Theorem (English: Theorem) is a statement that is proved to be true after logical restrictions. Generally speaking, only important or interesting statements are called theorems in mathematics. Proving theorem is the central activity of mathematics. A theorem states that all (full name) elements of a given class have an immutable relationship, which can be infinite, and they hold at any time without exception. (for example, there are some, so it is not a theorem). A conjecture is a mathematical statement that is considered true but not proved, or a proposition. Proved is the theorem. Conjecture is the source of theorem, but it is not the only source. A mathematical statement derived from other theorems can become a theorem without going through the process of becoming a conjecture. As mentioned above, theorems need some logical framework, and then form a set of axioms (axiom system). At the same time, the reasoning process allows new theorems and other previously discovered theorems to be derived from axioms. In propositional logic, all proved statements are called theorems.

Various mathematical narratives (in order of importance)

Lemma (also called auxiliary theorem and complementary theorem)-a part of theorem proving. This is not the main result. The proof of lemma is sometimes longer than theorem, such as Schur lemma.

Inference-a statement that appears immediately from a theorem. If proposition B can deduce proposition A quickly and simply, then proposition A is the inference of proposition B. ..

proposition

theorem

Mathematical principle

structure

Theorems generally have many conditions. Then there is the conclusion-a mathematical narrative that is established under conditions. Usually write "If the condition holds, then the conclusion holds." Writing with symbolic logic is a condition → conclusion. And the proof is not considered as a part of the theorem.

Inverse principle

If there is narrative behavior, its inverse narrative is. The reverse narrative is true, otherwise it is usually causal and unreasonable. If a statement is a theorem, its inverse statement is an inverse theorem.

If a statement and its converse statement are true, the condition is necessary and sufficient.

If a narrative is true, its reverse narrative is false and the conditions are sufficient.

If a statement is false, its reverse statement is true and the condition is necessary.

Theorem in logic

Theorem in logical language represents a set of formulas, and each formula in this set of formulas represents a piece of knowledge, so we can give a more accurate expression to the theorem (the theorem mentioned here