(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *)
(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **)
Require Bool.
(** A theory of boolean predicates and operators. A large part of this file is concerned with boolean reflection. Definitions and notations: is_true b == the coercion of b : bool to Prop (:= b = true). This is just input and displayed as `b''. reflect P b == the reflection inductive predicate, asserting that the logical proposition P : prop with the formula b : bool. Lemmas asserting reflect P b are often referred to as "views". iffP, appP, sameP, rwP :: lemmas for direct manipulation of reflection views: iffP is used to prove reflection from logical equivalence, appP to compose views, and sameP and rwP to perform boolean and setoid rewriting. elimT :: coercion reflect >-> Funclass, which allows the direct application of `reflect' views to boolean assertions. decidable P <-> P is effectively decidable (:= {P} + {~ P}. contra, contraL, ... :: contraposition lemmas. altP my_viewP :: natural alternative for reflection; given lemma myviewP: reflect my_Prop my_formula, have #[#myP | not_myP#]# := altP my_viewP. generates two subgoals, in which my_formula has been replaced by true and false, resp., with new assumptions myP : my_Prop and not_myP: ~~ my_formula. Caveat: my_formula must be an APPLICATION, not a variable, constant, let-in, etc. (due to the poor behaviour of dependent index matching). boolP my_formula :: boolean disjunction, equivalent to altP (idP my_formula) but circumventing the dependent index capture issue; destructing boolP my_formula generates two subgoals with assumptions my_formula and ~~ myformula. As with altP, my_formula must be an application. \unless C, P <-> we can assume property P when a something that holds under condition C (such as C itself). := forall G : Prop, (C -> G) -> (P -> G) -> G. This is just C \/ P or rather its impredicative encoding, whose usage better fits the above description: given a lemma UCP whose conclusion is \unless C, P we can assume P by writing: wlog hP: / P by apply/UCP; (prove C -> goal). or even apply: UCP id _ => hP if the goal is C. classically P <-> we can assume P when proving is_true b. := forall b : bool, (P -> b) -> b. This is equivalent to ~ (~ P) when P : Prop. implies P Q == wrapper variant type that coerces to P -> Q and can be used as a P -> Q view unambiguously. Useful to avoid spurious insertion of <-> views when Q is a conjunction of foralls, as in Lemma all_and2 below; conversely, avoids confusion in apply views for impredicative properties, such as \unless C, P. Also supports contrapositives. a && b == the boolean conjunction of a and b. a || b == the boolean disjunction of a and b. a ==> b == the boolean implication of b by a. ~~ a == the boolean negation of a. a (+) b == the boolean exclusive or (or sum) of a and b. #[# /\ P1 , P2 & P3 #]# == multiway logical conjunction, up to 5 terms. #[# \/ P1 , P2 | P3 #]# == multiway logical disjunction, up to 4 terms. #[#&& a, b, c & d#]# == iterated, right associative boolean conjunction with arbitrary arity. #[#|| a, b, c | d#]# == iterated, right associative boolean disjunction with arbitrary arity. #[#==> a, b, c => d#]# == iterated, right associative boolean implication with arbitrary arity. and3P, ... == specific reflection lemmas for iterated connectives. andTb, orbAC, ... == systematic names for boolean connective properties (see suffix conventions below). prop_congr == a tactic to move a boolean equality from its coerced form in Prop to the equality in bool. bool_congr == resolution tactic for blindly weeding out like terms from boolean equalities (can fail). This file provides a theory of boolean predicates and relations: pred T == the type of bool predicates (:= T -> bool). simpl_pred T == the type of simplifying bool predicates, based on the simpl_fun type from ssrfun.v. mem_pred T == a specialized form of simpl_pred for "collective" predicates (see below). rel T == the type of bool relations. := T -> pred T or T -> T -> bool. simpl_rel T == type of simplifying relations. := T -> simpl_pred T predType == the generic predicate interface, supported for for lists and sets. pred_sort == the predType >-> Type projection; pred_sort is itself a Coercion target class. Declaring a coercion to pred_sort is an alternative way of equiping a type with a predType structure, which interoperates better with coercion subtyping. This is used, e.g., for finite sets, so that finite groups inherit the membership operation by coercing to sets. {pred T} == a type convertible to pred T, but whose head constant is pred_sort. This type should be used for parameters that can be used as collective predicates (see below), as this will allow passing in directly collections that implement predType by coercion as described above, e.g., finite