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