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「SF-LC」1 Basics
These series of notes combined My notes on reading Software Foundation and (if any) watching on Coq intensive (m : nat) : nat := Fixpoint and Structrual Recursion This requirement is a fundamental feature of Coq ’s design: In particular, it guarantees that every function that can be defined in Coq will terminate However, Coq’s “decreasing analysis” is not very sophisticated. E.g. Any as clause could be ommited and Coq will fill in random var name auto-magically.
58720编辑于 2022-03-14 - 来自专栏KAAAsS's Blog
用了一段时间Agda的感想
和Coq相比，虽然Gallina也支持使用Unicode字符作为identifier，但是Coq并没有广泛使用。 在证明方面，Agda和Coq有本质的不同。 而Coq却完全相反。Coq使用了不同的Tactics来辅助证明。在Coq中进行证明的过程更加类似于一般的数学证明。以下是证明皮尔士定律与排中律等价的Agda、Coq程序片段。 相比之下，Coq的证明过程更加近似于人工证明。Coq的证明中自然而然的带入的证明的“顺序”，所以在一定程度上，阅读Coq的代码更容易得到证明的大致思路。 而且由于Tactics的应用是有序的，所以结合相关证明信息的说明，Coq代码的证明过程可以得到非常直观的展现。 Coq有官方的CoqIde，还有比如ProofAssistant也可以使用Coq。
2K10编辑于 2022-01-14 - 来自专栏R语言及实用科研软件
CoQ10强势逆转~
此外，研究团队发现补充辅酶Q10（CoQ10）能够有效抑制铁死亡、改善肌肉功能，为OA相关肌肉病变的治疗提供了全新思路✨。 图7N–P：通过AAV介导的HMGCR基因过表达，可以恢复股四头肌中CoQ10水平，提示靶向HMGCR–CoQ10轴可有效干预OA相关铁死亡。 研究结果八 CoQ10有效抑制铁死亡，改善OA相关肌肉病变 图8A–C：在OA小鼠模型中补充CoQ10后，股四头肌纤维横截面积（CSA）显著恢复，肌肉重量和肌力明显提升。 图8J–K：不仅肌肉病变得到缓解，CoQ10处理还减轻了OA小鼠的关节结构损伤，说明其具有全局保护作用。 该研究为 OA合并肌肉萎缩的治疗 提供了全新的干预思路：靶向衰老巨噬细胞或利用铁死亡抑制剂（如CoQ10）。
30810编辑于 2025-11-13 - 来自专栏零域Blog
「SF-LC」8 Maps
The Coq Standard Lib From now on, importing from std lib. (but should not notice much difference) From Coq Require Import Arith.Arith. From Coq Require Import Bool.Bool. Require Export Coq.Strings.String. From Coq Require Import Logic.FunctionalExtensionality. From Coq Require Import Lists.List. - Notation In Coq, we can leverage some meta programming: Notation "'_' '!
48340编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」9 ProofObjects
The fundamental idea of Coq is that: provability in Coq is represented by concrete evidence. (Coq has maken sure the type safety.) Wow… CoqI: What’s Coq logic? Forall + Inductive type (+ coinduction), that’s it. Coq Magics, “Implicit” Implicit and Overloading?? So what’s the problem? (since Coq is total!) Lemma four: 2 + 2 == 1 + 3. Proof. apply eq_refl. Qed.
75820编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」6 Logic
. (* ===> nat -> Prop *) Properties In Coq, functions that return propositions are said to define properties the type abstraction) works at type level, accepts type var, substitute and reture a type. forall in Coq Coq Intensive Q: why destruct can work on and? is and inductively defined? A: Yes. So I guess when we apply theorems, Coq implicitly use the type of the Proof Object. This is not built-in Coq, but we can add them as Axioms. Why not add everything?
