KoAT2 Proof Output

Initial Problem

Start: evalrsdstart
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalrsdbb1in, evalrsdbb2in, evalrsdbb3in, evalrsdbb4in, evalrsdbbin, evalrsdentryin, evalrsdreturnin, evalrsdstart, evalrsdstop
Transitions:
t₆: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb2in(X₀, X₁, X₂) :|: 1+D ≤ 0
t₇: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb2in(X₀, X₁, X₂) :|: 1 ≤ D
t₈: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb3in(X₀, X₁, X₂)
t₉: evalrsdbb2in(X₀, X₁, X₂) → evalrsdbb4in(X₀, X₁, X₂-1)
t₁₀: evalrsdbb3in(X₀, X₁, X₂) → evalrsdbb4in(X₀, X₁-1, X₁-1)
t₄: evalrsdbb4in(X₀, X₁, X₂) → evalrsdbb1in(X₀, X₁, X₂) :|: X₀ ≤ X₂
t₅: evalrsdbb4in(X₀, X₁, X₂) → evalrsdreturnin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀
t₃: evalrsdbbin(X₀, X₁, X₂) → evalrsdbb4in(X₀, 2⋅X₀, 2⋅X₀)
t₁: evalrsdentryin(X₀, X₁, X₂) → evalrsdbbin(X₀, X₁, X₂) :|: 0 ≤ X₀
t₂: evalrsdentryin(X₀, X₁, X₂) → evalrsdreturnin(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₁₁: evalrsdreturnin(X₀, X₁, X₂) → evalrsdstop(X₀, X₁, X₂)
t₀: evalrsdstart(X₀, X₁, X₂) → evalrsdentryin(X₀, X₁, X₂)

Preprocessing

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location evalrsdbb2in

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location evalrsdbb1in

Found invariant 0 ≤ X₀ for location evalrsdbbin

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₀ for location evalrsdbb3in

Found invariant 0 ≤ X₀ for location evalrsdbb4in

Problem after Preprocessing

Start: evalrsdstart
Program_Vars: X₀, X₁, X₂
Temp_Vars: D
Locations: evalrsdbb1in, evalrsdbb2in, evalrsdbb3in, evalrsdbb4in, evalrsdbbin, evalrsdentryin, evalrsdreturnin, evalrsdstart, evalrsdstop
Transitions:
t₆: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb2in(X₀, X₁, X₂) :|: 1+D ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₇: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb2in(X₀, X₁, X₂) :|: 1 ≤ D ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₈: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb3in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₉: evalrsdbb2in(X₀, X₁, X₂) → evalrsdbb4in(X₀, X₁, X₂-1) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₁₀: evalrsdbb3in(X₀, X₁, X₂) → evalrsdbb4in(X₀, X₁-1, X₁-1) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂
t₄: evalrsdbb4in(X₀, X₁, X₂) → evalrsdbb1in(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 0 ≤ X₀
t₅: evalrsdbb4in(X₀, X₁, X₂) → evalrsdreturnin(X₀, X₁, X₂) :|: 1+X₂ ≤ X₀ ∧ 0 ≤ X₀
t₃: evalrsdbbin(X₀, X₁, X₂) → evalrsdbb4in(X₀, 2⋅X₀, 2⋅X₀) :|: 0 ≤ X₀
t₁: evalrsdentryin(X₀, X₁, X₂) → evalrsdbbin(X₀, X₁, X₂) :|: 0 ≤ X₀
t₂: evalrsdentryin(X₀, X₁, X₂) → evalrsdreturnin(X₀, X₁, X₂) :|: 1+X₀ ≤ 0
t₁₁: evalrsdreturnin(X₀, X₁, X₂) → evalrsdstop(X₀, X₁, X₂)
t₀: evalrsdstart(X₀, X₁, X₂) → evalrsdentryin(X₀, X₁, X₂)

MPRF for transition t₈: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb3in(X₀, X₁, X₂) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 2:

new bound:

32⋅X₀+9 {O(n)}

MPRF:

• evalrsdbb1in: [X₁; 1+X₂]
• evalrsdbb2in: [X₁; 1+X₂]
• evalrsdbb3in: [X₁-1; X₁]
• evalrsdbb4in: [X₁; 1+X₂]

MPRF for transition t₁₀: evalrsdbb3in(X₀, X₁, X₂) → evalrsdbb4in(X₀, X₁-1, X₁-1) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 2:

new bound:

32⋅X₀+1 {O(n)}

MPRF:

• evalrsdbb1in: [X₁; X₂]
• evalrsdbb2in: [X₁; X₂]
• evalrsdbb3in: [X₁; 0]
• evalrsdbb4in: [X₁; X₂]

MPRF for transition t₄: evalrsdbb4in(X₀, X₁, X₂) → evalrsdbb1in(X₀, X₁, X₂) :|: X₀ ≤ X₂ ∧ 0 ≤ X₀ of depth 1:

new bound:

1088⋅X₀⋅X₀+164⋅X₀+6 {O(n^2)}

MPRF:

• evalrsdbb1in: [1+X₂]
• evalrsdbb2in: [1+X₂]
• evalrsdbb3in: [1]
• evalrsdbb4in: [2+X₂]

MPRF for transition t₆: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb2in(X₀, X₁, X₂) :|: 1+D ≤ 0 ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}

