)}
t₁₀: X₁+1 {O(n)}
t₁₁: 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: 1 {O(1)}
t₁, X₁: X₁ {O(n)}
t₁, X₂: X₂ {O(n)}
t₁, X₃: X₃ {O(n)}
t₂, X₀: X₁+2 {O(n)}
t₂, X₁: X₁ {O(n)}
t₂, X₃: X₁+X₃+4 {O(n)}
t₃, X₀: X₁+3 {O(n)}
t₃, X₁: 2⋅X₁ {O(n)}
t₃, X₃: X₁+X₃+4 {O(n)}
t₄, X₀: X₁+2 {O(n)}
t₄, X₁: X₁ {O(n)}
t₄, X₃: X₁+2 {O(n)}
t₅, X₀: X₁+2 {O(n)}
t₅, X₁: X₁ {O(n)}
t₅, X₃: 1 {O(1)}
t₆, X₀: X₁+2 {O(n)}
t₆, X₁: X₁ {O(n)}
t₆, X₃: X₁+3 {O(n)}
t₇, X₀: X₁+2 {O(n)}
t₇, X₁: X₁ {O(n)}
t₇, X₃: X₁+3 {O(n)}
t₈, X₀: X₁+2 {O(n)}
t₈, X₁: X₁ {O(n)}
t₈, X₃: X₁+3 {O(n)}
t₉, X₀: X₁+2 {O(n)}
t₉, X₁: X₁ {O(n)}
t₉, X₃: X₁+3 {O(n)}
t₁₀, X₀: X₁+2 {O(n)}
t₁₀, X₁: X₁ {O(n)}
t₁₀, X₃: X₁+4 {O(n)}
t₁₁, X₀: X₁+3 {O(n)}
t₁₁, X₁: 2⋅X₁ {O(n)}
t₁₁, X₃: X₁+X₃+4 {O(n)}