76720编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」4 Poly
Polymorphism Until today, We were living in the monomorphic world of Coq. Polymorphic Type and Constructors But of course Coq supports polymorphic type. Coq pick the ML way, not haskell way. Fluet https://stackoverflow.com/questions/32153710/what-does-error-universe-inconsistency-mean-in-coq The error message you saw means that Coq’s constraint solver for universe levels says that there can
1.5K40编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」5 Tactics
So we need to tell Coq explicitly which value should be picked for m: apply trans_eq with (m:=[c;d]). In Coq Intensive 2 (2018), someone proposed the below which works: Example trans_eq_example'' : forall apply trans_eq. (* Coq was able to match three at all at this time...hmm *) Qed. IMH, which one to use depends on semantics from Coq Intensive (not sure why it’s not the case in book Side question: could Coq derive equality function for inductive type? A: nope.
70530编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」3 List
Instead of SML/OCaml’s @, Coq chooses Haskell’s ++. hd with default Coq function (for some reason) has Search Search rev (* list all theorems of [rev] *) Coq Conditionals (if then else) Fixpoint nth_error One small generalization: since the boolean type in Coq is not built-in. Coq actually supports conditional expr over any inductive defined typewith two constructors.
60220编辑于 2022-03-14 - 来自专栏互联网杂技
用于数学的 10 个优秀编程语言
COQ / GALLINA Coq是一个交互式的定理证明工具。它允许表达数学断言，机械地检查这些断言的证明，帮助找到形式化的证明，并从其正式规范的建设性证明中提取认证程序。 Coq工作在归纳结构微积分理论的基础上，归纳结构微积分是结构微积分的一个衍生物。 作为编程语言，Coq实现了一种依赖类型的函数式编程语言，作为逻辑系统，Coq实现了一个更高阶的类型理论。 Coq提供了一种名为Gallina的规范语言。用Gallina编写的程序具有弱化的标准化属性 ——它们总是终止。 5. PROLOG Prolog是与人工智能和计算语言学相关的通用逻辑编程语言。 语言支持可与Coq媲美的交互式定理证明，包括策略，即使在定理证明之前，重点仍然放在通用编程上。 Idris的其他目标是“充足”性能，易于管理的副作用和支持实施嵌入式领域特定语言。 它结合了Haskell和Coq的元素。很有意思。 8. JULIA Julia是用于技术计算的高级和高性能的动态编程语言，具有用户熟悉的其他技术计算环境的语法。
4.2K100发布于 2018-04-03 - 来自专栏零域Blog
「SF-PLF」7 Stlc
. — Coq \x:T. t is always value — most FP lang, either CBV or CBN Coq 这么做挺奇怪的，不过对 Coq 来说: terms can be considered equiv up to the computation VM (在其项化简可以做到的范围内都算相等) this rich the notion of Coq’s value (所以 Coq 的值的概念是比一般要大的) Three ways to construct value (unary relation = predicate) Inductive value : tm → (in metalanguage, Coq here) 如果我们考虑用于替换掉某个变量的项 s 其本身也含有自由变量， 那么定义替换将会变得困难一点。 Provability in Coq witness decidabile operations. show term is “not typeable” Keep inversion till the
53520编辑于 2022-03-02 - 来自专栏零域Blog
「SF-LC」11 Rel
Back to Coq “relation” is a general idea. but in Coq standard lib it means “binary relation on a set X” Coq identifier relation will always refer to a binary relation between some set and itself. it’s defined generically about certain sorts of relations It’s pretty fun to see all mathematical things defined in Coq order under which every pair of elements is comparable is called a total order or linear order In the Coq Back to Coq Inductive clos_refl_trans {A: Type} (R: relation A) : relation A := | rt_step x y (H :
57320编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」15 Extraction
When Coq processes this command: The file imp1.ml has been created by extraction. Controlling Extraction of Specific Types 如果不做任何处理的话…生成的 ml 里的 nat 则都会是 Church Numeral… We can tell Coq how to extract certain Inductive definitions to specific OCaml types. we must say: how the Coq type 比如这么做很诱人……但是我们 Coq 的定义里 0 - 1 = 0, OCaml 的 int 则会有负数… Recursor 的理论与实现 - a “encoding” of case expression
74410编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」12 Imp
From Coq Intensive: when we simpl on APlus case. it’s not “simplified” but give us a pattern matching Coq Automation Tacticals “higher-order tactics”. try T and ; tacticals if T fail, try T successfully e.g. repeat simpl This does not affect Coq’s logical consistency, construction process diverges means (change auto-generated name that we don’t like…) the below three are very useful in Coq Automation (w probably has to be option (Coq is total!)
