From: Benjamin Barenblat Subject: Remove ssrmatching Forwarded: not-needed Last-Update: 2018-12-29 ssrmatching still has two files licensed under CeCILL-B, which I believe is a nonfree license. I’ve removed them from the Debian source package (see gbp.conf). This patch disables everything that depends on them. This patch is fortunately a stopgap: Upstream believes that the files should have had the license headers changed to LGPL when importing them and has created a pull request to change the headers now (https://github.com/coq/coq/pull/9282). Once that is merged, this patch should disappear and be replaced with a backport of that PR. --- a/Makefile.common +++ b/Makefile.common @@ -84,7 +84,7 @@ setoid_ring extraction fourier \ cc funind firstorder derive \ rtauto nsatz syntax btauto \ - ssrmatching ltac ssr + ltac SRCDIRS:=\ $(CORESRCDIRS) \ @@ -149,7 +149,7 @@ $(FOURIERCMO) $(EXTRACTIONCMO) \ $(CCCMO) $(FOCMO) $(RTAUTOCMO) $(BTAUTOCMO) \ $(FUNINDCMO) $(NSATZCMO) $(NATSYNTAXCMO) $(OTHERSYNTAXCMO) \ - $(DERIVECMO) $(SSRMATCHINGCMO) $(SSRCMO) + $(DERIVECMO) ifeq ($(HASNATDYNLINK)-$(BEST),false-opt) STATICPLUGINS:=$(PLUGINSCMO) --- a/test-suite/success/ssrpattern.v +++ /dev/null @@ -1,22 +0,0 @@ -Require Import ssrmatching. - -(*Set Debug SsrMatching.*) - -Tactic Notation "at" "[" ssrpatternarg(pat) "]" tactic(t) := - let name := fresh in - let def_name := fresh in - ssrpattern pat; - intro name; - pose proof (refl_equal name) as def_name; - unfold name at 1 in def_name; - t def_name; - [ rewrite <- def_name | idtac.. ]; - clear name def_name. - -Lemma test (H : True -> True -> 3 = 7) : 28 = 3 * 4. -Proof. -at [ X in X * 4 ] ltac:(fun place => rewrite -> H in place). -- reflexivity. -- trivial. -- trivial. -Qed. --- a/plugins/ssr/ssreflect.v +++ /dev/null @@ -1,453 +0,0 @@ -(************************************************************************) -(* * The Coq Proof Assistant / The Coq Development Team *) -(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) -(* R. *) -(* {type of c for s} == P s where c : forall x : T, P x. *) -(* phantom T v == singleton type with inhabitant Phantom T v. *) -(* phant T == singleton type with inhabitant Phant v. *) -(* =^~ r == the converse of rewriting rule r (e.g., in a *) -(* rewrite multirule). *) -(* unkeyed t == t, but treated as an unkeyed matching pattern by *) -(* the ssreflect matching algorithm. *) -(* nosimpl t == t, but on the right-hand side of Definition C := *) -(* nosimpl disables expansion of C by /=. *) -(* locked t == t, but locked t is not convertible to t. *) -(* locked_with k t == t, but not convertible to t or locked_with k' t *) -(* unless k = k' (with k : unit). Coq type-checking *) -(* will be much more efficient if locked_with with a *) -(* bespoke k is used for sealed definitions. *) -(* unlockable v == interface for sealed constant definitions of v. *) -(* Unlockable def == the unlockable that registers def : C = v. *) -(* [unlockable of C] == a clone for C of the canonical unlockable for the *) -(* definition of C (e.g., if it uses locked_with). *) -(* [unlockable fun C] == [unlockable of C] with the expansion forced to be *) -(* an explicit lambda expression. *) -(* -> The usage pattern for ADT operations is: *) -(* Definition foo_def x1 .. xn := big_foo_expression. *) -(* Fact foo_key : unit. Proof. by []. Qed. *) -(* Definition foo := locked_with foo_key foo_def. *) -(* Canonical foo_unlockable := [unlockable fun foo]. *) -(* This minimizes the comparison overhead for foo, while still allowing *) -(* rewrite unlock to expose big_foo_expression. *) -(* More information about these definitions