sets. := pred_sort (predPredType T) If P is a predicate the proposition "x satisfies P" can be written applicatively as (P x), or using an explicit connective as (x \in P); in the latter case we say that P is a "collective" predicate. We use A, B rather than P, Q for collective predicates: x \in A == x satisfies the (collective) predicate A. x \notin A == x doesn't satisfy the (collective) predicate A. The pred T type can be used as a generic predicate type for either kind, but the two kinds of predicates should not be confused. When a "generic" pred T value of one type needs to be passed as the other the following conversions should be used explicitly: SimplPred P == a (simplifying) applicative equivalent of P. mem A == an applicative equivalent of collective predicate A: mem A x simplifies to x \in A, as mem A has in fact type mem_pred T. --> In user notation collective predicates _only_ occur as arguments to mem: A only appears as (mem A). This is hidden by notation, e.g., x \in A := in_mem x (mem A) here, enum A := enum_mem (mem A) in fintype. This makes it possible to unify the various ways in which A can be interpreted as a predicate, for both pattern matching and display. Alternatively one can use the syntax for explicit simplifying predicates and relations (in the following x is bound in E): #[#pred x | E#]# == simplifying (see ssrfun) predicate x => E. #[#pred x : T | E#]# == predicate x => E, with a cast on the argument. #[#pred : T | P#]# == constant predicate P on type T. #[#pred x | E1 & E2#]# == #[#pred x | E1 && E2#]#; an x : T cast is allowed. #[#pred x in A#]# == #[#pred x | x in A#]#. #[#pred x in A | E#]# == #[#pred x | x in A & E#]#. #[#pred x in A | E1 & E2#]# == #[#pred x in A | E1 && E2#]#. #[#predU A & B#]# == union of two collective predicates A and B. #[#predI A & B#]# == intersection of collective predicates A and B. #[#predD A & B#]# == difference of collective predicates A and B. #[#predC A#]# == complement of the collective predicate A. #[#preim f of A#]# == preimage under f of the collective predicate A. predU P Q, ..., preim f P == union, etc of applicative predicates. pred0 == the empty predicate. predT == the total (always true) predicate. if T : predArgType, then T coerces to predT. {: T} == T cast to predArgType (e.g., {: bool * nat}). In the following, x and y are bound in E: #[#rel x y | E#]# == simplifying relation x, y => E. #[#rel x y : T | E#]# == simplifying relation with arguments cast. #[#rel x y in A & B | E#]# == #[#rel x y | #[#&& x \in A, y \in B & E#]# #]#. #[#rel x y in A & B#]# == #[#rel x y | (x \in A) && (y \in B) #]#. #[#rel x y in A | E#]# == #[#rel x y in A & A | E#]#. #[#rel x y in A#]# == #[#rel x y in A & A#]#. relU R S == union of relations R and S. relpre f R == preimage of relation R under f. xpredU, ..., xrelpre == lambda terms implementing predU, ..., etc. Explicit values of type pred T (i.e., lamdba terms) should always be used applicatively, while values of collection types implementing the predType interface, such as sequences or sets should always be used as collective predicates. Defined constants and functions of type pred T or simpl_pred T as well as the explicit simpl_pred T values described below, can generally be used either way. Note however that x \in A will not auto-simplify when A is an explicit simpl_pred T value; the generic simplification rule inE must be used (when A : pred T, the unfold_in rule can be used). Constants of type pred T with an explicit simpl_pred value do not auto-simplify when used applicatively, but can still be expanded with inE. This behavior can be controlled as follows: Let A : collective_pred T := #[#pred x | ... #]#. The collective_pred T type is just an alias for pred T, but this cast stops rewrite inE from expanding the definition of A, thus treating A into an abstract collection (unfold_in or in_collective can be used to expand manually). Let A : applicative_pred T := #[#pred x | ... #]#. This cast causes inE to turn x \in A into the applicative A x form; A will then have to unfolded explicitly with the /A rule. This will also apply to any definition that reduces to A (e.g., Let B := A). Canonical A_app_pred := ApplicativePred A. This declaration, given after definition of A, similarly causes inE to turn x \in A into A x, but in addition allows the app_predE rule to turn A x back into x \in A; it can be used for any definition of type pred T, which makes it especially useful for ambivalent predicates as the relational transitive closure connect, that are used in both applicative and collective styles. Purely for aesthetics, we provide a subtype of collective predicates: qualifier q T == a pred T pretty-printing wrapper. An A : qualifier q T coerces to pred_sort and thus behaves as a collective predicate, but x \in A and x \notin A are