(* now, we treat the case s = consn n e s with induction Then, we have two new goalsĬorresponding on the case analysis about s (either it is Theorem length_corr : forall (n : nat) (s : seq n), length n s = n. Using Σ-types, a sorting function over lists of natural numbers can be given the type: Aujourd’hui, je veux dire la vie non prévue, inimaginable à dix-huit ans, entre les bouillies, la vaccination au tétracoq, la culotte plastique à savonner, le sirop Delabarre sur les gencives. For instance, the pairing function of type Z -> Z * Z is written fun x => (x,x) and cons (S (S O)) (cons (S O) nil) (shortened to 2::1::nil in Coq) denotes a list of type list nat made of the two elements 2 and 1. Vaccin combiné contenant les vaccins contre la diphtérie, le tétanos, la coqueluche et la poliomyélite. the type of even natural numbers:Ĭoq implements a functional programming language supporting these types. Or a form of subset types called Σ-types, e.g. Op : dom -> dom -> dom where "x * y" := (op x y) Īssoc : ∀ x y z, x * (y * z) = (x * y) * z well-formed types or structures are of type Type. Type is the sort for datatypes and mathematical structures, i.e.Or by abstracting over other existing propositions, e.g.:ĭefinition divide (x y:N) := ∃ z, x * z = y.ĭefinition prime x := ∀ y, divide y x -> y = 1 \/ y = x. Typical propositions are:Īnd new predicates can be defined either inductively, e.g.: well-formed propositions are of type Prop. Prop is the sort for propositions, i.e.
Then, all logical judgments in Coq are typing judgments: the very heart of Coq is in fact a type-checking algorithm.Ĭoq objects are sorted into two categories: the Prop sort and the Type sort: It is the role of your organization’s leadership to instill a culture of quality. You will accomplish this by focusing on prevention. Your long-term goal is to not produce defective or non-conforming products. Same language called the Calculus of Inductive Constructions (CIC). The concept of COQ tells you that there are four costs of quality: prevention, appraisal, internal failures, and external failures. Properties, programs and proofs are formalized in the Interesting additional feature of Coq is that it can automatically extractĮxecutable programs from specifications, as either Objective Caml or Haskell Programs and proofs that programs comply to their specifications. Mathematical proofs, and especially to write formal specifications,
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