Inductive Predicates Unit vs True

Inductive unit : Set := tt : unit Inductive True : Prop := I : True Curry Howard Isomorphism But Coq distinguish Proof and Program

Efficiency? Proof Prevalence?

Propositional Logic not = fun A : Prop => A -> False

     : Prop -> Prop Inductive and (A : Prop) (B : Prop) : Prop :=

conj : A -> B -> A /\ B Inductive or (A : Prop) (B : Prop) : Prop :=

    or_introl : A -> A \/ B | or_intror : B -> A \/ B

Bool Vs Prop bool is datatype of two members, true, false Prop is primitive type in Coq, and True, False

are two members in Prop Coq use Constructive Logic, while P\/~P is not

allowed Issues

bool is decidable Constructive logic allow us to extract program

from proof We can write program by proof a theorem, though

maybe less efficient

First-Order Logic forall and exists

forall is built-in exists:

Inductive ex (A : Type) (P : A -> Prop) : Prop :=    ex_intro : forall x : A, P x -> ex P

Tactics Tauto

Solve intuition logic, only unfold not Intuition

Use tauto, and apply tactics Intuition=intuition auto.

Firstorder Extend tauto to firstorder logic

Trivial Restricted auto

auto Eauto

Auto using eapply, generate ‘_’

Basic dependant type Inductive isZero : nat -> Prop :=

| IsZero : isZero 0. Inductive even : nat -> Prop :=

| EvenO : even O| EvenSS : forall n, even n -> even (S (S n)).

Example of product: forall a:A,B (s,s’,s’’) 约束 分类(s,s’,s’) s,s’ {Set,Prop} 普通的类型(Type,Prop,Prop)

(s,Type,Type) Type denpends on a value

(Type,Type,Type) High order type