and their use can be found in the *) -(* ssreflect manual, and in specific comments below. *) -(******************************************************************************) - - -Set Implicit Arguments. -Unset Strict Implicit. -Unset Printing Implicit Defensive. - -Module SsrSyntax. - -(* Declare Ssr keywords: 'is' 'of' '//' '/=' and '//='. We also declare the *) -(* parsing level 8, as a workaround for a notation grammar factoring problem. *) -(* Arguments of application-style notations (at level 10) should be declared *) -(* at level 8 rather than 9 or the camlp5 grammar will not factor properly. *) - -Reserved Notation "(* x 'is' y 'of' z 'isn't' // /= //= *)" (at level 8). -Reserved Notation "(* 69 *)" (at level 69). - -(* Non ambiguous keyword to check if the SsrSyntax module is imported *) -Reserved Notation "(* Use to test if 'SsrSyntax_is_Imported' *)" (at level 8). - -Reserved Notation "" (at level 200). -Reserved Notation "T (* n *)" (at level 200, format "T (* n *)"). - -End SsrSyntax. - -Export SsrMatchingSyntax. -Export SsrSyntax. - -(* Make the general "if" into a notation, so that we can override it below. *) -(* The notations are "only parsing" because the Coq decompiler will not *) -(* recognize the expansion of the boolean if; using the default printer *) -(* avoids a spurrious trailing %GEN_IF. *) - -Delimit Scope general_if_scope with GEN_IF. - -Notation "'if' c 'then' v1 'else' v2" := - (if c then v1 else v2) - (at level 200, c, v1, v2 at level 200, only parsing) : general_if_scope. - -Notation "'if' c 'return' t 'then' v1 'else' v2" := - (if c return t then v1 else v2) - (at level 200, c, t, v1, v2 at level 200, only parsing) : general_if_scope. - -Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" := - (if c as x return t then v1 else v2) - (at level 200, c, t, v1, v2 at level 200, x ident, only parsing) - : general_if_scope. - -(* Force boolean interpretation of simple if expressions. *) - -Delimit Scope boolean_if_scope with BOOL_IF. - -Notation "'if' c 'return' t 'then' v1 'else' v2" := - (if c%bool is true in bool return t then v1 else v2) : boolean_if_scope. - -Notation "'if' c 'then' v1 'else' v2" := - (if c%bool is true in bool return _ then v1 else v2) : boolean_if_scope. - -Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" := - (if c%bool is true as x in bool return t then v1 else v2) : boolean_if_scope. - -Open Scope boolean_if_scope. - -(* To allow a wider variety of notations without reserving a large number of *) -(* of identifiers, the ssreflect library systematically uses "forms" to *) -(* enclose complex mixfix syntax. A "form" is simply a mixfix expression *) -(* enclosed in square brackets and introduced by a keyword: *) -(* [keyword ... ] *) -(* Because the keyword follows a bracket it does not need to be reserved. *) -(* Non-ssreflect libraries that do not respect the form syntax (e.g., the Coq *) -(* Lists library) should be loaded before ssreflect so that their notations *) -(* do not mask all ssreflect forms. *) -Delimit Scope form_scope with FORM. -Open Scope form_scope. - -(* Allow overloading of the cast (x : T) syntax, put whitespace around the *) -(* ":" symbol to avoid lexical clashes (and for consistency with the parsing *) -(* precedence of the notation, which binds less tightly than application), *) -(* and put printing boxes that print the type of a long definition on a *) -(* separate line rather than force-fit it at the right margin. *) -Notation "x : T" := (x : T) - (at level 100, right associativity, - format "'[hv' x '/ ' : T ']'") : core_scope. - -(* Allow the casual use of notations