Question

I have a question about the: Record and Definition

I have this definition:

Definition rule := term -> term.

and I write a boolean function for it.

Definition beq_rule a b := beq_term a && beq_term b.

where beq_term : term -> term -> bool.

so my definition of beq_rule actually return exactly type of beq_term which is not what I want here. I want it return for me a type: rule -> rule -> bool.

So I changed a definition of rule by Record:

Record rule := mkRule {lhs : term; rhs : term}.

and

Definition beq_rule (a b : rule) : bool :=
 beq_term (lhs a) (lhs b) && beq_term (rhs a) (rhs b).

My question is that:

1) What is the different between my first defined rule used Definition and another used Record?

2) If I want to define rule by Definition can I give an alias lhs and rhs likes in Record definition?

Answer 1

Your two definitions of rule are saying totally different things

Definition rule := term -> term

is defining rule as a type (or Prop) alias of the function type term -> term. Hence

Definition not_what_you_meant : rule := fun t => t.

will happily compile.

As to the relation between Record and Definition. Record is just a macro that converts into an Inductive. So

Record rule := mkRule {lhs : term; rhs : term}.

is the same as

Inductive rule := mkRule : term -> term -> rule.

plus accessor functions

Definition lhs (r : rule) : term := 
 match r with
  mkRule l _ => l
 end.

etc.

You should think of Inductive as being fundamentally different from Definition. Definition defines an alias. Another way f saying this is Definitions are “referentially transparent”, you can (up to variable renaming) always substitute the right hand side of a definition for any occurrence of its name.

Inductive on the other hand defines type (elements of Coqs universes) by listing off a set of constructors. In more logical way of thinking, Inductive defines a logical proposition in terms of its elimination/introduction rules in a way that ensures “harmony”.