displayed as: x \is A and x \isn't A when q = 0, x \is a A and x \isn't a A when q = 1, x \is an A and x \isn't an A when q = 2, respectively. #[#qualify x | P#]# := Qualifier 0 (fun x => P), constructor for the above. #[#qualify x : T | P#]#, #[#qualify a x | P#]#, #[#qualify an X | P#]#, etc. variants of the above with type constraints and different values of q. We provide an internal interface to support attaching properties (such as being multiplicative) to predicates: pred_key p == phantom type that will serve as a support for properties to be attached to p : {pred _}; instances should be created with Fact/Qed so as to be opaque. KeyedPred k_p == an instance of the interface structure that attaches (k_p : pred_key P) to P; the structure projection is a coercion to pred_sort. KeyedQualifier k_q == an instance of the interface structure that attaches (k_q : pred_key q) to (q : qualifier n T). DefaultPredKey p == a default value for pred_key p; the vernacular command Import DefaultKeying attaches this key to all predicates that are not explicitly keyed. Keys can be used to attach properties to predicates, qualifiers and generic nouns in a way that allows them to be used transparently. The key projection of a predicate property structure such as unsignedPred should be a pred_key, not a pred, and corresponding lemmas will have the form Lemma rpredN R S (oppS : @opprPred R S) (kS : keyed_pred oppS) : {mono -%%R: x / x \in kS}. Because x \in kS will be displayed as x \in S (or x \is S, etc), the canonical instance of opprPred will not normally be exposed (it will also be erased by /= simplification). In addition each predicate structure should have a DefaultPredKey Canonical instance that simply issues the property as a proof obligation (which can be caught by the Prop-irrelevant feature of the ssreflect plugin). Some properties of predicates and relations: A =i B <-> A and B are extensionally equivalent. {subset A <= B} <-> A is a (collective) subpredicate of B. subpred P Q <-> P is an (applicative) subpredicate or Q. subrel R S <-> R is a subrelation of S. In the following R is in rel T: reflexive R <-> R is reflexive. irreflexive R <-> R is irreflexive. symmetric R <-> R (in rel T) is symmetric (equation). pre_symmetric R <-> R is symmetric (implication). antisymmetric R <-> R is antisymmetric. total R <-> R is total. transitive R <-> R is transitive. left_transitive R <-> R is a congruence on its left hand side. right_transitive R <-> R is a congruence on its right hand side. equivalence_rel R <-> R is an equivalence relation. Localization of (Prop) predicates; if P1 is convertible to forall x, Qx, P2 to forall x y, Qxy and P3 to forall x y z, Qxyz : {for y, P1} <-> Qx{y / x}. {in A, P1} <-> forall x, x \in A -> Qx. {in A1 & A2, P2} <-> forall x y, x \in A1 -> y \in A2 -> Qxy. {in A &, P2} <-> forall x y, x \in A -> y \in A -> Qxy. {in A1 & A2 & A3, Q3} <-> forall x y z, x \in A1 -> y \in A2 -> z \in A3 -> Qxyz. {in A1 & A2 &, Q3} := {in A1 & A2 & A2, Q3}. {in A1 && A3, Q3} := {in A1 & A1 & A3, Q3}. {in A &&, Q3} := {in A & A & A, Q3}. {in A, bijective f} <-> f has a right inverse in A. {on C, P1} <-> forall x, (f x) \in C -> Qx when P1 is also convertible to Pf f, e.g., {on C, involutive f}. {on C &, P2} == forall x y, f x \in C -> f y \in C -> Qxy when P2 is also convertible to Pf f, e.g., {on C &, injective f}. {on C, P1' & g} == forall x, (f x) \in cd -> Qx when P1' is convertible to Pf f and P1' g is convertible to forall x, Qx, e.g., {on C, cancel f & g}. {on C, bijective f} == f has a right inverse on C. This file extends the lemma name suffix conventions of ssrfun as follows: A -- associativity, as in andbA : associative andb. AC -- right commutativity. ACA -- self-interchange (inner commutativity), e.g., orbACA : (a || b) || (c || d) = (a || c) || (b || d). b -- a boolean argument, as in andbb : idempotent andb. C -- commutativity, as in andbC : commutative andb, or predicate complement, as in predC. CA -- left commutativity. D -- predicate difference, as in predD. E -- elimination, as in negbFE : ~~ b = false -> b. F or f -- boolean false, as in andbF : b && false = false. I -- left/right injectivity, as in addbI : right_injective addb, or predicate intersection, as in predI. l -- a left-hand operation, as andb_orl : left_distributive andb orb. N or n -- boolean negation, as in andbN : a && (~~ a) = false. P -- a characteristic property, often a reflection lemma, as in andP : reflect (a /\ b) (a && b). r -- a right-hand operation, as orb_andr : rightt_distributive orb andb. T or t -- boolean truth, as in andbT: right_id true andb. U -- predicate union, as in predU. W -- weakening, as in in1W : {in D, forall x, P} -> forall x, P.          **)
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Set Warnings "-projection-no-head-constant".
