coq

According to this course , all constructors (for inductive types) are injective and disjoint: ...Similar principles apply to all inductively defined types: all constructors are injective, and the values built from distinct constructors are never equal. For lists, the cons constructor is injective and nil is different from every non-empty list. For booleans, true and false are unequal. (And the inversion tactic based on this assumption) I am just wondering how do we know this assumption holds? How do we know that, e.g., we cannot define natural numbers based on 1) a Successor and maybe a

I would like to have Ltac tactics for these simple inference rules. In Modus Ponens, if I have H:P->Q and H1:P , Ltac mp H H1 will add Q to the context as H2 : Q . In Modus Tollens, if I have H:P->Q and H1:~Q , then Ltac mt H H1 will add H2:~P to the context. I have written one for the modus ponens, but it does not work in the complicated cases where the precedent is also an implication. Ltac mp H0 H1 := let H := fresh "H" in apply H0 in H1 as H. Edit : I have found the answer to Modus Ponens in another seemingly unrelated question ( Rewrite hypothesis in Coq, keeping implication ), where a

I want to make a Ltac tactic in coq which would take either 1 or 3 arguments. I have read about ltac_No_arg in the LibTactics module but if I understood it correctly I would have to invoke my tactic with : Coq < mytactic arg_1 ltac_no_arg ltac_no_arg. which is not very convenient. Is there any way to get a result like this ? : Coq < mytactic arg_1. Coq < mytactic arg_1 arg_2 arg_3. We can use the Tactic Notation mechanism to try to solve your issue because it can handle variadic arguments. Let's reuse ltac_No_arg and define a dummy tactic mytactic for the purposes of demonstration Inductive

$ coqtop -nois Welcome to Coq 8.7.0 (October 2017) Coq < Ltac i := idtac. Toplevel input, characters 0-4: > Ltac i := idtac. > ^^^^ Error: Syntax error: illegal begin of vernac. I am redeveloping "Coq.Init.Prelude" and "HoTT.Basics.Overture" under "coqtop -nois" for pratice. I find it hard to write expressions directly. That's why I want to use tactics. I wonder why I can not use "Ltac". Ltac is now provided as a plugin, which you’ll need to load to use: Declare ML Module "ltac_plugin". 来源： https://stackoverflow.com/questions/48837345/can-i-define-a-tactic-under-coqtop-nois

I am trying to write a function for computing natural division in Coq and I am having some trouble defining it since it is not structural recursion. My code is: Inductive N : Set := | O : N | S : N -> N. Inductive Bool : Set := | True : Bool | False : Bool. Fixpoint sum (m :N) (n : N) : N := match m with | O => n | S x => S ( sum x n) end. Notation "m + n" := (sum m n) (at level 50, left associativity). Fixpoint mult (m :N) (n : N) : N := match m with | O => O | S x => n + (mult x n) end. Notation "m * n" := (mult m n) (at level 40, left associativity). Fixpoint pred (m : N) : N := match m

I was working is this simple exercise about writing certified function using subset types. The idea is to first write a predecessor function pred : forall (n : {n : nat | n > 0}), {m : nat | S m = n.1}. and then using this definition give a funtion pred2 : forall (n : {n : nat | n > 1}), {m : nat | S (S m) = n.1}. I have no problem with the first one. Here is my code Program Definition pred (n : {n : nat | n > 0}) : {m : nat | S m = n.1} := match n with | O => _ | S n' => n' end. Next Obligation. elimtype False. compute in H. inversion H. Qed. But I cannot workout the second definition. I

In my coq development I am learning how to create new tactics tailored to my problem domain, a la Prof. Adam Chlipala . On that page he describes how to create powerful custom tactics by e.g. combining repeat with match . Now, I already have a powerful one-shot tactic in use, auto . It strings together chains of steps found from hint databases. I have invested some effort in curating those hint databases, so I'd like to continue using it as well. However this presents a problem. It isn't clear what the "right" way is to incorporate auto 's functionality into customized tactics. For example,

I'm having some troubles in Coq when trying to perform case analysis on the result of a function (which returns an inductive type). When using the usual tactics, like elim , induction , destroy , etc, the information gets lost. I'll put an example: We first have a function like so: Definition f(n:nat): bool := (* definition *) Now, imagine we are at this step in the proof of a specific theorem: n: nat H: f n = other_stuff ------ P (f n ) When I apply a tactic, like let's say, induction (f n) , this happens: Subgoal 1 n:nat H: true = other_stuff ------ P true Subgoal 2 n:nat H: false = other