like nat * nat for explicit Type *) -(* declarations. Note that (nat * nat : Type) is NOT equivalent to *) -(* (nat * nat)%type, whose inferred type is legacy type "Set". *) -Notation "T : 'Type'" := (T%type : Type) - (at level 100, only parsing) : core_scope. -(* Allow similarly Prop annotation for, e.g., rewrite multirules. *) -Notation "P : 'Prop'" := (P%type : Prop) - (at level 100, only parsing) : core_scope. - -(* Constants for abstract: and [: name ] intro pattern *) -Definition abstract_lock := unit. -Definition abstract_key := tt. - -Definition abstract (statement : Type) (id : nat) (lock : abstract_lock) := - let: tt := lock in statement. - -Notation "" := (abstract _ n _). -Notation "T (* n *)" := (abstract T n abstract_key). - -(* Constants for tactic-views *) -Inductive external_view : Type := tactic_view of Type. - -(* Syntax for referring to canonical structures: *) -(* [the struct_type of proj_val by proj_fun] *) -(* This form denotes the Canonical instance s of the Structure type *) -(* struct_type whose proj_fun projection is proj_val, i.e., such that *) -(* proj_fun s = proj_val. *) -(* Typically proj_fun will be A record field accessors of struct_type, but *) -(* this need not be the case; it can be, for instance, a field of a record *) -(* type to which struct_type coerces; proj_val will likewise be coerced to *) -(* the return type of proj_fun. In all but the simplest cases, proj_fun *) -(* should be eta-expanded to allow for the insertion of implicit arguments. *) -(* In the common case where proj_fun itself is a coercion, the "by" part *) -(* can be omitted entirely; in this case it is inferred by casting s to the *) -(* inferred type of proj_val. Obviously the latter can be fixed by using an *) -(* explicit cast on proj_val, and it is highly recommended to do so when the *) -(* return type intended for proj_fun is "Type", as the type inferred for *) -(* proj_val may vary because of sort polymorphism (it could be Set or Prop). *) -(* Note when using the [the _ of _] form to generate a substructure from a *) -(* telescopes-style canonical hierarchy (implementing inheritance with *) -(* coercions), one should always project or coerce the value to the BASE *) -(* structure, because Coq will only find a Canonical derived structure for *) -(* the Canonical base structure -- not for a base structure that is specific *) -(* to proj_value. *) - -Module TheCanonical. - -CoInductive put vT sT (v1 v2 : vT) (s : sT) := Put. - -Definition get vT sT v s (p : @put vT sT v v s) := let: Put _ _ _ := p in s. - -Definition get_by vT sT of sT -> vT := @get vT sT. - -End TheCanonical. - -Import TheCanonical. (* Note: no export. *) - -Local Arguments get_by _%type_scope _%type_scope _ _ _ _. - -Notation "[ 'the' sT 'of' v 'by' f ]" := - (@get_by _ sT f _ _ ((fun v' (s : sT) => Put v' (f s) s) v _)) - (at level 0, only parsing) : form_scope. - -Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v (*coerce*)s s) _)) - (at level 0, only parsing) : form_scope. - -(* The following are "format only" versions of the above notations. Since Coq *) -(* doesn't provide this facility, we fake it by splitting the "the" keyword. *) -(* We need to do this to prevent the formatter from being be thrown off by *) -(* application collapsing, coercion insertion and beta reduction in the right *) -(* hand side of the notations above. *) - -Notation "[ 'th' 'e' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _) - (at level 0, format "[ 'th' 'e' sT 'of' v 'by' f ]") : form_scope. - -Notation "[ 'th' 'e' sT 'of' v ]" := (@get _ sT v _ _) - (at level 0, format "[ 'th' 'e' sT 