Notation reflect := Bool.reflect.
Notation ReflectT := Bool.ReflectT.
Notation ReflectF := Bool.ReflectF.
Reserved Notation "~~ b" (at level 35, right associativity).
Reserved Notation "b ==> c" (at level 55, right associativity).
Reserved Notation "b1 (+) b2" (at level 50, left associativity).
Reserved Notation "x \in A" (at level 70, no associativity,
format "'[hv' x '/ '  \in  A ']'").
Reserved Notation "x \notin A" (at level 70, no associativity,
format "'[hv' x '/ '  \notin  A ']'").
Reserved Notation "x \is A" (at level 70, no associativity,
format "'[hv' x '/ '  \is  A ']'").
Reserved Notation "x \isn't A" (at level 70, no associativity,
format "'[hv' x '/ '  \isn't  A ']'").
Reserved Notation "x \is 'a' A" (at level 70, no associativity,
format "'[hv' x '/ '  \is  'a'  A ']'").
Reserved Notation "x \isn't 'a' A" (at level 70, no associativity,
format "'[hv' x '/ '  \isn't  'a'  A ']'").
Reserved Notation "x \is 'an' A" (at level 70, no associativity,
format "'[hv' x '/ '  \is  'an'  A ']'").
Reserved Notation "x \isn't 'an' A" (at level 70, no associativity,
format "'[hv' x '/ '  \isn't  'an'  A ']'").
Reserved Notation "p1 =i p2" (at level 70, no associativity,
format "'[hv' p1 '/ '  =i  p2 ']'").
Reserved Notation "{ 'subset' A <= B }" (at level 0, A, B at level 69,
format "'[hv' { 'subset'  A '/    '  <=  B } ']'").
Reserved Notation "{ : T }" (at level 0, format "{ :  T }").
Reserved Notation "{ 'pred' T }" (at level 0, format "{ 'pred'  T }").
Reserved Notation "[ 'predType' 'of' T ]" (at level 0,
format "[ 'predType'  'of'  T ]").
Reserved Notation "[ 'pred' : T | E ]" (at level 0,
format "'[hv' [ 'pred' :  T  | '/ '  E ] ']'").
Reserved Notation "[ 'pred' x | E ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  | '/ '  E ] ']'").
Reserved Notation "[ 'pred' x : T | E ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  :  T  | '/ '  E ] ']'").
Reserved Notation "[ 'pred' x | E1 & E2 ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  | '/ '  E1  & '/ '  E2 ] ']'").
Reserved Notation "[ 'pred' x : T | E1 & E2 ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  :  T  | '/ '  E1  &  E2 ] ']'").
Reserved Notation "[ 'pred' x 'in' A ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  'in'  A ] ']'").
Reserved Notation "[ 'pred' x 'in' A | E ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  'in'  A  | '/ '  E ] ']'").
Reserved Notation "[ 'pred' x 'in' A | E1 & E2 ]" (at level 0, x ident,
format "'[hv' [ 'pred'  x  'in'  A  | '/ '  E1  & '/ '  E2 ] ']'").
Reserved Notation "[ 'qualify' x | P ]" (at level 0, x at level 99,
format "'[hv' [  'qualify'  x  | '/ '  P ] ']'").
Reserved Notation "[ 'qualify' x : T | P ]" (at level 0, x at level 99,
format "'[hv' [  'qualify'  x  :  T  | '/ '  P ] ']'").
