(* Copyright (c) 2008, 2011, 2015, Adam Chlipala * * This work is licensed under a * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 * Unported License. * The license text is available at: * http://creativecommons.org/licenses/by-nc-nd/3.0/ *) (* Types and notations presented in Chapter 6 *) Set Implicit Arguments. Set Asymmetric Patterns. Notation "!" := (False_rec _ _) : specif_scope. Notation "[ e ]" := (exist _ e _) : specif_scope. Notation "x <== e1 ; e2" := (let (x, _) := e1 in e2) (right associativity, at level 60) : specif_scope. Local Open Scope specif_scope. Delimit Scope specif_scope with specif. Notation "'Yes'" := (left _ _) : specif_scope. Notation "'No'" := (right _ _) : specif_scope. Notation "'Reduce' x" := (if x then Yes else No) (at level 50) : specif_scope. Notation "x || y" := (if x then Yes else Reduce y) (at level 50, left associativity) : specif_scope. Section sumbool_and. Variables P1 Q1 P2 Q2 : Prop. Variable x1 : {P1} + {Q1}. Variable x2 : {P2} + {Q2}. Definition sumbool_and : {P1 /\ P2} + {Q1 \/ Q2} := match x1 with | left HP1 => match x2 with | left HP2 => left _ (conj HP1 HP2) | right HQ2 => right _ (or_intror _ HQ2) end | right HQ1 => right _ (or_introl _ HQ1) end. End sumbool_and. Infix "&&" := sumbool_and (at level 40, left associativity) : specif_scope. Inductive maybe (A : Set) (P : A -> Prop) : Set := | Unknown : maybe P | Found : forall x : A, P x -> maybe P. Notation "{{ x | P }}" := (maybe (fun x => P)) : specif_scope. Notation "??" := (Unknown _) : specif_scope. Notation "[| x |]" := (Found _ x _) : specif_scope. Notation "x <- e1 ; e2" := (match e1 with | Unknown => ?? | Found x _ => e2 end) (right associativity, at level 60) : specif_scope. Notation "!!" := (inright _ _) : specif_scope. Notation "[|| x ||]" := (inleft _ [x]) : specif_scope. Notation "x <-- e1 ; e2" := (match e1 with | inright _ => !! | inleft (exist x _) => e2 end) (right associativity, at level 60) : specif_scope. Notation "e1 ;; e2" := (if e1 then e2 else ??) (right associativity, at level 60) : specif_scope. Notation "e1 ;;; e2" := (if e1 then e2 else !!) (right associativity, at level 60) : specif_scope. Section partial. Variable P : Prop. Inductive partial : Set := | Proved : P -> partial | Uncertain : partial. End partial. Notation "[ P ]" := (partial P) : type_scope. Notation "'Yes'" := (Proved _) : partial_scope. Notation "'No'" := (Uncertain _) : partial_scope. Local Open Scope partial_scope. Delimit Scope partial_scope with partial. Notation "'Reduce' v" := (if v then Yes else No) : partial_scope. Notation "x || y" := (if x then Yes else Reduce y) : partial_scope. Notation "x && y" := (if x then Reduce y else No) : partial_scope.