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203 lines
3.1 KiB
Coq
203 lines
3.1 KiB
Coq
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
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* Supplementary Coq material: subset types
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* Author: Adam Chlipala
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* License: https://creativecommons.org/licenses/by-nc-nd/4.0/
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* Much of the material comes from CPDT <http://adam.chlipala.net/cpdt/> by the same author. *)
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Require Import FrapWithoutSets.
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(* We import a pared-down version of the book library, to avoid notations that
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* clash with some we want to use here. *)
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Set Implicit Arguments.
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Set Asymmetric Patterns.
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(** * Introducing Subset Types *)
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Definition pred (n : nat) : nat :=
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match n with
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| O => O
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| S n' => n'
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end.
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Extraction pred.
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(** * Decidable Proposition Types *)
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Print sumbool.
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Notation "'Yes'" := (left _ _).
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Notation "'No'" := (right _ _).
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Notation "'Reduce' x" := (if x then Yes else No) (at level 50).
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Definition eq_nat_dec : forall n m : nat, {n = m} + {n <> m}.
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Admitted.
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Compute eq_nat_dec 2 2.
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Compute eq_nat_dec 2 3.
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Extraction eq_nat_dec.
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Section In_dec.
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Variable A : Set.
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Variable A_eq_dec : forall x y : A, {x = y} + {x <> y}.
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(* The final function is easy to write using the techniques we have developed
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* so far. *)
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Definition In_dec : forall (x : A) (ls : list A), {In x ls} + {~ In x ls}.
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Admitted.
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End In_dec.
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Compute In_dec eq_nat_dec 2 (1 :: 2 :: nil).
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Compute In_dec eq_nat_dec 3 (1 :: 2 :: nil).
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Extraction In_dec.
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(** * Partial Subset Types *)
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Inductive maybe (A : Set) (P : A -> Prop) : Set :=
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| Unknown : maybe P
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| Found : forall x : A, P x -> maybe P.
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Notation "{{ x | P }}" := (maybe (fun x => P)).
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Notation "??" := (Unknown _).
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Notation "[| x |]" := (Found _ x _).
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Print sumor.
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Notation "!!" := (inright _ _).
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Notation "[|| x ||]" := (inleft _ [x]).
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(** * Monadic Notations *)
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Notation "x <- e1 ; e2" := (match e1 with
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| Unknown => ??
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| Found x _ => e2
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end)
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(right associativity, at level 60).
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Definition doublePred : forall n1 n2 : nat, {{p | n1 = S (fst p) /\ n2 = S (snd p)}}.
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Admitted.
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Notation "x <-- e1 ; e2" := (match e1 with
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| inright _ => !!
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| inleft (exist x _) => e2
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end)
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(right associativity, at level 60).
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Definition doublePred' : forall n1 n2 : nat,
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{p : nat * nat | n1 = S (fst p) /\ n2 = S (snd p)}.
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Admitted.
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(** * A Type-Checking Example *)
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Inductive exp :=
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| Nat (n : nat)
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| Plus (e1 e2 : exp)
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| Bool (b : bool)
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| And (e1 e2 : exp).
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Inductive type := TNat | TBool.
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Inductive hasType : exp -> type -> Prop :=
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| HtNat : forall n,
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hasType (Nat n) TNat
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| HtPlus : forall e1 e2,
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hasType e1 TNat
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-> hasType e2 TNat
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-> hasType (Plus e1 e2) TNat
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| HtBool : forall b,
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hasType (Bool b) TBool
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| HtAnd : forall e1 e2,
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hasType e1 TBool
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-> hasType e2 TBool
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-> hasType (And e1 e2) TBool.
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Definition typeCheck : forall e : exp, {{t | hasType e t}}.
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Admitted.
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Compute typeCheck (Nat 0).
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Compute typeCheck (Plus (Nat 1) (Nat 2)).
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Compute typeCheck (Plus (Nat 1) (Bool false)).
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Extraction typeCheck.
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