Reserved Notation "[ 'qualify' 'a' x | P ]" (at level 0, x at level 99,
format "'[hv' [ 'qualify'  'a'  x  | '/ '  P ] ']'").
Reserved Notation "[ 'qualify' 'a' x : T | P ]" (at level 0, x at level 99,
format "'[hv' [ 'qualify'  'a'  x  :  T  | '/ '  P ] ']'").
Reserved Notation "[ 'qualify' 'an' x | P ]" (at level 0, x at level 99,
format "'[hv' [ 'qualify'  'an'  x  | '/ '  P ] ']'").
Reserved Notation "[ 'qualify' 'an' x : T | P ]" (at level 0, x at level 99,
format "'[hv' [ 'qualify'  'an'  x  :  T  | '/ '  P ] ']'").
Reserved Notation "[ 'rel' x y | E ]"  (at level 0, x ident, y ident,
format "'[hv' [ 'rel'  x  y  | '/ '  E ] ']'").
Reserved Notation "[ 'rel' x y : T | E ]" (at level 0, x ident, y ident,
format "'[hv' [ 'rel'  x  y  :  T  | '/ '  E ] ']'").
Reserved Notation "[ 'rel' x y 'in' A & B | E ]" (at level 0, x ident, y ident,
format "'[hv' [ 'rel'  x  y  'in'  A  &  B  | '/ '  E ] ']'").
Reserved Notation "[ 'rel' x y 'in' A & B ]" (at level 0, x ident, y ident,
format "'[hv' [ 'rel'  x  y  'in'  A  &  B ] ']'").
Reserved Notation "[ 'rel' x y 'in' A | E ]" (at level 0, x ident, y ident,
format "'[hv' [ 'rel'  x  y  'in'  A  | '/ '  E ] ']'").
Reserved Notation "[ 'rel' x y 'in' A ]" (at level 0, x ident, y ident,
format "'[hv' [ 'rel'  x  y  'in'  A ] ']'").
Reserved Notation "[ 'mem' A ]" (at level 0, format "[ 'mem'  A ]").
Reserved Notation "[ 'predI' A & B ]" (at level 0,
format "[ 'predI'  A  &  B ]").
Reserved Notation "[ 'predU' A & B ]" (at level 0,
format "[ 'predU'  A  &  B ]").
Reserved Notation "[ 'predD' A & B ]" (at level 0,
format "[ 'predD'  A  &  B ]").
Reserved Notation "[ 'predC' A ]" (at level 0,
format "[ 'predC'  A ]").
Reserved Notation "[ 'preim' f 'of' A ]" (at level 0,
format "[ 'preim'  f  'of'  A ]").
Reserved Notation "\unless C , P" (at level 200, C at level 100,
format "'[hv' \unless  C , '/ '  P ']'").
Reserved Notation "{ 'for' x , P }" (at level 0,
format "'[hv' { 'for'  x , '/ '  P } ']'").
Reserved Notation "{ 'in' d , P }" (at level 0,
format "'[hv' { 'in'  d , '/ '  P } ']'").
Reserved Notation "{ 'in' d1 & d2 , P }" (at level 0,
format "'[hv' { 'in'  d1  &  d2 , '/ '  P } ']'").
Reserved Notation "{ 'in' d & , P }" (at level 0,
format "'[hv' { 'in'  d  & , '/ '  P } ']'").
Reserved Notation "{ 'in' d1 & d2 & d3 , P }" (at level 0,
format "'[hv' { 'in'  d1  &  d2  &  d3 , '/ '  P } ']'").
Reserved Notation "{ 'in' d1 & & d3 , P }" (at level 0,
format "'[hv' { 'in'  d1  &  &  d3 , '/ '  P } ']'").
Reserved Notation "{ 'in' d1 & d2 & , P }" (at level 0,
format "'[hv' { 'in'  d1  &  d2  & , '/ '  P } ']'").
Reserved Notation "{ 'in' d & & , P }" (at level 0,
format "'[hv' { 'in'  d  &  & , '/ '  P } ']'").
Reserved Notation "{ 'on' cd , P }" (at level 0,
format "'[hv' { 'on'  cd , '/ '  P } ']'").
Reserved Notation "{ 'on' cd & , P }" (at level 0,
format "'[hv' { 'on'  cd  & , '/ '  P } ']'").