2K20编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」7 Ind Prop
In particular, we can use Coq’s apply tactic with the rule names to prove even for particular numbers Introducing even with an Inductive declaration tells Coq that these two constructors are the only ways But Coq provide the inversion tactics that does more! simply trust what we declared… that’s why there is even Quick Check for Coq. Require Export Coq.Strings.Ascii. Definition string := list ascii.
88720编辑于 2022-03-14 - 来自专栏零域Blog
「SF-LC」10 IndPrinciples
Basic 每次我们使用 Inductive 来声明数据类型时，Coq 会自动为这个类型生成 归纳原理。 Every time we declare a new Inductive datatype, Coq automatically generates an induction principle for S n) ------------------------- ∀ n : nat, P n induction tactic is wrapper of apply t_ind Coq 为每一个 Inductive 定义的数据类型生成了归纳原理，包括那些非递归的 Coq generates induction principles for every datatype defined Induction Principles in Prop 理解依赖类型的归纳假设 与 Coq 排除证据参数的原因 除了集合 Set，命题 Prop 也可以是归纳定义与 induction on 得.
1.1K30编辑于 2022-03-14 铁死亡机制全解析：五大核心通路 | MCE
相反，System xc- -GSH-GPX4，AIFM2-CoQ10，GCH1-BH4 轴的激活限制了铁死亡中的脂质过氧化。 通路CoQ10 的还原形式泛醇（Ubiquinol）可捕获脂质过氧化物自由基，抑制脂质过氧化，而 AIFM2（FSP1） 可催化 CoQ10 的再生，也可通过激活 ESCRT-III 膜修复系统来预防癌细胞中的铁死亡 GCH1-BH4 轴 可控制 BH4 的内源性产生、CoQ10 丰度，以及异常磷脂的过氧化，从而抑制铁死亡[11]。 DHODH 负责催化嘧啶 核苷酸合成途径中二氢乳清酸（DHO）氧化为乳清酸（OA），同时将内膜中的CoQ10 还原为泛醇 （Ubiquinol）来抑制线粒体内膜的铁死亡。 因此，在线粒体 GPX4 功能缺失时，DHODH 能通过维持 CoQ10/泛醇抗氧化系统来部分 补偿，从而延缓铁死亡的发生 [12]。
2.9K21编辑于 2025-06-24- 来自专栏零域Blog
「SF-LC」13 ImpParser
the parser relies on some “monadic” programming idioms basically, parser combinator (But 非常麻烦 in Coq) [] -> [] [(v,out)] -> parse (f v) out) combinator many Coq
49810编辑于 2022-03-14 - 来自专栏EmacsTalk
007. 编辑器专家的 Emacs 世界
[14] • 如何防止小指痛：上机械键盘，用手掌边沿去按 Ctrl 比用小指按更容易更舒服 •嘉宾推荐 • Coq[15] • Dart + Flutter •主播推荐 Get in Touch • 收听方式见 : https://blog.csdn.net/DelphiNew/article/details/80112823 [15] Coq: https://coq.inria.fr/ [16] 530146104
74210编辑于 2022-07-26 - 来自专栏汇智网教程
收藏贴 ：2019年必备43种区块链开发工具 原
官网：https://github.com/seeseplusplus/velma 41、Coq Coq是一个形式化证明管理系统，可用来验证你的代码没有bug。 ? 官网：https://coq.inria.fr 42、Isabelle Isabelle类似于COq，用来证明你的代码没有bug。 ?
2.1K50发布于 2019-05-14
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