(************************************************************************) (* * The Coq Proof Assistant / The Coq Development Team *) (* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) (* -> 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 *) (* generates two subgoals, in which my_formula has *) (* been replaced by true and false, resp., with *) (* new assumptions myP : my_Prop and *) (* not_myP: ~~ my_formula.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 *) (* assumtions 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 coinductive type that coerces to P -> Q *) (* and can be used as a P -> Q view unambigously. *) (* 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, using *) (* the simpl_fun from ssrfun.v. *) (* rel T == the type of bool relations. *) (* := T -> pred T or T -> T -> bool. *) (* simpl_rel T == type of simplifying relations. *) (* predType == the generic predicate interface, supported for *) (* for lists and sets. *) (* pred_class == a coercion class for the predType projection to *) (* pred; declaring a coercion to pred_class is an *) (* alternative way of equipping 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. *) (* 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 A: *) (* mem A x simplifies to x \in A. *) (* 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, ... == 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. *) (* 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_class 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_class; 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_class. *) (* 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. *) (* {on C &, P2} == forall x y, f x \in C -> f y \in C -> Qxy *) (* when P2 is also convertible to Pf 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. *) (* {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, format "'[hv' x '/ ' \in A ']'", no associativity). Reserved Notation "x \notin A" (at level 70, format "'[hv' x '/ ' \notin A ']'", no associativity). Reserved Notation "p1 =i p2" (at level 70, format "'[hv' p1 '/ ' =i p2 ']'", no associativity). (* 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 ] ']'"). Reserved Notation "[ 'pred' : T => E ]" (at level 0, format "'[hv' [ 'pred' : T => '/ ' E ] ']'"). Reserved Notation "[ 'pred' x => E ]" (at level 0, x at level 8, format "'[hv' [ 'pred' x => '/ ' E ] ']'"). Reserved Notation "[ 'pred' x : T => E ]" (at level 0, x at level 8, format "'[hv' [ 'pred' x : T => '/ ' E ] ']'"). Reserved Notation "[ 'rel' x y => E ]" (at level 0, x, y at level 8, format "'[hv' [ 'rel' x y => '/ ' E ] ']'"). Reserved Notation "[ 'rel' x y : T => E ]" (at level 0, x, y at level 8, format "'[hv' [ 'rel' x y : T => '/ ' E ] ']'"). (* 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. Proof. by move=> b b' ->. Qed. Ltac prop_congr := apply: prop_congr. (* Lemmas for trivial. *) Lemma is_true_true : true. Proof. by []. Qed. Lemma not_false_is_true : ~ false. Proof. by []. Qed. Lemma is_true_locked_true : locked true. Proof. by unlock. Qed. Hint Resolve is_true_true not_false_is_true is_true_locked_true. (* Shorter names. *) Definition isT := is_true_true. Definition notF := not_false_is_true. (* Negation lemmas. *) (* We generally take NEGATION as the standard form of a false condition: *) (* negative boolean hypotheses should be of the form ~~ b, rather than ~ b or *) (* b = false, as much as possible. *) Lemma negbT b : b = false -> ~~ b. Proof. by case: b. Qed. Lemma negbTE b : ~~ b -> b = false. Proof. by case: b. Qed. Lemma negbF b : (b : bool) -> ~~ b = false. Proof. by case: b. Qed. Lemma negbFE b : ~~ b = false -> b. Proof. by case: b. Qed. Lemma negbK : involutive negb. Proof. by case. Qed. Lemma negbNE b : ~~ ~~ b -> b. Proof. by case: b. Qed. Lemma negb_inj : injective negb. Proof. exact: can_inj negbK. Qed. Lemma negbLR b c : b = ~~ c -> ~~ b = c. Proof. exact: canLR negbK. Qed. Lemma negbRL b c : ~~ b = c -> b = ~~ c. Proof. exact: canRL negbK. Qed. Lemma contra (c b : bool) : (c -> b) -> ~~ b -> ~~ c. Proof. by case: b => //; case: c. Qed. Definition contraNN := contra. Lemma contraL (c b : bool) : (c -> ~~ b) -> b -> ~~ c. Proof. by case: b => //; case: c. Qed. Definition contraTN := contraL. Lemma contraR (c b : bool) : (~~ c -> b) -> ~~ b -> c. Proof. by case: b => //; case: c. Qed. Definition contraNT := contraR. Lemma contraLR (c b : bool) : (~~ c -> ~~ b) -> b -> c. Proof. by case: b => //; case: c. Qed. Definition contraTT := contraLR. Lemma contraT b : (~~ b -> false) -> b. Proof. by case: b => // ->. Qed. Lemma wlog_neg b : (~~ b -> b) -> b. Proof. by case: b => // ->. Qed. Lemma contraFT (c b : bool) : (~~ c -> b) -> b = false -> c. Proof. by move/contraR=> notb_c /negbT. Qed. Lemma contraFN (c b : bool) : (c -> b) -> b = false -> ~~ c. Proof. by move/contra=> notb_notc /negbT. Qed. Lemma contraTF (c b : bool) : (c -> ~~ b) -> b -> c = false. Proof. by move/contraL=> b_notc /b_notc/negbTE. Qed. Lemma contraNF (c b : bool) : (c -> b) -> ~~ b -> c = false. Proof. by move/contra=> notb_notc /notb_notc/negbTE. Qed. Lemma contraFF (c b : bool) : (c -> b) -> b = false -> c = false. Proof. by move/contraFN=> bF_notc /bF_notc/negbTE. Qed. (* Coercion of sum-style datatypes into bool, which makes it possible *) (* to use ssr's boolean if rather than Coq's "generic" if. *) Coercion isSome T (u : option T) := if u is Some _ then true else false. Coercion is_inl A B (u : A + B) := if u is inl _ then true else false. Coercion is_left A B (u : {A} + {B}) := if u is left _ then true else false. Coercion is_inleft A B (u : A + {B}) := if u is inleft _ then true else false. Prenex Implicits isSome is_inl is_left is_inleft. Definition decidable P := {P} + {~ P}. (* Lemmas for ifs with large conditions, which allow reasoning about the *) (* condition without repeating it inside the proof (the latter IS *) (* preferable when the condition is short). *) (* Usage : *) (* if the goal contains (if cond then ...) = ... *) (* case: ifP => Hcond. *) (* generates two subgoal, with the assumption Hcond : cond = true/false *) (* Rewrite if_same eliminates redundant ifs *) (* Rewrite (fun_if f) moves a function f inside an if *) (* Rewrite if_arg moves an argument inside a function-valued if *) Section BoolIf. Variables (A B : Type) (x : A) (f : A -> B) (b : bool) (vT vF : A). CoInductive if_spec (not_b : Prop) : bool -> A -> Set := | IfSpecTrue of b : if_spec not_b true vT | IfSpecFalse of not_b : if_spec not_b false vF. Lemma ifP : if_spec (b = false) b (if b then vT else vF). Proof. by case def_b: b; constructor. Qed. Lemma ifPn : if_spec (~~ b) b (if b then vT else vF). Proof. by case def_b: b; constructor; rewrite ?def_b. Qed. Lemma ifT : b -> (if b then vT else vF) = vT. Proof. by move->. Qed. Lemma ifF : b = false -> (if b then vT else vF) = vF. Proof. by move->. Qed. Lemma ifN : ~~ b -> (if b then vT else vF) = vF. Proof. by move/negbTE->. Qed. Lemma if_same : (if b then vT else vT) = vT. Proof. by case b. Qed. Lemma if_neg : (if ~~ b then vT else vF) = if b then vF else vT. Proof. by case b. Qed. Lemma fun_if : f (if b then vT else vF) = if b then f vT else f vF. Proof. by case b. Qed. Lemma if_arg (fT fF : A -> B) : (if b then fT else fF) x = if b then fT x else fF x. Proof. by case b. Qed. (* Turning a boolean "if" form into an application. *) Definition if_expr := if b then vT else vF. Lemma ifE : (if b then vT else vF) = if_expr. Proof. by []. Qed. End BoolIf. (* Core (internal) reflection lemmas, used for the three kinds of views. *) Section ReflectCore. Variables (P Q : Prop) (b c : bool). Hypothesis Hb : reflect