'of' v ]") : form_scope. - -(* We would like to recognize -Notation "[ 'th' 'e' sT 'of' v : 'Type' ]" := (@get Type sT v _ _) - (at level 0, format "[ 'th' 'e' sT 'of' v : 'Type' ]") : form_scope. -*) - -(* Helper notation for canonical structure inheritance support. *) -(* This is a workaround for the poor interaction between delta reduction and *) -(* canonical projections in Coq's unification algorithm, by which transparent *) -(* definitions hide canonical instances, i.e., in *) -(* Canonical a_type_struct := @Struct a_type ... *) -(* Definition my_type := a_type. *) -(* my_type doesn't effectively inherit the struct structure from a_type. Our *) -(* solution is to redeclare the instance as follows *) -(* Canonical my_type_struct := Eval hnf in [struct of my_type]. *) -(* The special notation [str of _] must be defined for each Strucure "str" *) -(* with constructor "Str", typically as follows *) -(* Definition clone_str s := *) -(* let: Str _ x y ... z := s return {type of Str for s} -> str in *) -(* fun k => k _ x y ... z. *) -(* Notation "[ 'str' 'of' T 'for' s ]" := (@clone_str s (@Str T)) *) -(* (at level 0, format "[ 'str' 'of' T 'for' s ]") : form_scope. *) -(* Notation "[ 'str' 'of' T ]" := (repack_str (fun x => @Str T x)) *) -(* (at level 0, format "[ 'str' 'of' T ]") : form_scope. *) -(* The notation for the match return predicate is defined below; the eta *) -(* expansion in the second form serves both to distinguish it from the first *) -(* and to avoid the delta reduction problem. *) -(* There are several variations on the notation and the definition of the *) -(* the "clone" function, for telescopes, mixin classes, and join (multiple *) -(* inheritance) classes. We describe a different idiom for clones in ssrfun; *) -(* it uses phantom types (see below) and static unification; see fintype and *) -(* ssralg for examples. *) - -Definition argumentType T P & forall x : T, P x := T. -Definition dependentReturnType T P & forall x : T, P x := P. -Definition returnType aT rT & aT -> rT := rT. - -Notation "{ 'type' 'of' c 'for' s }" := (dependentReturnType c s) - (at level 0, format "{ 'type' 'of' c 'for' s }") : type_scope. - -(* A generic "phantom" type (actually, a unit type with a phantom parameter). *) -(* This type can be used for type definitions that require some Structure *) -(* on one of their parameters, to allow Coq to infer said structure so it *) -(* does not have to be supplied explicitly or via the "[the _ of _]" notation *) -(* (the latter interacts poorly with other Notation). *) -(* The definition of a (co)inductive type with a parameter p : p_type, that *) -(* needs to use the operations of a structure *) -(* Structure p_str : Type := p_Str {p_repr :> p_type; p_op : p_repr -> ...} *) -(* should be given as *) -(* Inductive indt_type (p : p_str) := Indt ... . *) -(* Definition indt_of (p : p_str) & phantom p_type p := indt_type p. *) -(* Notation "{ 'indt' p }" := (indt_of (Phantom p)). *) -(* Definition indt p x y ... z : {indt p} := @Indt p x y ... z. *) -(* Notation "[ 'indt' x y ... z ]" := (indt x y ... z). *) -(* That is, the concrete type and its constructor should be shadowed by *) -(* definitions that use a phantom argument to infer and display the true *) -(* value of p (in practice, the "indt" constructor often performs additional *) -(* functions, like "locking" the representation -- see below). *) -(* We also define a simpler version ("phant" / "Phant") of phantom for the *) -(* common case where p_type is Type. *) - -CoInductive phantom T (p : T) := Phantom. -Arguments phantom : clear implicits. -Arguments Phantom : clear