Reserved Notation "{ 'on' cd , P & g }" (at level 0, g at level 8,
format "'[hv' { 'on'  cd , '/ '  P  &  g } ']'").
Reserved Notation "{ 'in' d , 'bijective' f }" (at level 0, f at level 8,
format "'[hv' { 'in'  d , '/ '  'bijective'  f } ']'").
Reserved Notation "{ 'on' cd , 'bijective' f }" (at level 0, f at level 8,
format "'[hv' { 'on'  cd , '/ '  'bijective'  f } ']'").
(** We introduce a number of n-ary "list-style" notations that share a common format, namely #[#op arg1, arg2, ... last_separator last_arg#]# This usually denotes a right-associative applications of op, e.g., #[#&& a, b, c & d#]# denotes a && (b && (c && d)) The last_separator must be a non-operator token. Here we use &, | or =>; our default is &, but we try to match the intended meaning of op. The separator is a workaround for limitations of the parsing engine; the same limitations mean the separator cannot be omitted even when last_arg can. The Notation declarations are complicated by the separate treatment for some fixed arities (binary for bool operators, and all arities for Prop operators). We also use the square brackets in comprehension-style notations #[#type var separator expr#]# where "type" is the type of the comprehension (e.g., pred) and "separator" is | or => . It is important that in other notations a leading square bracket #[# is always followed by an operator symbol or a fixed identifier.   **)
Reserved Notation "[ /\ P1 & P2 ]" (at level 0, only parsing).
Reserved Notation "[ /\ P1 , P2 & P3 ]" (at level 0, format
"'[hv' [ /\ '['  P1 , '/'  P2 ']' '/ '  &  P3 ] ']'").
Reserved Notation "[ /\ P1 , P2 , P3 & P4 ]" (at level 0, format
"'[hv' [ /\ '['  P1 , '/'  P2 , '/'  P3 ']' '/ '  &  P4 ] ']'").
Reserved Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" (at level 0, format
"'[hv' [ /\ '['  P1 , '/'  P2 , '/'  P3 , '/'  P4 ']' '/ '  &  P5 ] ']'").
Reserved Notation "[ \/ P1 | P2 ]" (at level 0, only parsing).
Reserved Notation "[ \/ P1 , P2 | P3 ]" (at level 0, format
"'[hv' [ \/ '['  P1 , '/'  P2 ']' '/ '  |  P3 ] ']'").
Reserved Notation "[ \/ P1 , P2 , P3 | P4 ]" (at level 0, format
"'[hv' [ \/ '['  P1 , '/'  P2 , '/'  P3 ']' '/ '  |  P4 ] ']'").
Reserved Notation "[ && b1 & c ]" (at level 0, only parsing).
Reserved Notation "[ && b1 , b2 , .. , bn & c ]" (at level 0, format
"'[hv' [ && '['  b1 , '/'  b2 , '/'  .. , '/'  bn ']' '/ '  &  c ] ']'").
Reserved Notation "[ || b1 | c ]" (at level 0, only parsing).
Reserved Notation "[ || b1 , b2 , .. , bn | c ]" (at level 0, format
"'[hv' [ || '['  b1 , '/'  b2 , '/'  .. , '/'  bn ']' '/ '  |  c ] ']'").
Reserved Notation "[ ==> b1 => c ]" (at level 0, only parsing).
Reserved Notation "[ ==> b1 , b2 , .. , bn => c ]" (at level 0, format
"'[hv' [ ==> '['  b1 , '/'  b2 , '/'  .. , '/'  bn ']' '/'  =>  c ] ']'").
(**  Shorter delimiter  **)
Delimit Scope bool_scope with B.
Open Scope bool_scope.
(**  An alternative to xorb that behaves somewhat better wrt simplification. **)
Definition addb b := if b then negb else id.
(**  Notation for && and || is declared in Init.Datatypes.  **)
Notation "~~ b" := (negb b) : bool_scope.
Notation "b ==> c" := (implb b c) : bool_scope.
Notation "b1 (+) b2" := (addb b1 b2) : bool_scope.
(**  Constant is_true b := b = true is defined in Init.Datatypes.  **)
Coercion is_true : bool >-> Sortclass. (* Prop *)

Lemma prop_congr : forall b b' : bool, b = b' -> b = b' :> Prop.