Nondestructive decomposition and judging whether it is a non-destructive decomposition method

Lossless decomposition
For any table T and one of its function-dependent set f,t, a decomposition is a set of tables {t1,t2, ... Tk}, the collection has two properties:
(1) For each table in this set Ti,head (Ti) is a subset of Head (T);
(2) Head (T) =head (T1) u head (T2) u head (Tk).
If any of the contents of table T, the function dependency in F guarantees that the following relationship is established, it is said that a decomposition of table t relative to the function dependency set F is a lossless decomposition, sometimes referred to as a lossless connection decomposition. That satisfies:
T=T1 Join T2 join ... join TK

Note:
The definition of non-destructive decomposition requires that the concatenated table's connection be able to get the information from the original table, and that this should be true for any future possible content of the original table.

theorem 1
Given table T and attribute set x,x belong to head (T), the following two statements are equivalent:
(1) x is a super-key of T.
(2) X->head (t), which is the attribute set X function, determines all the properties in T. The equivalent expression is x+ =head (T).
Proof:
Known (1), the definition of a super-key allows you to know that the X attribute in T uniquely identifies one row, that is, any two rows are the same on the X property, and the values of the remaining properties must be the same. That is, if any two rows have the same x attribute, the two rows are identical, that is, the two rows are the same on all properties. This means X->head (T).
Known as (2), X->head (t), the definition of a function dependency indicates that two rows of table T cannot have the same value on X and a different value on the Head (T) property at the same time. This means that two different rows cannot take the same value on X, and all x uniquely identifies a row, where X is a super-key of T.

theorem 2
Given table t and its valid function dependency set F, a decomposition that divides t into two tables {T1,T2} is a lossless decomposition of T, when and only if head (T1) and head (T2) are true subsets of head (T), head (t) =head (T1) U Head (T2) (i.e. , all properties of T are repeated in T1 or T2), and one of the following function dependencies can be deduced by F:
(1) Head (T1) turn head (T2)->head (T1)
Or
(2) Head (T1) turn head (T2)->head (T2)

Example:
Given table T,head (T) =a B C, there is a function dependent b->c. Decomposed into Tables T1 and T2,head (T1) =a b,head (T2) =b C. Using theorem 2, we have Head (T1) intersection head (T2)->head (T2), which is B->BC, which is obvious from B->c. So the decomposition is non-destructive decomposition.