implicits. -CoInductive phant (p : Type) := Phant. - -(* Internal tagging used by the implementation of the ssreflect elim. *) - -Definition protect_term (A : Type) (x : A) : A := x. - -(* The ssreflect idiom for a non-keyed pattern: *) -(* - unkeyed t wiil match any subterm that unifies with t, regardless of *) -(* whether it displays the same head symbol as t. *) -(* - unkeyed t a b will match any application of a term f unifying with t, *) -(* to two arguments unifying with with a and b, repectively, regardless of *) -(* apparent head symbols. *) -(* - unkeyed x where x is a variable will match any subterm with the same *) -(* type as x (when x would raise the 'indeterminate pattern' error). *) - -Notation unkeyed x := (let flex := x in flex). - -(* Ssreflect converse rewrite rule rule idiom. *) -Definition ssr_converse R (r : R) := (Logic.I, r). -Notation "=^~ r" := (ssr_converse r) (at level 100) : form_scope. - -(* Term tagging (user-level). *) -(* The ssreflect library uses four strengths of term tagging to restrict *) -(* convertibility during type checking: *) -(* nosimpl t simplifies to t EXCEPT in a definition; more precisely, given *) -(* Definition foo := nosimpl bar, foo (or foo t') will NOT be expanded by *) -(* the /= and //= switches unless it is in a forcing context (e.g., in *) -(* match foo t' with ... end, foo t' will be reduced if this allows the *) -(* match to be reduced). Note that nosimpl bar is simply notation for a *) -(* a term that beta-iota reduces to bar; hence rewrite /foo will replace *) -(* foo by bar, and rewrite -/foo will replace bar by foo. *) -(* CAVEAT: nosimpl should not be used inside a Section, because the end of *) -(* section "cooking" removes the iota redex. *) -(* locked t is provably equal to t, but is not convertible to t; 'locked' *) -(* provides support for selective rewriting, via the lock t : t = locked t *) -(* Lemma, and the ssreflect unlock tactic. *) -(* locked_with k t is equal but not convertible to t, much like locked t, *) -(* but supports explicit tagging with a value k : unit. This is used to *) -(* mitigate a flaw in the term comparison heuristic of the Coq kernel, *) -(* which treats all terms of the form locked t as equal and conpares their *) -(* arguments recursively, leading to an exponential blowup of comparison. *) -(* For this reason locked_with should be used rather than locked when *) -(* defining ADT operations. The unlock tactic does not support locked_with *) -(* but the unlock rewrite rule does, via the unlockable interface. *) -(* we also use Module Type ascription to create truly opaque constants, *) -(* because simple expansion of constants to reveal an unreducible term *) -(* doubles the time complexity of a negative comparison. Such opaque *) -(* constants can be expanded generically with the unlock rewrite rule. *) -(* See the definition of card and subset in fintype for examples of this. *) - -Notation nosimpl t := (let: tt := tt in t). - -Lemma master_key : unit. Proof. exact tt. Qed. -Definition locked A := let: tt := master_key in fun x : A => x. - -Lemma lock A x : x = locked x :> A. Proof. unlock; reflexivity. Qed. - -(* Needed for locked predicates, in particular for eqType's. *) -Lemma not_locked_false_eq_true : locked false <> true. -Proof. unlock; discriminate. Qed. - -(* The basic closing tactic "done". *) -Ltac done := - trivial; hnf; intros; solve - [ do ![solve [trivial | apply: sym_equal; trivial] - | discriminate | contradiction | split] - | case not_locked_false_eq_true; assumption - | match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. - -(* Quicker