P b. Lemma introNTF : (if c then ~ P else P) -> ~~ b = c. Proof. by case c; case Hb. Qed. Lemma introTF : (if c then P else ~ P) -> b = c. Proof. by case c; case Hb. Qed. Lemma elimNTF : ~~ b = c -> if c then ~ P else P. Proof. by move <-; case Hb. Qed. Lemma elimTF : b = c -> if c then P else ~ P. Proof. by move <-; case Hb. Qed. Lemma equivPif : (Q -> P) -> (P -> Q) -> if b then Q else ~ Q. Proof. by case Hb; auto. Qed. Lemma xorPif : Q \/ P -> ~ (Q /\ P) -> if b then ~ Q else Q. Proof. by case Hb => [? _ H ? | ? H _]; case: H. Qed. End ReflectCore. (* Internal negated reflection lemmas *) Section ReflectNegCore. Variables (P Q : Prop) (b c : bool). Hypothesis Hb : reflect P (~~ b). Lemma introTFn : (if c then ~ P else P) -> b = c. Proof. by move/(introNTF Hb) <-; case b. Qed. Lemma elimTFn : b = c -> if c then ~ P else P. Proof. by move <-; apply: (elimNTF Hb); case b. Qed. Lemma equivPifn : (Q -> P) -> (P -> Q) -> if b then ~ Q else Q. Proof. by rewrite -if_neg; apply: equivPif. Qed. Lemma xorPifn : Q \/ P -> ~ (Q /\ P) -> if b then Q else ~ Q. Proof. by rewrite -if_neg; apply: xorPif. Qed. End ReflectNegCore. (* User-oriented reflection lemmas *) Section Reflect. Variables (P Q : Prop) (b b' c : bool). Hypotheses (Pb : reflect P b) (Pb' : reflect P (~~ b')). Lemma introT : P -> b. Proof. exact: introTF true _. Qed. Lemma introF : ~ P -> b = false. Proof. exact: introTF false _. Qed. Lemma introN : ~ P -> ~~ b. Proof. exact: introNTF true _. Qed. Lemma introNf : P -> ~~ b = false. Proof. exact: introNTF false _. Qed. Lemma introTn : ~ P -> b'. Proof. exact: introTFn true _. Qed. Lemma introFn : P -> b' = false. Proof. exact: introTFn false _. Qed. Lemma elimT : b -> P. Proof. exact: elimTF true _. Qed. Lemma elimF : b = false -> ~ P. Proof. exact: elimTF false _. Qed. Lemma elimN : ~~ b -> ~P. Proof. exact: elimNTF true _. Qed. Lemma elimNf : ~~ b = false -> P. Proof. exact: elimNTF false _. Qed. Lemma elimTn : b' -> ~ P. Proof. exact: elimTFn true _. Qed. Lemma elimFn : b' = false -> P. Proof. exact: elimTFn false _. Qed. Lemma introP : (b -> Q) -> (~~ b -> ~ Q) -> reflect Q b. Proof. by case b; constructor; auto. Qed. Lemma iffP : (P -> Q) -> (Q -> P) -> reflect Q b. Proof. by case: Pb; constructor; auto. Qed. Lemma equivP : (P <-> Q) -> reflect Q b. Proof. by case; apply: iffP. Qed. Lemma sumboolP (decQ : decidable Q) : reflect Q decQ. Proof. by case: decQ; constructor. Qed. Lemma appP : reflect Q b -> P -> Q. Proof. by move=> Qb; move/introT; case: Qb. Qed. Lemma sameP : reflect P c -> b = c. Proof. by case; [apply: introT | apply: introF]. Qed. Lemma decPcases : if b then P else ~ P. Proof. by case Pb. Qed. Definition decP : decidable P. by case: b decPcases; [left | right]. Defined. Lemma rwP : P <-> b. Proof. by split; [apply: introT | apply: elimT]. Qed. Lemma rwP2 : reflect Q b -> (P <-> Q). Proof. by move=> Qb; split=> ?; [apply: appP | apply: elimT; case: Qb]. Qed. (* Predicate family to reflect excluded middle in bool. *) CoInductive alt_spec : bool -> Type := | AltTrue of P : alt_spec true | AltFalse of ~~ b : alt_spec false. Lemma altP : alt_spec b. Proof. by case def_b: b / Pb; constructor; rewrite ?def_b. Qed. End Reflect. Hint View for move/ elimTF|3 elimNTF|3 elimTFn|3 introT|2 introTn|2 introN|2. Hint View for apply/ introTF|3 introNTF|3 introTFn|3 elimT|2 elimTn|2 elimN|2. Hint View for apply// equivPif|3 xorPif|3 equivPifn|3 xorPifn|3. (* Allow the direct application of a reflection lemma to a boolean assertion. *) Coercion elimT : reflect >-> Funclass. CoInductive implies P Q := Implies of P -> Q. Lemma impliesP P Q : implies P Q -> P -> Q. Proof. by case. Qed. Lemma impliesPn (P Q : Prop) : implies P Q -> ~ Q -> ~ P. Proof. by case=> iP ? /iP. Qed. Coercion impliesP : implies >-> Funclass. Hint View for move/ impliesPn|2 impliesP|2. Hint View for apply/ impliesPn|2 impliesP|2. (* Impredicative or, which can emulate a classical not-implies. *) Definition unless condition property : Prop := forall goal : Prop, (condition -> goal) -> (property -> goal) -> goal. Notation "\unless C , P" := (unless C P) (at level 200, C at level 100, format "'[' \unless C , '/ ' P ']'") : type_scope. Lemma unlessL C P : implies C (\unless C, P). Proof. by split=> hC G /(_ hC). Qed. Lemma unlessR C P : implies P (\unless C, P). Proof. by split=> hP G _ /(_ hP). Qed. Lemma unless_sym C P : implies (\unless C, P) (\unless P, C). Proof. by split; apply; [apply/unlessR | apply/unlessL]. Qed. Lemma unlessP (C P : Prop) : (\unless C, P) <-> C \/ P. Proof. by split=> [|[/unlessL | /unlessR]]; apply; [left | right]. Qed. Lemma bind_unless C P {Q} : implies (\unless C, P) (\unless (\unless C, Q), P). Proof. by split; apply=> [hC|hP]; [apply/unlessL/unlessL | apply/unlessR]. Qed. Lemma unless_contra b C : implies (~~ b -> C) (\unless C, b). Proof. by split; case: b => [_ | hC]; [apply/unlessR | apply/unlessL/hC]. Qed. (* Classical reasoning becomes directly accessible for any bool subgoal. *) (* Note that we cannot use "unless" here for lack of universe polymorphism. *) Definition classically P : Prop := forall b : bool, (P -> b) -> b. Lemma classicP (P : Prop) : classically P <-> ~ ~ P. Proof. split=> [cP nP | nnP [] // nP]; last by case nnP; move/nP. by have: P -> false; [move/nP | move/cP]. Qed. Lemma classicW P : P -> classically P. Proof. by move=> hP _ ->. Qed. Lemma classic_bind P Q : (P -> classically Q) -> classically P -> classically Q. Proof. by move=> iPQ cP b /iPQ-/cP. Qed. Lemma classic_EM P : classically (decidable P). Proof. by case=> // undecP; apply/undecP; right=> notP; apply/notF/undecP; left. Qed. Lemma classic_pick T P : classically ({x : T | P x} + (forall x, ~ P x)). Proof. case=> // undecP; apply/undecP; right=> x Px. by apply/notF/undecP; left; exists x. Qed. Lemma classic_imply P Q : (P -> classically Q) -> classically (P -> Q). Proof. move=> iPQ []// notPQ; apply/notPQ=> /iPQ-cQ. by case: notF; apply: cQ => hQ; apply: notPQ. Qed. (* List notations for wider connectives; the Prop connectives have a fixed *) (* width so as to avoid iterated destruction (we go up to width 5 for /\, and *) (* width 4 for or). The bool connectives have arbitrary widths, but denote *) (* expressions that associate to the RIGHT. This is consistent with the right *) (* associativity of list expressions and thus more convenient in most proofs. *) Inductive and3 (P1 P2 P3 : Prop) : Prop := And3 of P1 & P2 & P3. Inductive and4 (P1 P2 P3 P4 : Prop) : Prop := And4 of P1 & P2 & P3 & P4. Inductive and5 (P1 P2 P3 P4 P5 : Prop) : Prop := And5 of P1 & P2 & P3 & P4 & P5. Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3. Inductive or4 (P1 P2 P3 P4 : Prop) : Prop := Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4. Notation "[ /\ P1 & P2 ]" := (and P1 P2) (only parsing) : type_scope. Notation "[ /\ P1 , P2 & P3 ]" := (and3 P1 P2 P3) : type_scope. Notation "[ /\ P1 , P2 , P3 & P4 ]" := (and4 P1 P2 P3 P4) : type_scope. Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" := (and5 P1 P2 P3 P4 P5) : type_scope. Notation "[ \/ P1 | P2 ]" := (or P1 P2) (only parsing) : type_scope. Notation "[ \/ P1 , P2 | P3 ]" := (or3 P1 P2 P3) : type_scope. Notation "[ \/ P1 , P2 , P3 | P4 ]" := (or4 P1 P2 P3 P4) : type_scope. Notation "[ && b1 & c ]" := (b1 && c) (only parsing) : bool_scope. Notation "[ && b1 , b2 , .. , bn & c ]" := (b1 && (b2 && .. (bn && c) .. )) : bool_scope. Notation "[ || b1 | c ]" := (b1 || c) (only parsing) : bool_scope. Notation "[ || b1 , b2 , .. , bn | c ]" := (b1 || (b2 || .. (bn || c) .. )) : bool_scope. Notation "[ ==> b1 , b2 , .. , bn => c ]" := (b1 ==> (b2 ==> .. (bn ==> c) .. )) : bool_scope. Notation "[ ==> b1 => c ]" := (b1 ==> c) (only parsing) : bool_scope. Section AllAnd. Variables (T : Type) (P1 P2 P3 P4 P5 : T -> Prop). Local Notation a P := (forall x, P x). Lemma all_and2 : implies (forall x, [/\ P1 x & P2 x]) [/\ a P1 & a P2]. Proof. by split=> haveP; split=> x; case: (haveP x). Qed. Lemma all_and3 : implies (forall x, [/\ P1 x, P2 x & P3 x]) [/\ a P1, a P2 & a P3]. Proof. by split=> haveP; split=> x; case: (haveP x). Qed. Lemma all_and4 : implies (forall x, [/\ P1 x, P2 x, P3 x & P4 x]) [/\ a P1, a P2, a P3 & a P4]. Proof. by split=> haveP; split=> x; case: (haveP x). Qed. Lemma all_and5 : implies (forall x, [/\ P1 x, P2 x, P3 x, P4 x & P5 x]) [/\ a P1, a P2, a P3, a P4 & a P5]. Proof. by split=> haveP; split=> x; case: (haveP x). Qed. End AllAnd. Arguments all_and2 {T P1 P2}. Arguments all_and3 {T P1 P2 P3}. Arguments all_and4 {T P1 P2 P3 P4}. Arguments all_and5 {T P1 P2 P3 P4 P5}. Lemma pair_andP P Q : P /\ Q <-> P * Q. Proof. by split; case. Qed. Section ReflectConnectives. Variable b1 b2 b3 b4 b5 : bool. Lemma idP : reflect b1 b1. Proof. by case b1; constructor. Qed. Lemma boolP : alt_spec b1 b1 b1. Proof. exact: (altP idP). Qed. Lemma idPn : reflect (~~ b1) (~~ b1). Proof. by case b1; constructor. Qed. Lemma negP : reflect (~ b1) (~~ b1). Proof. by case b1; constructor; auto. Qed. Lemma negPn : reflect b1 (~~ ~~ b1). Proof. by case b1; constructor. Qed. Lemma negPf : reflect (b1 = false) (~~ b1). Proof. by case b1; constructor. Qed. Lemma andP : reflect (b1 /\ b2) (b1 && b2). Proof. by case b1; case b2; constructor=> //; case. Qed. Lemma and3P : reflect [/\ b1, b2 & b3] [&& b1, b2 & b3]. Proof. by case b1; case b2; case b3; constructor; try by case. Qed. Lemma and4P : reflect [/\ b1, b2, b3 & b4] [&& b1, b2, b3 & b4]. Proof. by case b1; case b2; case b3; case b4; constructor; try by case. Qed. Lemma and5P : reflect [/\ b1, b2, b3, b4 & b5] [&& b1, b2, b3, b4 & b5]. Proof. by case b1; case b2; case b3; case b4; case b5; constructor; try by case. Qed. Lemma orP : reflect (b1 \/ b2) (b1 || b2). Proof. by case b1; case b2; constructor; auto; case. Qed. Lemma or3P : reflect [\/ b1, b2 | b3] [|| b1, b2 | b3]. Proof. case b1; first by constructor; constructor 1. case b2; first by constructor; constructor 2. case b3; first by constructor; constructor 3. by constructor; case. Qed. Lemma or4P : reflect [\/ b1, b2, b3 | b4] [|| b1, b2, b3 | b4]. Proof. case b1; first by constructor; constructor 1. case b2; first by constructor; constructor 2. case b3; first by constructor; constructor 3. case b4; first by constructor; constructor 4. by constructor; case. Qed. Lemma nandP : reflect (~~ b1 \/ ~~ b2) (~~ (b1 && b2)). Proof. by case b1; case b2; constructor; auto; case; auto. Qed. Lemma norP : reflect (~~ b1 /\ ~~ b2) (~~ (b1 || b2)). Proof. by case b1; case b2; constructor; auto; case; auto. Qed. Lemma implyP : reflect (b1 -> b2) (b1 ==> b2). Proof. by case b1; case b2; constructor; auto. Qed. End ReflectConnectives. Arguments idP [b1]. Arguments idPn [b1]. Arguments negP [b1]. Arguments negPn [b1]. Arguments negPf [b1]. Arguments andP [b1 b2]. Arguments and3P [b1 b2 b3]. Arguments and4P [b1 b2 b3 b4]. Arguments and5P [b1 b2 b3 b4 b5]. Arguments orP [b1 b2]. Arguments or3P [b1 b2 b3]. Arguments or4P [b1 b2 b3 b4]. Arguments nandP [b1 b2]. Arguments norP [b1 b2]. Arguments implyP [b1 b2]. Prenex Implicits idP idPn negP negPn negPf. Prenex Implicits andP and3P and4P and5P orP or3P or4P nandP norP implyP. (* Shorter, more systematic names for the boolean connectives laws. *) Lemma andTb : left_id true andb. Proof. by []. Qed. Lemma andFb : left_zero false andb. Proof. by []. Qed. Lemma andbT : right_id true andb. Proof. by case. Qed. Lemma andbF : right_zero false andb. Proof. by case. Qed. Lemma andbb : idempotent andb. Proof. by case. Qed. Lemma andbC : commutative andb. Proof. by do 2!case. Qed. Lemma andbA : associative andb. Proof. by do 3!case. Qed. Lemma andbCA : left_commutative andb. Proof. by do 3!case. Qed. Lemma andbAC : right_commutative andb. Proof. by do 3!case. Qed. Lemma andbACA : interchange andb andb. Proof. by do 4!case. Qed. Lemma orTb : forall b, true || b. Proof. by []. Qed. Lemma orFb : left_id false orb. Proof. by []. Qed. Lemma orbT : forall b, b || true. Proof. by case. Qed. Lemma orbF : right_id false orb. Proof. by case. Qed. Lemma orbb : idempotent orb. Proof. by case. Qed. Lemma orbC : commutative orb. Proof. by do 2!case. Qed. Lemma orbA : associative orb. Proof. by do 3!case. Qed. Lemma orbCA : left_commutative orb. Proof. by do 3!case. Qed. Lemma orbAC : right_commutative orb. Proof. by do 3!case. Qed. Lemma orbACA : interchange orb orb. Proof. by do 4!case. Qed. Lemma andbN b : b && ~~ b = false. Proof. by case: b. Qed. Lemma andNb b : ~~ b && b = false. Proof. by case: b. Qed. Lemma orbN b : b || ~~ b = true. Proof. by case: b. Qed. Lemma orNb b : ~~ b || b = true. Proof. by case: b. Qed. Lemma andb_orl : left_distributive andb orb. Proof. by do 3!case. Qed. Lemma andb_orr : right_distributive andb orb. Proof. by do 3!case. Qed. Lemma orb_andl : left_distributive orb andb. Proof. by do 3!case. Qed. Lemma orb_andr : right_distributive orb andb. Proof. by do 3!case. Qed. Lemma andb_idl (a b : bool) : (b -> a) -> a && b = b. Proof. by case: a; case: b => // ->. Qed. Lemma andb_idr (a b : bool) : (a -> b) -> a && b = a. Proof. by case: a; case: b => // ->. Qed. Lemma andb_id2l (a b c : bool) : (a -> b = c) -> a && b = a && c. Proof. by case: a; case: b; case: c => // ->. Qed. Lemma andb_id2r (a b c : bool) : (b -> a = c) -> a && b = c && b. Proof. by case: a; case: b; case: c => // ->. Qed. Lemma orb_idl (a b : bool) : (a -> b) -> a || b = b. Proof. by case: a; case: b => // ->. Qed. Lemma orb_idr (a b : bool) : (b -> a) -> a || b = a. Proof. by case: a; case: b => // ->. Qed. Lemma orb_id2l (a b c : bool) : (~~ a -> b = c) -> a || b = a || c. Proof. by case: a; case: b; case: c => // ->. Qed. Lemma orb_id2r (a b c : bool) : (~~ b -> a = c) -> a || b = c || b. Proof. by case: a; case: b; case: c => // ->. Qed. Lemma negb_and (a b : bool) : ~~ (a && b) = ~~ a || ~~ b. Proof. by case: a; case: b. Qed. Lemma negb_or (a b : bool) : ~~ (a || b) = ~~ a && ~~ b. Proof. by case: a; case: b. Qed. (* Pseudo-cancellation -- i.e, absorbtion *) Lemma andbK a b : a && b || a = a. Proof. by case: a; case: b. Qed. Lemma andKb a b : a || b && a = a. Proof. by case: a; case: b. Qed. Lemma orbK a b : (a || b) && a = a. Proof. by case: a; case: b. Qed. Lemma orKb a b : a && (b || a) = a. Proof. by case: a; case: b. Qed. (* Imply *) Lemma implybT b : b ==> true. Proof. by case: b. Qed. Lemma implybF b : (b ==> false) = ~~ b. Proof. by case: b. Qed. Lemma implyFb b : false ==> b. Proof. by []. Qed. Lemma implyTb b : (true ==> b) = b. Proof. by []. Qed. Lemma implybb b : b ==> b. Proof. by case: b. Qed. Lemma negb_imply a b : ~~ (a ==> b) = a && ~~ b. Proof. by case: a; case: b. Qed. Lemma implybE a b : (a ==> b) = ~~ a || b. Proof. by case: a; case: b. Qed. Lemma implyNb a b : (~~ a ==> b) = a || b. Proof. by case: a; case: b. Qed. Lemma implybN a b : (a ==> ~~ b) = (b ==> ~~ a). Proof. by case: a; case: b. Qed. Lemma implybNN a b : (~~ a ==> ~~ b) = b ==> a. Proof. by case: a; case: b. Qed. Lemma implyb_idl (a b : bool) : (~~ a -> b) -> (a ==> b) = b. Proof. by case: a; case: b => // ->. Qed. Lemma implyb_idr (a b : bool) : (b -> ~~ a) -> (a ==> b) = ~~ a. Proof. by case: a; case: b => // ->. Qed. Lemma implyb_id2l (a b c : bool) : (a -> b = c) -> (a ==> b) = (a ==> c). Proof. by case: a; case: b; case: c => // ->. Qed. (* Addition (xor) *) Lemma