What is a constructor in Coq? -

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i having trouble understanding principles of constructor , how works.

for example, in coq, have been taught define natural numbers this:

inductive nat : type :=    | o : nat    | s : nat -> nat.

and have been told s constructor, mean?

if do:

check (s (s (s (s o)))).

i 4 , of type nat.

how work, , how coq know (s (s (s (s o)))) represents 4?

i guess answer extremely clever background magic in coq.

inductive naturals : type :=    | z : naturals    | n : naturals -> naturals.

says following things:

  1. z term of type naturals

  2. when e term of type naturals, n e term of type naturals.

  3. applying z or n 2 ways create natural. when given arbitrary natural, know either made 1 or other.

  4. two terms e1 , e2 of type naturals equal if , if both z or if respectively n t1 , n t2 t1 equal t2.

you can see how these rules generalize arbitrary inductive type definitions. however, turns out when type definition constructors of shape of z , n, these properties correspond more or less peano's axioms natural numbers. why type named nat pre-defined in coq constructors of these shapes used represent natural numbers. type receives special treatment because gets tiring manipulate raw form, special treatments there convenience.


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