done tactic not including split, syntax: /0/ *) -Ltac ssrdone0 := - trivial; hnf; intros; solve - [ do ![solve [trivial | apply: sym_equal; trivial] - | discriminate | contradiction ] - | case not_locked_false_eq_true; assumption - | match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. - -(* To unlock opaque constants. *) -Structure unlockable T v := Unlockable {unlocked : T; _ : unlocked = v}. -Lemma unlock T x C : @unlocked T x C = x. Proof. by case: C. Qed. - -Notation "[ 'unlockable' 'of' C ]" := (@Unlockable _ _ C (unlock _)) - (at level 0, format "[ 'unlockable' 'of' C ]") : form_scope. - -Notation "[ 'unlockable' 'fun' C ]" := (@Unlockable _ (fun _ => _) C (unlock _)) - (at level 0, format "[ 'unlockable' 'fun' C ]") : form_scope. - -(* Generic keyed constant locking. *) - -(* The argument order ensures that k is always compared before T. *) -Definition locked_with k := let: tt := k in fun T x => x : T. - -(* This can be used as a cheap alternative to cloning the unlockable instance *) -(* below, but with caution as unkeyed matching can be expensive. *) -Lemma locked_withE T k x : unkeyed (locked_with k x) = x :> T. -Proof. by case: k. Qed. - -(* Intensionaly, this instance will not apply to locked u. *) -Canonical locked_with_unlockable T k x := - @Unlockable T x (locked_with k x) (locked_withE k x). - -(* More accurate variant of unlock, and safer alternative to locked_withE. *) -Lemma unlock_with T k x : unlocked (locked_with_unlockable k x) = x :> T. -Proof. exact: unlock. Qed. - -(* The internal lemmas for the have tactics. *) - -Definition ssr_have Plemma Pgoal (step : Plemma) rest : Pgoal := rest step. -Arguments ssr_have Plemma [Pgoal]. - -Definition ssr_have_let Pgoal Plemma step - (rest : let x : Plemma := step in Pgoal) : Pgoal := rest. -Arguments ssr_have_let [Pgoal]. - -Definition ssr_suff Plemma Pgoal step (rest : Plemma) : Pgoal := step rest. -Arguments ssr_suff Plemma [Pgoal]. - -Definition ssr_wlog := ssr_suff. -Arguments ssr_wlog Plemma [Pgoal]. - -(* Internal N-ary congruence lemmas for the congr tactic. *) - -Fixpoint nary_congruence_statement (n : nat) - : (forall B, (B -> B -> Prop) -> Prop) -> Prop := - match n with - | O => fun k => forall B, k B (fun x1 x2 : B => x1 = x2) - | S n' => - let k' A B e (f1 f2 : A -> B) := - forall x1 x2, x1 = x2 -> (e (f1 x1) (f2 x2) : Prop) in - fun k => forall A, nary_congruence_statement n' (fun B e => k _ (k' A B e)) - end. - -Lemma nary_congruence n (k := fun B e => forall y : B, (e y y : Prop)) : - nary_congruence_statement n k. -Proof. -have: k _ _ := _; rewrite {1}/k. -elim: n k => [|n IHn] k k_P /= A; first exact: k_P. -by apply: IHn => B e He; apply: k_P => f x1 x2 <-. -Qed. - -Lemma ssr_congr_arrow Plemma Pgoal : Plemma = Pgoal -> Plemma -> Pgoal. -Proof. by move->. Qed. -Arguments ssr_congr_arrow : clear implicits. - -(* View lemmas that don't use reflection. *) - -Section ApplyIff. - -Variables P Q : Prop. -Hypothesis eqPQ : P <-> Q. - -Lemma iffLR : P -> Q. Proof. by case: eqPQ. Qed. -Lemma iffRL : Q -> P. Proof. by case: eqPQ. Qed. - -Lemma iffLRn : ~P -> ~Q. Proof. by move=> nP tQ; case: nP; case: eqPQ tQ. Qed. -Lemma iffRLn : ~Q -> ~P. Proof. by move=> nQ tP; case: nQ; case: eqPQ tP. Qed. - -End ApplyIff. - -Hint View for move/ iffLRn|2 iffRLn|2 iffLR|2 iffRL|2. -Hint View for apply/ iffRLn|2 iffLRn|2 iffRL|2 iffLR|2. - -(* To focus non-ssreflect tactics on a subterm, eg vm_compute. *) -(* Usage: *) -(* elim/abstract_context: (pattern) => G defG. *) -(* vm_compute; rewrite {}defG {G}. *) -(* Note that vm_cast are not stored in the proof term *) -(* for reductions occuring