addFb : left_id false addb. Proof. by []. Qed. Lemma addbF : right_id false addb. Proof. by case. Qed. Lemma addbb : self_inverse false addb. Proof. by case. Qed. Lemma addbC : commutative addb. Proof. by do 2!case. Qed. Lemma addbA : associative addb. Proof. by do 3!case. Qed. Lemma addbCA : left_commutative addb. Proof. by do 3!case. Qed. Lemma addbAC : right_commutative addb. Proof. by do 3!case. Qed. Lemma addbACA : interchange addb addb. Proof. by do 4!case. Qed. Lemma andb_addl : left_distributive andb addb. Proof. by do 3!case. Qed. Lemma andb_addr : right_distributive andb addb. Proof. by do 3!case. Qed. Lemma addKb : left_loop id addb. Proof. by do 2!case. Qed. Lemma addbK : right_loop id addb. Proof. by do 2!case. Qed. Lemma addIb : left_injective addb. Proof. by do 3!case. Qed. Lemma addbI : right_injective addb. Proof. by do 3!case. Qed. Lemma addTb b : true (+) b = ~~ b. Proof. by []. Qed. Lemma addbT b : b (+) true = ~~ b. Proof. by case: b. Qed. Lemma addbN a b : a (+) ~~ b = ~~ (a (+) b). Proof. by case: a; case: b. Qed. Lemma addNb a b : ~~ a (+) b = ~~ (a (+) b). Proof. by case: a; case: b. Qed. Lemma addbP a b : reflect (~~ a = b) (a (+) b). Proof. by case: a; case: b; constructor. Qed. Arguments addbP [a b]. (* Resolution tactic for blindly weeding out common terms from boolean *) (* equalities. When faced with a goal of the form (andb/orb/addb b1 b2) = b3 *) (* they will try to locate b1 in b3 and remove it. This can fail! *) Ltac bool_congr := match goal with | |- (?X1 && ?X2 = ?X3) => first [ symmetry; rewrite -1?(andbC X1) -?(andbCA X1); congr 1 (andb X1); symmetry | case: (X1); [ rewrite ?andTb ?andbT // | by rewrite ?andbF /= ] ] | |- (?X1 || ?X2 = ?X3) => first [ symmetry; rewrite -1?(orbC X1) -?(orbCA X1); congr 1 (orb X1); symmetry | case: (X1); [ by rewrite ?orbT //= | rewrite ?orFb ?orbF ] ] | |- (?X1 (+) ?X2 = ?X3) => symmetry; rewrite -1?(addbC X1) -?(addbCA X1); congr 1 (addb X1); symmetry | |- (~~ ?X1 = ?X2) => congr 1 negb end. (******************************************************************************) (* Predicates, i.e., packaged functions to bool. *) (* - pred T, the basic type for predicates over a type T, is simply an alias *) (* for T -> bool. *) (* We actually distinguish two kinds of predicates, which we call applicative *) (* and collective, based on the syntax used to test them at some x in T: *) (* - For an applicative predicate P, one uses prefix syntax: *) (* P x *) (* Also, most operations on applicative predicates use prefix syntax as *) (* well (e.g., predI P Q). *) (* - For a collective predicate A, one uses infix syntax: *) (* x \in A *) (* and all operations on collective predicates use infix syntax as well *) (* (e.g., [predI A & B]). *) (* There are only two kinds of applicative predicates: *) (* - pred T, the alias for T -> bool mentioned above *) (* - simpl_pred T, an alias for simpl_fun T bool with a coercion to pred T *) (* that auto-simplifies on application (see ssrfun). *) (* On the other hand, the set of collective predicate types is open-ended via *) (* - predType T, a Structure that can be used to put Canonical collective *) (* predicate interpretation on other types, such as lists, tuples, *) (* finite sets, etc. *) (* Indeed, we define such interpretations for applicative predicate types, *) (* which can therefore also be used with the infix syntax, e.g., *) (* x \in predI P Q *) (* Moreover these infix forms are convertible to their prefix counterpart *) (* (e.g., predI P Q x which in turn simplifies to P x && Q x). The converse *) (* is not true, however; collective predicate types cannot, in general, be *) (* general, be used applicatively, because of the "uniform inheritance" *) (* restriction on implicit coercions. *) (* However, we do define an explicit generic coercion *) (* - mem : forall (pT : predType), pT -> mem_pred T *) (* where mem_pred T is a variant of simpl_pred T that preserves the infix *) (* syntax, i.e., mem A x auto-simplifies to x \in A. *) (* Indeed, the infix "collective" operators are notation for a prefix *) (* operator with arguments of type mem_pred T or pred T, applied to coerced *) (* collective predicates, e.g., *) (* Notation "x \in A" := (in_mem x (mem A)). *) (* This prevents the variability in the predicate type from interfering with *) (* the application of generic lemmas. Moreover this also makes it much easier *) (* to define generic lemmas, because the simplest type -- pred T -- can be *) (* used as the type of generic collective predicates, provided one takes care *) (* not to use it applicatively; this avoids the burden of having to declare a *) (* different predicate type for each predicate parameter of each section or *) (* lemma. *) (* This trick is made possible by the fact that the constructor of the *) (* mem_pred T type aligns the unification process, forcing a generic *) (* "collective" predicate A : pred T to unify with the actual collective B, *) (* which mem has coerced to pred T via an internal, hidden implicit coercion, *) (* supplied by the predType structure for B. Users should take care not to *) (* inadvertently "strip" (mem B) down to the coerced B, since this will *) (* expose the internal coercion: Coq will display a term B x that cannot be *) (* typed as such. The topredE lemma can be used to restore the x \in B *) (* syntax in this case. While -topredE can conversely be used to change *) (* x \in P into P x, it is safer to use the inE and memE lemmas instead, as *) (* they do not run the risk of exposing internal coercions. As a consequence *) (* it is better to explicitly cast a generic applicative pred T to simpl_pred *) (* using the SimplPred constructor, when it is used as a collective predicate *) (* (see, e.g., Lemma eq_big in bigop). *) (* We also sometimes "instantiate" the predType structure by defining a *) (* coercion to the sort of the predPredType structure. This works better for *) (* types such as {set T} that have subtypes that coerce to them, since the *) (* same coercion will be inserted by the application of mem. It also lets us *) (* turn any Type aT : predArgType into the total predicate over that type, *) (* i.e., fun _: aT => true. This allows us to write, e.g., #|'I_n| for the *) (* cardinal of the (finite) type of integers less than n. *) (* Collective predicates have a specific extensional equality, *) (* - A =i B, *) (* while applicative predicates use the extensional equality of functions, *) (* - P =1 Q *) (* The two forms are convertible, however. *) (* We lift boolean operations to predicates, defining: *) (* - predU (union), predI (intersection), predC (complement), *) (* predD (difference), and preim (preimage, i.e., composition) *) (* For each operation we define three forms, typically: *) (* - predU : pred T -> pred T -> simpl_pred T *) (* - [predU A & B], a Notation for predU (mem A) (mem B) *) (* - xpredU, a Notation for the lambda-expression inside predU, *) (* which is mostly useful as an argument of =1, since it exposes the head *) (* head constant of the expression to the ssreflect matching algorithm. *) (* The syntax for the preimage of a collective predicate A is *) (* - [preim f of A] *) (* Finally, the generic syntax for defining a simpl_pred T is *) (* - [pred x : T | P(x)], [pred x | P(x)], [pred x in A | P(x)], etc. *) (* We also support boolean relations, but only the applicative form, with *) (* types *) (* - rel T, an alias for T -> pred T *) (* - simpl_rel T, an auto-simplifying version, and syntax *) (* [rel x y | P(x,y)], [rel x y in A & B | P(x,y)], etc. *) (* The notation [rel of fA] can be used to