MPRF:

• evalrsdbb1in: [1+X₂]
• evalrsdbb2in: [X₂]
• evalrsdbb3in: [1]
• evalrsdbb4in: [1+X₂]

MPRF for transition t₇: evalrsdbb1in(X₀, X₁, X₂) → evalrsdbb2in(X₀, X₁, X₂) :|: 1 ≤ D ∧ 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}

MPRF:

• evalrsdbb1in: [1+X₂]
• evalrsdbb2in: [X₂]
• evalrsdbb3in: [1]
• evalrsdbb4in: [1+X₂]

MPRF for transition t₉: evalrsdbb2in(X₀, X₁, X₂) → evalrsdbb4in(X₀, X₁, X₂-1) :|: 0 ≤ X₀ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₂ of depth 1:

new bound:

1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}

MPRF:

• evalrsdbb1in: [1+X₂]
• evalrsdbb2in: [1+X₂]
• evalrsdbb3in: [1+X₂]
• evalrsdbb4in: [1+X₂]

Cut unsatisfiable transition [t₅: evalrsdbb4in→evalrsdreturnin; t₆₂: evalrsdbb4in→evalrsdreturnin]

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb1in_v2

Found invariant X₂ ≤ X₁ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 2+X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ X₀ ≤ 1+X₁ ∧ 0 ≤ X₀ for location evalrsdbb4in_v5

Found invariant 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb4in_v1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb4in_v4

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb1in_v1

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb3in_v1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 2+X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ 1+X₁ ∧ 0 ≤ 1+X₀+X₁ ∧ X₀ ≤ 1+X₁ ∧ 0 ≤ X₀ for location evalrsdbb4in_v2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ 1+X₂ ∧ 0 ≤ 1+X₁+X₂ ∧ 0 ≤ 1+X₀+X₂ ∧ X₀ ≤ 1+X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb4in_v3

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb3in_v2

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb3in_v3

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb1in_v3

Found invariant 1+X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 1 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 1 ≤ X₁ ∧ 1 ≤ X₀+X₁ ∧ 1+X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb2in_v3

Found invariant 0 ≤ X₀ for location evalrsdbbin

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb2in_v1

Found invariant X₂ ≤ X₁ ∧ 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ X₁ ≤ X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb2in_v2

Found invariant 0 ≤ X₂ ∧ 0 ≤ X₁+X₂ ∧ 0 ≤ X₀+X₂ ∧ X₀ ≤ X₂ ∧ 0 ≤ X₁ ∧ 0 ≤ X₀+X₁ ∧ X₀ ≤ X₁ ∧ 0 ≤ X₀ for location evalrsdbb4in

All Bounds

Timebounds

Overall timebound:4352⋅X₀⋅X₀+624⋅X₀+34 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1088⋅X₀⋅X₀+164⋅X₀+6 {O(n^2)}
t₅: 1 {O(1)}
t₆: 1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}
t₇: 1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}
t₈: 32⋅X₀+9 {O(n)}
t₉: 1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}
t₁₀: 32⋅X₀+1 {O(n)}
t₁₁: 1 {O(1)}

Costbounds

Overall costbound: 4352⋅X₀⋅X₀+624⋅X₀+34 {O(n^2)}
t₀: 1 {O(1)}
t₁: 1 {O(1)}
t₂: 1 {O(1)}
t₃: 1 {O(1)}
t₄: 1088⋅X₀⋅X₀+164⋅X₀+6 {O(n^2)}
t₅: 1 {O(1)}
t₆: 1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}
t₇: 1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}
t₈: 32⋅X₀+9 {O(n)}
t₉: 1088⋅X₀⋅X₀+132⋅X₀+4 {O(n^2)}
t₁₀: 32⋅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: X₁ {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: 2⋅X₀ {O(n)}
t₃, X₂: 2⋅X₀ {O(n)}
t₄, X₀: X₀ {O(n)}
t₄, X₁: 34⋅X₀+1 {O(n)}
t₄, X₂: 36⋅X₀+3 {O(n)}
t₅, X₀: 2⋅X₀ {O(n)}
t₅, X₁: 68⋅X₀+2 {O(n)}
t₅, X₂: 70⋅X₀+5 {O(n)}
t₆, X₀: X₀ {O(n)}
t₆, X₁: 34⋅X₀+1 {O(n)}
t₆, X₂: 36⋅X₀+3 {O(n)}
t₇, X₀: X₀ {O(n)}
t₇, X₁: 34⋅X₀+1 {O(n)}
t₇, X₂: 36⋅X₀+3 {O(n)}
t₈, X₀: X₀ {O(n)}
t₈, X₁: 34⋅X₀+1 {O(n)}
t₈, X₂: 36⋅X₀+3 {O(n)}
t₉, X₀: X₀ {O(n)}
t₉, X₁: 34⋅X₀+1 {O(n)}
t₉, X₂: 36⋅X₀+3 {O(n)}
t₁₀, X₀: X₀ {O(n)}
t₁₀, X₁: 34⋅X₀+1 {O(n)}
t₁₀, X₂: 34⋅X₀+2 {O(n)}
t₁₁, X₀: 3⋅X₀ {O(n)}
t₁₁, X₁: 68⋅X₀+X₁+2 {O(n)}
t₁₁, X₂: 70⋅X₀+X₂+5 {O(n)}