coq

I have a type, say Inductive Tt := a | b | c. What's the easiest and/or best way to define a subtype of it? Suppose I want the subtype to contain only constructors a and b . A way would be to parametrize on a two-element type, e.g. bool: Definition filt (x:bool): Tt := match x with | true => a | false => b end. Check filt true: Tt. This works but is very awkward if your expression has several (possibly interdependent) subtypes defined this way. Besides, it works only half way, as no subtype is defined. For this I must additionally define e.g. Notation _Tt := ltac: (let T := type of (forall {x

I am learning Coq. I am stuck on a quite silly problem (which has no motivation, it is really silly). I want to build a function from ]2,+oo] to the set of integers mapping x to x-3. That should be simple... In any language I know, it is simple. But not in Coq. First, I write (I explain with a lot of details so that someone can explain what I don't understand in the behaviour of Coq) Definition f : forall n : nat, n > 2 -> nat. I get a subgoal ============================ forall n : nat, n > 2 -> nat which means that Coq wants a map from a proof of n>2 to the set of integers. Fine. So I want

I am experimenting with Coq Coinductive types. I use the lazy list type form the Coq'Art book (sect. 13.1.4): Set Implicit Arguments. CoInductive LList (A:Set) : Set := | LNil : LList A | LCons : A -> LList A -> LList A. Implicit Arguments LNil [A]. CoFixpoint LAppend (A:Set) (u v:LList A) : LList A := match u with | LNil => v | LCons a u' => LCons a (LAppend u' v) end. In order to match the guard condition I also use the following decomposition functions form this book: Definition LList_decomp (A:Set) (l:LList A) : LList A := match l with | LNil => LNil | LCons a l' => LCons a l' end. Lemma

I'd like to have standard notation like "x ∈ { x }" in Coq. But there are problems: 1) Curly braces has special meaning in Coq, so the following happens: Notation " x ∈ y " :=(tin x y) (at level 50). Notation " { x } ":=(Sing x). Check fun x => (x ∈ { x }). (*error: Unknown interpretation for notation "_ ∈ { _ }". *) How to define this notation correctly? 2) If the first problem cannot be solved, there is another. (Here I decided to use the additional symbol '`' in the notation.) Notation " { x }` ":=(Sing x). Check fun x => (x ∈ { x }`). (* fun x : Ens => x ∈ {x }` *) Now I should a) either

Is it possible to define a single notation for multiple constructors in Coq? If the constructors differ by their argument types, they might be inferrable from them. A minimal (non-)working example: Inductive A : Set := a | b | c: C -> A | d: D -> A with C: Set := c1 | c2 with D: Set := d1 | d2. Notation "' x" := (_ x) (at level 19). Check 'c1. (*?6 c1 : ?8*) In this case, constructor inference doesn't work. Maybe there's another way to specify a constructor as a variable? You can create a typeclass with the constructors as instances and let the instance resolution mechanism infer the

I want to change the goal from S x = S y to x = y . It's like inversion , but for the goal instead of a hypothesis. Such a tactic seems legit, because when we have x = y , we can simply use rewrite and reflexivity to prove the goal. Currently I always find myself using assert (x = y) to introduce a new subgoal, but it's tedious to write when x and y are complex expression. The tactic apply f_equal. will do what you want, for any constructor or function. The lema f_equal shows that for any function f , you always have x = y -> f x = f y . This allows you to reduce the goal from f x = f y to x =

I was going through IndProp in software foundations and Adam Chlipala's chapter 4 book and I was having difficulties understanding inductive propositions. For the sake of a running example, lets use: Inductive ev : nat -> Prop := | ev_0 : ev 0 | ev_SS : forall n : nat, ev n -> ev (S (S n)). I think I do understand "normal" types using Set like: Inductive nat : Set := | O : nat | S : nat -> nat. and things like O just return the single object of type nat and S O is taking an object of type nat and returning another different one of the same type nat . By different object I guess I mean they