coerce a function returning a *) (* collective predicate to one returning pred T. *) (* Finally, note that there is specific support for ambivalent predicates *) (* that can work in either style, as per this file's head descriptor. *) (******************************************************************************) Definition pred T := T -> bool. Identity Coercion fun_of_pred : pred >-> Funclass. Definition rel T := T -> pred T. Identity Coercion fun_of_rel : rel >-> Funclass. Notation xpred0 := (fun _ => false). Notation xpredT := (fun _ => true). Notation xpredI := (fun (p1 p2 : pred _) x => p1 x && p2 x). Notation xpredU := (fun (p1 p2 : pred _) x => p1 x || p2 x). Notation xpredC := (fun (p : pred _) x => ~~ p x). Notation xpredD := (fun (p1 p2 : pred _) x => ~~ p2 x && p1 x). Notation xpreim := (fun f (p : pred _) x => p (f x)). Notation xrelU := (fun (r1 r2 : rel _) x y => r1 x y || r2 x y). Section Predicates. Variables T : Type. Definition subpred (p1 p2 : pred T) := forall x, p1 x -> p2 x. Definition subrel (r1 r2 : rel T) := forall x y, r1 x y -> r2 x y. Definition simpl_pred := simpl_fun T bool. Definition applicative_pred := pred T. Definition collective_pred := pred T. Definition SimplPred (p : pred T) : simpl_pred := SimplFun p. Coercion pred_of_simpl (p : simpl_pred) : pred T := fun_of_simpl p. Coercion applicative_pred_of_simpl (p : simpl_pred) : applicative_pred := fun_of_simpl p. Coercion collective_pred_of_simpl (p : simpl_pred) : collective_pred := fun x => (let: SimplFun f := p in fun _ => f x) x. (* Note: applicative_of_simpl is convertible to pred_of_simpl, while *) (* collective_of_simpl is not. *) Definition pred0 := SimplPred xpred0. Definition predT := SimplPred xpredT. Definition predI p1 p2 := SimplPred (xpredI p1 p2). Definition predU p1 p2 := SimplPred (xpredU p1 p2). Definition predC p := SimplPred (xpredC p). Definition predD p1 p2 := SimplPred (xpredD p1 p2). Definition preim rT f (d : pred rT) := SimplPred (xpreim f d). Definition simpl_rel := simpl_fun T (pred T). Definition SimplRel (r : rel T) : simpl_rel := [fun x => r x]. Coercion rel_of_simpl_rel (r : simpl_rel) : rel T := fun x y => r x y. Definition relU r1 r2 := SimplRel (xrelU r1 r2). Lemma subrelUl r1 r2 : subrel r1 (relU r1 r2). Proof. by move=> *; apply/orP; left. Qed. Lemma subrelUr r1 r2 : subrel r2 (relU r1 r2). Proof. by move=> *; apply/orP; right. Qed. CoInductive mem_pred := Mem of pred T. Definition isMem pT topred mem := mem = (fun p : pT => Mem [eta topred p]). Structure predType := PredType { pred_sort :> Type; topred : pred_sort -> pred T; _ : {mem | isMem topred mem} }. Definition mkPredType pT toP := PredType (exist (@isMem pT toP) _ (erefl _)). Canonical predPredType := Eval hnf in @mkPredType (pred T) id. Canonical simplPredType := Eval hnf in mkPredType pred_of_simpl. Canonical boolfunPredType := Eval hnf in @mkPredType (T -> bool) id. Coercion pred_of_mem mp : pred_sort predPredType := let: Mem p := mp in [eta p]. Canonical memPredType := Eval hnf in mkPredType pred_of_mem. Definition clone_pred U := fun pT & pred_sort pT -> U => fun a mP (pT' := @PredType U a mP) & phant_id pT' pT => pT'. End Predicates. Arguments pred0 [T]. Arguments predT [T]. Prenex Implicits pred0 predT predI predU predC predD preim relU. Notation "[ 'pred' : T | E ]" := (SimplPred (fun _ : T => E%B)) (at level 0, format "[ 'pred' : T | E ]") : fun_scope. Notation "[ 'pred' x | E ]" := (SimplPred (fun x => E%B)) (at level 0, x ident, format "[ 'pred' x | E ]") : fun_scope. Notation "[ 'pred' x | E1 & E2 ]" := [pred x | E1 && E2 ] (at level 0, x ident, format "[ 'pred' x | E1 & E2 ]") : fun_scope. Notation "[ 'pred' x : T | E ]" := (SimplPred (fun x : T => E%B)) (at level 0, x ident, only parsing) : fun_scope. Notation "[ 'pred' x : T | E1 & E2 ]" := [pred x : T | E1 && E2 ] (at level 0, x ident, only parsing) : fun_scope. Notation "[ 'rel' x y | E ]" := (SimplRel (fun x y => E%B)) (at level 0, x ident, y ident, format "[ 'rel' x y | E ]") : fun_scope. Notation "[ 'rel' x y : T | E ]" := (SimplRel (fun x y : T => E%B)) (at level 0, x ident, y ident, only parsing) : fun_scope. Notation "[ 'predType' 'of' T ]" := (@clone_pred _ T _ id _ _ id) (at level 0, format "[ 'predType' 'of' T ]") : form_scope. (* This redundant coercion lets us "inherit" the simpl_predType canonical *) (* instance by declaring a coercion to simpl_pred. This hack is the only way *) (* to put a predType structure on a predArgType. We use simpl_pred rather *) (* than pred to ensure that /= removes the identity coercion. Note that the *) (* coercion will never be used directly for simpl_pred, since the canonical *) (* instance should always be resolved. *) Notation pred_class := (pred_sort (predPredType _)). Coercion sort_of_simpl_pred T (p : simpl_pred T) : pred_class := p : pred T. (* This lets us use some types as a synonym for their universal predicate. *) (* Unfortunately, this won't work for existing types like bool, unless we *) (* redefine bool, true, false and all bool ops. *) Definition predArgType := Type. Bind Scope type_scope with predArgType. Identity Coercion sort_of_predArgType : predArgType >-> Sortclass. Coercion pred_of_argType (T : predArgType) : simpl_pred T := predT. Notation "{ : T }" := (T%type : predArgType) (at level 0, format "{ : T }") : type_scope. (* These must be defined outside a Section because "cooking" kills the *) (* nosimpl tag. *) Definition mem T (pT : predType T) : pT -> mem_pred T := nosimpl (let: @PredType _ _ _ (exist _ mem _) := pT return pT -> _ in mem). Definition in_mem T x mp := nosimpl pred_of_mem T mp x. Prenex Implicits mem. Coercion pred_of_mem_pred T mp := [pred x : T | in_mem x mp]. Definition eq_mem T p1 p2 := forall x : T, in_mem x p1 = in_mem x p2. Definition sub_mem T p1 p2 := forall x : T, in_mem x p1 -> in_mem x p2. Typeclasses Opaque eq_mem. Lemma sub_refl T (p : mem_pred T) : sub_mem p p. Proof. by []. Qed. Arguments sub_refl {T p}. Notation "x \in A" := (in_mem x (mem A)) : bool_scope. Notation "x \in A" := (in_mem x (mem A)) : bool_scope. Notation "x \notin A" := (~~ (x \in A)) : bool_scope. Notation "A =i B" := (eq_mem (mem A) (mem B)) : type_scope. Notation "{ 'subset' A <= B }" := (sub_mem (mem A) (mem B)) (at level 0, A, B at level 69, format "{ '[hv' 'subset' A '/ ' <= B ']' }") : type_scope. Notation "[ 'mem' A ]" := (pred_of_simpl (pred_of_mem_pred (mem A))) (at level 0, only parsing) : fun_scope. Notation "[ 'rel' 'of' fA ]" := (fun x => [mem (fA x)]) (at level 0, format "[ 'rel' 'of' fA ]") : fun_scope. Notation "[ 'predI' A & B ]" := (predI [mem A] [mem B]) (at level 0, format "[ 'predI' A & B ]") : fun_scope. Notation "[ 'predU' A & B ]" := (predU [mem A] [mem B]) (at level 0, format "[ 'predU' A & B ]") : fun_scope. Notation "[ 'predD' A & B ]" := (predD [mem A] [mem B]) (at level 0, format "[ 'predD' A & B ]") : fun_scope. Notation "[ 'predC' A ]" := (predC [mem A]) (at level 0, format "[ 'predC' A ]") : fun_scope. Notation "[ 'preim' f 'of' A ]" := (preim f [mem A]) (at level 0, format "[ 'preim' f 'of' A ]") : fun_scope. Notation "[ 'pred' x 'in' A ]" := [pred x | x \in A] (at level 0, x ident, format "[ 'pred' x 'in' A ]") : fun_scope. Notation "[ 'pred' x 'in' A | E ]" := [pred x | x \in A & E] (at level 0, x ident, format "[ 'pred' x 'in' A | E ]") : fun_scope. Notation "[ 'pred' x 'in' A | E1 & E2 ]" := [pred x | x \in A & E1 && E2 ] (at level 0, x ident, format "[ 'pred' x 'in' A | E1 & E2 ]") : fun_scope. Notation "[ 'rel' x y 'in' A & B | E ]" := [rel x y | (x \in A) && (y \in B) && E] (at level 0, x ident, y ident, format "[ 'rel' x y 'in' A & B | E ]") : fun_scope. Notation "[ 'rel' x y 'in' A & B ]" := [rel x y | (x \in A) && (y \in B)] (at level 0, x ident, y ident, format "[ 'rel' x y 'in' A & B ]") : fun_scope. Notation "[ 'rel' x y 'in' A | E ]" := [rel x y in A & A | E] (at level 0, x ident, y ident, format "[ 'rel' x y 'in' A | E ]") : fun_scope. Notation "[ 'rel' x y 'in' A ]" := [rel x y in A & A] (at level 0, x ident, y ident, format "[ 'rel' x y 'in' A ]") : fun_scope. Section simpl_mem. Variables (T : Type) (pT : predType T). Implicit Types (x : T) (p : pred T) (sp : simpl_pred T) (pp : pT). (* Bespoke structures that provide fine-grained control over matching the *) (* various forms of the \in predicate; note in particular the different forms *) (* of hoisting that are used. We had to work around several bugs in the *) (* implementation of unification, notably improper expansion of telescope *) (* projections and overwriting of a variable assignment by a later *) (* unification (probably due to conversion cache cross-talk). *) Structure manifest_applicative_pred p := ManifestApplicativePred { manifest_applicative_pred_value :> pred T; _ : manifest_applicative_pred_value = p }. Definition ApplicativePred p := ManifestApplicativePred (erefl p). Canonical applicative_pred_applicative sp := ApplicativePred (applicative_pred_of_simpl sp). Structure manifest_simpl_pred p := ManifestSimplPred { manifest_simpl_pred_value :> simpl_pred T; _ : manifest_simpl_pred_value = SimplPred p }. Canonical expose_simpl_pred p := ManifestSimplPred (erefl (SimplPred p)). Structure manifest_mem_pred p := ManifestMemPred { manifest_mem_pred_value :> mem_pred T; _ : manifest_mem_pred_value= Mem [eta p] }. Canonical expose_mem_pred p := @ManifestMemPred p _ (erefl _). Structure applicative_mem_pred p := ApplicativeMemPred {applicative_mem_pred_value :> manifest_mem_pred p}. Canonical check_applicative_mem_pred p (ap : manifest_applicative_pred p) mp := @ApplicativeMemPred ap mp. Lemma mem_topred (pp : pT) : mem (topred pp) = mem pp. Proof. by rewrite /mem; case: pT pp => T1 app1 [mem1 /= ->]. Qed. Lemma topredE x (pp : pT) : topred pp x = (x \in pp). Proof. by rewrite -mem_topred. Qed. Lemma app_predE x p (ap : manifest_applicative_pred p) : ap x = (x \in p). Proof. by case: ap => _ /= ->. Qed. Lemma in_applicative x p (amp : applicative_mem_pred p) : in_mem x amp = p x. Proof. by case: amp => [[_ /= ->]]. Qed. Lemma in_collective x p (msp : manifest_simpl_pred p) : (x \in collective_pred_of_simpl msp) = p x. Proof. by case: msp => _ /= ->. Qed. Lemma in_simpl x p (msp : manifest_simpl_pred p) : in_mem x (Mem [eta fun_of_simpl (msp : simpl_pred T)]) = p x. Proof. by case: msp => _ /= ->. Qed. (* Because of the explicit eta expansion in the left-hand side, this lemma *) (* should only be used in a right-to-left direction. The 8.3 hack allowing *) (* partial right-to-left use does not work with the improved expansion *) (* heuristics in 8.4. *) Lemma unfold_in x p : (x \in ([eta p] : pred T)) = p x. Proof. by []. Qed. Lemma simpl_predE p : SimplPred p =1 p. Proof. by []. Qed. Definition inE := (in_applicative, in_simpl, simpl_predE). (* to be extended *) Lemma mem_simpl sp : mem sp = sp :> pred T. Proof. by []. Qed. Definition memE := mem_simpl. (* could be extended *) Lemma mem_mem (pp : pT) : (mem (mem pp) = mem pp) * (mem [mem pp] = mem pp). Proof. by rewrite -mem_topred. Qed. End simpl_mem. (* Qualifiers and keyed predicates. *) CoInductive qualifier (q : nat) T := Qualifier of predPredType T. Coercion has_quality n T (q : qualifier n T) : pred_class := fun x => let: Qualifier _ p := q in p x. Arguments has_quality n [T]. Lemma qualifE n T p x : (x \in @Qualifier n T p) = p x. Proof. by []. Qed. Notation "x \is A" := (x \in has_quality 0 A) (at level 70, no associativity, format "'[hv' x '/ ' \is A ']'") : bool_scope. Notation "x \is 'a' A" := (x \in has_quality 1 A) (at level 70, no associativity, format "'[hv' x '/ ' \is 'a' A ']'") : bool_scope. Notation "x \is 'an' A" := (x \in has_quality 2 A) (at level 70, no associativity, format "'[hv' x '/ ' \is 'an' A ']'") : bool_scope. Notation "x \isn't A" := (x \notin has_quality 0 A) (at level 70, no associativity, format "'[hv' x '/ ' \isn't A ']'") : bool_scope. Notation "x \isn't 'a' A" := (x \notin has_quality 1 A) (at level 70, no associativity, format "'[hv' x '/ ' \isn't 'a' A ']'") : bool_scope. Notation "x \isn't 'an' A" := (x \notin has_quality 2 A) (at level 70, no associativity, format "'[hv' x '/ ' \isn't 'an' A ']'") : bool_scope. Notation "[ 'qualify' x | P ]" := (Qualifier 0 (fun x => P%B)) (at level 0, x at level 99, format "'[hv' [ 'qualify' x | '/ ' P ] ']'") : form_scope. Notation "[ 'qualify' x : T | P ]" := (Qualifier 0 (fun x : T => P%B)) (at level 0, x at level 99, only parsing) : form_scope. Notation "[ 'qualify' 'a' x | P ]" := (Qualifier 1 (fun x => P%B)) (at level 0, x at level 99, format "'[hv' [ 'qualify' 'a' x | '/ ' P ] ']'") : form_scope. Notation "[ 'qualify' 'a' x : T | P ]" := (Qualifier 1 (fun x : T => P%B)) (at level 0, x at level 99, only parsing) : form_scope. Notation "[ 'qualify' 'an' x | P ]" := (Qualifier 2 (fun x => P%B)) (at level 0, x at level 99, format "'[hv' [ 'qualify' 'an' x | '/ ' P ] ']'") : form_scope. Notation "[ 'qualify' 'an' x : T | P ]" := (Qualifier 2 (fun x : T => P%B)) (at level 0, x at level 99, only parsing) : form_scope. (* Keyed predicates: support for property-bearing predicate interfaces. *) Section KeyPred. Variable T : Type. CoInductive pred_key (p : predPredType T) := DefaultPredKey. Variable p : predPredType T. Structure keyed_pred (k : pred_key p) := PackKeyedPred {unkey_pred :> pred_class; _ : unkey_pred =i p}. Variable k : pred_key p. Definition KeyedPred := @PackKeyedPred k p (frefl _). Variable k_p : keyed_pred k. Lemma keyed_predE : k_p =i p. Proof. by case: k_p. Qed. (* Instances that strip the mem cast; the first one has "pred_of_mem" as its *) (* projection head value, while the second has "pred_of_simpl". The latter *) (* has the side benefit of preempting accidental misdeclarations. *) (* Note: pred_of_mem is the registered mem >-> pred_class coercion, while *) (* simpl_of_mem; pred_of_simpl is the mem >-> pred >=> Funclass coercion. We *) (* must write down the coercions explicitly as the Canonical head constant *) (* computation does not strip casts !! *) Canonical keyed_mem := @PackKeyedPred k (pred_of_mem (mem k_p)) keyed_predE. Canonical keyed_mem_simpl := @PackKeyedPred k (pred_of_simpl (mem k_p)) keyed_predE. End KeyPred. Notation "x \i 'n' S" := (x \in @unkey_pred _ S _ _) (at level 70, format "'[hv' x '/ ' \i 'n' S ']'") : bool_scope. Section KeyedQualifier. Variables (T : Type) (n : nat) (q : qualifier n T). Structure keyed_qualifier (k : pred_key q) := PackKeyedQualifier {unkey_qualifier; _ : unkey_qualifier = q}. Definition KeyedQualifier k := PackKeyedQualifier k (erefl q). Variables (k : pred_key q) (k_q : keyed_qualifier k). Fact keyed_qualifier_suproof : unkey_qualifier k_q =i q. Proof. by case: k_q => /= _ ->. Qed. Canonical keyed_qualifier_keyed := PackKeyedPred k keyed_qualifier_suproof. End KeyedQualifier. Notation "x \i 's' A" := (x \i n has_quality 0 A) (at level 70, format "'[hv' x '/ ' \i 's' A ']'") : bool_scope. Notation "x \i 's' 'a' A" := (x \i n has_quality 1 A) (at level 70, format "'[hv' x '/ ' \i 's' 'a' A ']'") : bool_scope. Notation "x \i 's' 'an' A" := (x \i n has_quality 2 A) (at level 70, format "'[hv' x '/ ' \i 's' 'an' A ']'") : bool_scope. Module DefaultKeying. Canonical default_keyed_pred T p := KeyedPred (@DefaultPredKey T p). Canonical default_keyed_qualifier T n (q : qualifier