彭罗斯台阶数学专业术语 定理证明过程中常见的短语和句子

2020年11月26日发(作者：盛国胜)
定理证明过程中常见的短语和句子
1.下面的句型可用来表达“根据什么即可得到什么”的意思
According to definition , it follows …
According to hypothesis , it follows …
According to asssumptions, it follows …
According to theorem(N), it follows …
According to lemma (A) , it follows …
According to corollary (B) , it follows …
According to the remark , it follows …
According to the fact that … , it follows …
(可以把上面的“according to ”换成“ by” )
Since …, it follows …

2. 如果一个论断可以通过一 些简单运算或简单推理而获得，由于这些运算或推理比较简单，读者可
以自行推算，因而只需直接写出论 断来，这时可用下面句型：
(1) It is easy to see that …
It is easy to show that …
It is easy to prove that …
It is easy to verify that …
It is easy to check that …
(2) It can easily be seen that …
It can easily be shown that …
It can easily be proved that …
It can easily be verified that …
It can easily be checked that …

3.如果所要提及的结论比较显浅，或是众所周知，无需作进一步的证明，这时可用下面句型：
(1) It is clear that …
It is obvious that …
It is evident that …
It is well- known that …
(2) Clearly, …
Obviously, …
Evidently,…

4.为了证明一个定理有时需要引进辅助函数，这时可用下面句型：
Let us first define the function…
Let us introduce a new function…
Let us consider the function…
Let us first investigate the function…
Let …
Set…
Define…
Put…
Consider…

5. 在一个定理中，有几个结论需要证明，其中有些结论 比较明显，可不用证明，仅需证明余下结论
即可，这时可用下面句型：
Since (A) and (B) are obvious, we need only prove (C).
Since (A) and (B) are trivial, we need only prove (C).
Since (A) and (B) are trivial, it suffices to prove (C)

6. 为了证明一个定理，有时我们并不 是直接去证明，而是证明一个新的论断，一旦新的论断得到证
明，已给定理不难由此而得证，这时可用下 面句型：
以下各句用于新的论断被证明之前：
The theorem will be proved if we can show…
The result will be proved if we can show…
The theorem will be proved by showing that…
If we can prove…then the theorem follows immediately.
以下各句用于新的论断被证明之后：
The theorem is now a direct consequence of what we have proved.