n T) := KeyedQualifier (DefaultPredKey q). End DefaultKeying. (* Skolemizing with conditions. *) Lemma all_tag_cond_dep I T (C : pred I) U : (forall x, T x) -> (forall x, C x -> {y : T x & U x y}) -> {f : forall x, T x & forall x, C x -> U x (f x)}. Proof. move=> f0 fP; apply: all_tag (fun x y => C x -> U x y) _ => x. by case Cx: (C x); [case/fP: Cx => y; exists y | exists (f0 x)]. Qed. Lemma all_tag_cond I T (C : pred I) U : T -> (forall x, C x -> {y : T & U x y}) -> {f : I -> T & forall x, C x -> U x (f x)}. Proof. by move=> y0; apply: all_tag_cond_dep. Qed. Lemma all_sig_cond_dep I T (C : pred I) P : (forall x, T x) -> (forall x, C x -> {y : T x | P x y}) -> {f : forall x, T x | forall x, C x -> P x (f x)}. Proof. by move=> f0 /(all_tag_cond_dep f0)[f]; exists f. Qed. Lemma all_sig_cond I T (C : pred I) P : T -> (forall x, C x -> {y : T | P x y}) -> {f : I -> T | forall x, C x -> P x (f x)}. Proof. by move=> y0; apply: all_sig_cond_dep. Qed. Section RelationProperties. (* Caveat: reflexive should not be used to state lemmas, as auto and trivial *) (* will not expand the constant. *) Variable T : Type. Variable R : rel T. Definition total := forall x y, R x y || R y x. Definition transitive := forall y x z, R x y -> R y z -> R x z. Definition symmetric := forall x y, R x y = R y x. Definition antisymmetric := forall x y, R x y && R y x -> x = y. Definition pre_symmetric := forall x y, R x y -> R y x. Lemma symmetric_from_pre : pre_symmetric -> symmetric. Proof. by move=> symR x y; apply/idP/idP; apply: symR. Qed. Definition reflexive := forall x, R x x. Definition irreflexive := forall x, R x x = false. Definition left_transitive := forall x y, R x y -> R x =1 R y. Definition right_transitive := forall x y, R x y -> R^~ x =1 R^~ y. Section PER. Hypotheses (symR : symmetric) (trR : transitive). Lemma sym_left_transitive : left_transitive. Proof. by move=> x y Rxy z; apply/idP/idP; apply: trR; rewrite // symR. Qed. Lemma sym_right_transitive : right_transitive. Proof. by move=> x y /sym_left_transitive Rxy z; rewrite !(symR z) Rxy. Qed. End PER. (* We define the equivalence property with prenex quantification so that it *) (* can be localized using the {in ..., ..} form defined below. *) Definition equivalence_rel := forall x y z, R z z * (R x y -> R x z = R y z). Lemma equivalence_relP : equivalence_rel <-> reflexive /\ left_transitive. Proof. split=> [eqiR | [Rxx trR] x y z]; last by split=> [|/trR->]. by split=> [x | x y Rxy z]; [rewrite (eqiR x x x) | rewrite (eqiR x y z)]. Qed. End RelationProperties. Lemma rev_trans T (R : rel T) : transitive R -> transitive (fun x y => R y x). Proof. by move=> trR x y z Ryx Rzy; apply: trR Rzy Ryx. Qed. (* Property localization *) Local Notation "{ 'all1' P }" := (forall x, P x : Prop) (at level 0). Local Notation "{ 'all2' P }" := (forall x y, P x y : Prop) (at level 0). Local Notation "{ 'all3' P }" := (forall x y z, P x y z: Prop) (at level 0). Local Notation ph := (phantom _). Section LocalProperties. Variables T1 T2 T3 : Type. Variables (d1 : mem_pred T1) (d2 : mem_pred T2) (d3 : mem_pred T3). Local Notation ph := (phantom Prop). Definition prop_for (x : T1) P & ph {all1 P} := P x. Lemma forE x P phP : @prop_for x P phP = P x. Proof. by []. Qed. Definition prop_in1 P & ph {all1 P} := forall x, in_mem x d1 -> P x. Definition prop_in11 P & ph {all2 P} := forall x y, in_mem x d1 -> in_mem y d2 -> P x y. Definition prop_in2 P & ph {all2 P} := forall x y, in_mem x d1 -> in_mem y d1 -> P x y. Definition prop_in111 P & ph {all3 P} := forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d3 -> P x y z. Definition prop_in12 P & ph {all3 P} := forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d2 -> P x y z. Definition prop_in21 P & ph {all3 P} := forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d2 -> P x y z. Definition prop_in3 P & ph {all3 P} := forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d1 -> P x y z. Variable f : T1 -> T2. Definition prop_on1 Pf P & phantom T3 (Pf f) & ph {all1 P} := forall x, in_mem (f x) d2 -> P x. Definition prop_on2 Pf P & phantom T3 (Pf f) & ph {all2 P} := forall x y, in_mem (f x) d2 -> in_mem (f y) d2 -> P x y. End LocalProperties. Definition inPhantom := Phantom Prop. Definition onPhantom T P (x : T) := Phantom Prop (P x). Definition bijective_in aT rT (d : mem_pred aT) (f : aT -> rT) := exists2 g, prop_in1 d (inPhantom (cancel f g)) & prop_on1 d (Phantom _ (cancel g)) (onPhantom (cancel g) f). Definition bijective_on aT rT (cd : mem_pred rT) (f : aT -> rT) := exists2 g, prop_on1 cd (Phantom _ (cancel f)) (onPhantom (cancel f) g) & prop_in1 cd (inPhantom (cancel g f)). Notation "{ 'for' x , P }" := (prop_for x (inPhantom P)) (at level 0, format "{ 'for' x , P }") : type_scope. Notation "{ 'in' d , P }" := (prop_in1 (mem d) (inPhantom P)) (at level 0, format "{ 'in' d , P }") : type_scope. Notation "{ 'in' d1 & d2 , P }" := (prop_in11 (mem d1) (mem d2) (inPhantom P)) (at level 0, format "{ 'in' d1 & d2 , P }") : type_scope. Notation "{ 'in' d & , P }" := (prop_in2 (mem d) (inPhantom P)) (at level 0, format "{ 'in' d & , P }") : type_scope. Notation "{ 'in' d1 & d2 & d3 , P }" := (prop_in111 (mem d1) (mem d2) (mem d3) (inPhantom P)) (at level 0, format "{ 'in' d1 & d2 & d3 , P }") : type_scope. Notation "{ 'in' d1 & & d3 , P }" := (prop_in21 (mem d1) (mem d3) (inPhantom P)) (at level 0, format "{ 'in' d1 & & d3 , P }") : type_scope. Notation "{ 'in' d1 & d2 & , P }" := (prop_in12 (mem d1) (mem d2) (inPhantom P)) (at level 0, format "{ 'in' d1 & d2 & , P }") : type_scope. Notation "{ 'in' d & & , P }" := (prop_in3 (mem d) (inPhantom P)) (at level 0, format "{ 'in' d & & , P }") : type_scope. Notation "{ 'on' cd , P }" := (prop_on1 (mem cd) (inPhantom P) (inPhantom P)) (at level 0, format "{ 'on' cd , P }") : type_scope. Notation "{ 'on' cd & , P }" := (prop_on2 (mem cd) (inPhantom P) (inPhantom P)) (at level 0, format "{ 'on' cd & , P }") : type_scope. Local Arguments onPhantom {_%type_scope} _ _. Notation "{ 'on' cd , P & g }" := (prop_on1 (mem cd) (Phantom (_ -> Prop) P) (onPhantom P g)) (at level 0, format "{ 'on' cd , P & g }") : type_scope. Notation "{ 'in' d , 'bijective' f }" := (bijective_in (mem d) f) (at level 0, f at level 8, format "{ 'in' d , 'bijective' f }") : type_scope. Notation "{ 'on' cd , 'bijective' f }" := (bijective_on (mem cd) f) (at level 0, f at level 8, format "{ 'on' cd , 'bijective' f }") : type_scope. (* Weakening and monotonicity lemmas for localized predicates. *) (* Note that using these lemmas in backward reasoning will force expansion of *) (* the predicate definition, as Coq needs to expose the quantifier to apply *) (* these lemmas. We define a few specialized variants to avoid this for some *) (* of the ssrfun predicates. *) Section LocalGlobal. Variables T1 T2 T3 : predArgType. Variables (D1 : pred T1) (D2 : pred T2) (D3 : pred T3). Variables (d1 d1' : mem_pred T1) (d2 d2' : mem_pred T2) (d3 d3' : mem_pred T3). Variables (f f' : T1 -> T2) (g : T2 -> T1) (h : T3). Variables (P1 : T1 -> Prop) (P2 : T1 -> T2 -> Prop). Variable P3 : T1 -> T2 -> T3 -> Prop. Variable Q1 : (T1 -> T2) -> T1 -> Prop. Variable Q1l : (T1 -> T2) -> T3 -> T1 -> Prop. Variable Q2 : (T1 -> T2) -> T1 -> T1 -> Prop. Hypothesis sub1 : sub_mem d1 d1'. Hypothesis sub2 : sub_mem d2 d2'. Hypothesis sub3 : sub_mem d3 d3'. Lemma in1W : {all1 P1} -> {in D1, {all1 P1}}. Proof. by move=> ? ?. Qed. Lemma in2W : {all2 P2} -> {in D1 & D2, {all2 P2}}. Proof. by move=> ? ?. Qed. Lemma in3W : {all3 P3} -> {in D1 & D2 & D3, {all3 P3}}. Proof. by move=> ? ?. Qed. Lemma in1T : {in T1, {all1 P1}} -> {all1 P1}. Proof. by move=> ? ?; auto. Qed. Lemma in2T : {in T1 & T2, {all2 P2}} -> {all2 P2}. Proof. by move=> ? ?; auto. Qed. Lemma in3T : {in T1 & T2 & T3, {all3 P3}} -> {all3 P3}. Proof. by move=> ? ?; auto. Qed. Lemma sub_in1 (Ph : ph {all1 P1}) : prop_in1 d1' Ph -> prop_in1 d1 Ph. Proof. by move=> allP x /sub1; apply: allP. Qed. Lemma sub_in11 (Ph : ph {all2 P2}) : prop_in11 d1' d2' Ph -> prop_in11 d1 d2 Ph. Proof. by move=> allP x1 x2 /sub1 d1x1 /sub2; apply: allP. Qed. Lemma sub_in111 (Ph : ph {all3 P3}) : prop_in111 d1' d2' d3' Ph -> prop_in111 d1 d2 d3 Ph. Proof. by move=> allP x1 x2 x3 /sub1 d1x1 /sub2 d2x2 /sub3; apply: allP. Qed. Let allQ1 f'' := {all1 Q1 f''}. Let allQ1l f'' h' := {all1 Q1l f'' h'}. Let allQ2 f'' := {all2 Q2 f''}. Lemma on1W : allQ1 f -> {on D2, allQ1 f}. Proof. by move=> ? ?. Qed. Lemma on1lW : allQ1l f h -> {on D2, allQ1l f & h}. Proof. by move=> ? ?. Qed. Lemma on2W : allQ2 f -> {on D2 &, allQ2 f}. Proof. by move=> ? ?. Qed. Lemma on1T : {on T2, allQ1 f} -> allQ1 f. Proof. by move=> ? ?; auto. Qed. Lemma on1lT : {on T2, allQ1l f & h} -> allQ1l f h. Proof. by move=> ? ?; auto. Qed. Lemma on2T : {on T2 &, allQ2 f} -> allQ2 f. Proof. by move=> ? ?; auto. Qed. Lemma subon1 (Phf : ph (allQ1 f)) (Ph : ph (allQ1 f)) : prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph. Proof. by move=> allQ x /sub2; apply: allQ. Qed. Lemma subon1l (Phf : ph (allQ1l f)) (Ph : ph (allQ1l f h)) : prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph. Proof. by move=> allQ x /sub2; apply: allQ. Qed. Lemma subon2 (Phf : ph (allQ2 f)) (Ph : ph (allQ2 f)) : prop_on2 d2' Phf Ph -> prop_on2 d2 Phf Ph. Proof. by move=> allQ x y /sub2=> d2fx /sub2; apply: allQ. Qed. Lemma can_in_inj : {in D1, cancel f g} -> {in D1 &, injective f}. Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. Lemma canLR_in x y : {in D1, cancel f g} -> y \in D1 -> x = f y -> g x = y. Proof. by move=> fK D1y ->; rewrite fK. Qed. Lemma canRL_in x y : {in D1, cancel f g} -> x \in D1 -> f x = y -> x = g y. Proof. by move=> fK D1x <-; rewrite fK. Qed. Lemma on_can_inj : {on D2, cancel f & g} -> {on D2 &, injective f}. Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. Lemma canLR_on x y : {on D2, cancel f & g} -> f y \in D2 -> x = f y -> g x = y. Proof. by move=> fK D2fy ->; rewrite fK. Qed. Lemma canRL_on x y : {on D2, cancel f & g} -> f x \in D2 -> f x = y -> x = g y. Proof. by move=> fK D2fx <-; rewrite fK. Qed. Lemma inW_bij : bijective f -> {in D1, bijective f}. Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. Lemma onW_bij : bijective f -> {on D2, bijective f}. Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. Lemma inT_bij : {in T1, bijective f} -> bijective f. Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. Lemma onT_bij : {on T2, bijective f} -> bijective f. Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. Lemma sub_in_bij (D1' : pred T1) : {subset D1 <= D1'} -> {in D1', bijective f} -> {in D1, bijective f}. Proof. by move=> subD [g' fK g'K]; exists g' => x; move/subD; [apply: fK | apply: g'K]. Qed. Lemma subon_bij (D2' : pred T2) : {subset D2 <= D2'} -> {on D2', bijective f} -> {on D2, bijective f}. Proof. by move=> subD [g' fK g'K]; exists g' => x; move/subD; [apply: fK | apply: g'K]. Qed. End LocalGlobal. Lemma sub_in2 T d d' (P : T -> T -> Prop) : sub_mem d d' -> forall Ph : ph {all2 P}, prop_in2 d' Ph -> prop_in2 d Ph. Proof. by move=> /= sub_dd'; apply: sub_in11. Qed. Lemma sub_in3 T d d' (P : T -> T -> T -> Prop) : sub_mem d d' -> forall Ph : ph {all3 P}, prop_in3 d' Ph -> prop_in3 d Ph. Proof. by move=> /= sub_dd'; apply: sub_in111. Qed. Lemma sub_in12 T1 T d1 d1' d d' (P : T1 -> T -> T -> Prop) : sub_mem d1 d1' -> sub_mem d d' -> forall Ph : ph {all3 P}, prop_in12 d1' d' Ph -> prop_in12 d1 d Ph. Proof. by move=> /= sub1 sub; apply: sub_in111. Qed. Lemma sub_in21 T T3 d d' d3 d3' (P : T -> T -> T3 -> Prop) : sub_mem d d' -> sub_mem d3 d3' -> forall Ph : ph {all3 P}, prop_in21 d' d3' Ph -> prop_in21 d d3 Ph. Proof. by move=> /= sub sub3; apply: sub_in111. Qed. Lemma equivalence_relP_in T (R : rel T) (A : pred T) : {in A & &, equivalence_rel R} <-> {in A, reflexive R} /\ {in A &, forall x y, R x y -> {in A, R x =1 R y}}. Proof. split=> [eqiR | [Rxx trR] x y z *]; last by split=> [|/trR-> //]; apply: Rxx. by split=> [x Ax|x y Ax Ay Rxy z Az]; [rewrite (eqiR x x) | rewrite (eqiR x y)]. Qed. Section MonoHomoMorphismTheory. Variables (aT rT sT : Type) (f : aT -> rT) (g : rT -> aT). Variables (aP : pred aT) (rP : pred rT) (aR : rel aT) (rR : rel rT). Lemma monoW : {mono f : x / aP x >-> rP x} -> {homo f : x / aP x >-> rP x}. Proof. by move=> hf x ax; rewrite hf. Qed. Lemma mono2W : {mono f : x y / aR x y >-> rR x y} -> {homo f : x y / aR x y >-> rR x y}. Proof. by move=> hf x y axy; rewrite hf. Qed. Hypothesis fgK : cancel g f. Lemma homoRL : {homo f : x y / aR x y >-> rR x y} -> forall x y, aR (g x) y -> rR x (f y). Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. Lemma homoLR : {homo f : x y / aR x y >-> rR x y} -> forall x y, aR x (g y) -> rR (f x) y. Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. Lemma homo_mono : {homo f : x y / aR x y >-> rR x y} -> {homo g : x y / rR x y >-> aR x y} -> {mono g : x y / rR x y >-> aR x y}. Proof. move=> mf mg x y; case: (boolP (rR _ _))=> [/mg //|]. by apply: contraNF=> /mf; rewrite !fgK. Qed. Lemma monoLR : {mono f : x y / aR x y >-> rR x y} -> forall x y, rR (f x) y = aR x (g y). Proof. by move=> mf x y; rewrite -{1}[y]fgK mf. Qed. Lemma monoRL : {mono f : x y / aR x y >-> rR x y} -> forall x y, rR x (f y) = aR (g x) y. Proof. by move=> mf x y; rewrite -{1}[x]fgK mf. Qed. Lemma can_mono : {mono f : x y / aR x y >-> rR x y} -> {mono g : x y / rR x y >-> aR x y}. Proof. by move=> mf x y /=; rewrite -mf !fgK. Qed. End MonoHomoMorphismTheory. Section MonoHomoMorphismTheory_in. Variables (aT rT sT : predArgType) (f : aT -> rT) (g : rT -> aT). Variable (aD : pred aT). Variable (aP : pred aT) (rP : pred rT) (aR : rel aT) (rR : rel rT). Notation rD := [pred x | g x \in aD]. Lemma monoW_in : {in aD &, {mono f : x y / aR x y >-> rR x y}} -> {in aD &, {homo f : x y / aR x y >-> rR x y}}. Proof. by move=> hf x y hx hy axy; rewrite hf. Qed. Lemma mono2W_in : {in aD, {mono f : x / aP x >-> rP x}} -> {in aD, {homo f : x / aP x >-> rP x}}. Proof. by move=> hf x hx ax; rewrite hf. Qed. Hypothesis fgK_on : {on aD, cancel g & f}. Lemma homoRL_in : {in aD &, {homo f : x y / aR x y >-> rR x y}} -> {in rD & aD, forall x y, aR (g x) y -> rR x (f y)}. Proof. by move=> Hf x y hx hy /Hf; rewrite fgK_on //; apply. Qed. Lemma homoLR_in : {in aD &, {homo f : x y / aR x y >-> rR x y}} -> {in aD & rD, forall x y, aR x (g y) -> rR (f x) y}. Proof. by move=> Hf x y hx hy /Hf; rewrite fgK_on //; apply. Qed. Lemma homo_mono_in : {in aD &, {homo f : x y / aR x y >-> rR x y}} -> {in rD &, {homo g : x y / rR x y >-> aR x y}} -> {in rD &, {mono g : x y / rR x y >-> aR x y}}. Proof. move=> mf mg x y hx hy; case: (boolP (rR _ _))=> [/mg //|]; first exact. by apply: contraNF=> /mf; rewrite !fgK_on //; apply. Qed. Lemma monoLR_in : {in aD &, {mono f : x y / aR x y >-> rR x y}} -> {in aD & rD, forall x y, rR (f x) y = aR x (g y)}. Proof. by move=> mf x y hx hy; rewrite -{1}[y]fgK_on // mf. Qed. Lemma monoRL_in : {in aD &, {mono f : x y / aR x y >-> rR x y}} -> {in rD & aD, forall x y, rR x (f y) = aR (g x) y}. Proof. by move=> mf x y hx hy; rewrite -{1}[x]fgK_on // mf. Qed. Lemma can_mono_in : {in aD &, {mono f : x y / aR x y >-> rR x y}} -> {in rD &, {mono g : x y / rR x y >-> aR x y}}. Proof. by move=> mf x y hx hy /=; rewrite -mf // !fgK_on. Qed. End MonoHomoMorphismTheory_in.