in the context, hence *) -(* set here := pattern; vm_compute in (value of here) *) -(* blows up at Qed time. *) -Lemma abstract_context T (P : T -> Type) x : - (forall Q, Q = P -> Q x) -> P x. -Proof. by move=> /(_ P); apply. Qed. --- a/plugins/ssr/ssrbool.v +++ /dev/null @@ -1,1873 +0,0 @@ -(************************************************************************) -(* * 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. --- a/plugins/ssr/ssrfun.v +++ /dev/null @@ -1,796 +0,0 @@ -(************************************************************************) -(* * The Coq Proof Assistant / The Coq Development Team *) -(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) -(* -> pair coercion. *) -(* *) -(* - Simplifying functions, beta-reduced by /= and simpl: *) -(* [fun : T => E] == constant function from type T that returns E *) -(* [fun x => E] == unary function *) -(* [fun x : T => E] == unary function with explicit domain type *) -(* [fun x y => E] == binary function *) -(* [fun x y : T => E] == binary function with common domain type *) -(* [fun (x : T) y => E] \ *) -(* [fun (x : xT) (y : yT) => E] | == binary function with (some) explicit, *) -(* [fun x (y : T) => E] / independent domain types for each argument *) -(* *) -(* - Partial functions using option type: *) -(* oapp f d ox == if ox is Some x returns f x, d otherwise *) -(* odflt d ox == if ox is Some x returns x, d otherwise *) -(* obind f ox == if ox is Some x returns f x, None otherwise *) -(* omap f ox == if ox is Some x returns Some (f x), None otherwise *) -(* *) -(* - Singleton types: *) -(* all_equal_to x0 == x0 is the only value in its type, so any such value *) -(* can be rewritten to x0. *) -(* *) -(* - A generic wrapper type: *) -(* wrapped T == the inductive type with values Wrap x for x : T. *) -(* unwrap w == the projection of w : wrapped T on T. *) -(* wrap x == the canonical injection of x : T into wrapped T; it is *) -(* equivalent to Wrap x, but is declared as a (default) *) -(* Canonical Structure, which lets the Coq HO unification *) -(* automatically expand x into unwrap (wrap x). The delta *) -(* reduction of wrap x to Wrap can be exploited to *) -(* introduce controlled nondeterminism in Canonical *) -(* Structure inference, as in the implementation of *) -(* the mxdirect predicate in matrix.v. *) -(* *) -(* - Sigma types: *) -(* tag w == the i of w : {i : I & T i}. *) -(* tagged w == the T i component of w : {i : I & T i}. *) -(* Tagged T x == the {i : I & T i} with component x : T i. *) -(* tag2 w == the i of w : {i : I & T i & U i}. *) -(* tagged2 w == the T i component of w : {i : I & T i & U i}. *) -(* tagged2' w == the U i component of w : {i : I & T i & U i}. *) -(* Tagged2 T U x y == the {i : I & T i} with components x : T i and y : U i. *) -(* sval u == the x of u : {x : T | P x}. *) -(* s2val u == the x of u : {x : T | P x & Q x}. *) -(* The properties of sval u, s2val u are given by lemmas svalP, s2valP, and *) -(* s2valP'. We provide coercions sigT2 >-> sigT and sig2 >-> sig >-> sigT. *) -(* A suite of lemmas (all_sig, ...) let us skolemize sig, sig2, sigT, sigT2 *) -(* and pair, e.g., *) -(* have /all_sig[f fP] (x : T): {y : U | P y} by ... *) -(* yields an f : T -> U such that fP : forall x, P (f x). *) -(* - Identity functions: *) -(* id == NOTATION for the explicit identity function fun x => x. *) -(* @id T == notation for the explicit identity at type T. *) -(* idfun == an expression with a head constant, convertible to id; *) -(* idfun x simplifies to x. *) -(* @idfun T == the expression above, specialized to type T. *) -(* phant_id x y == the function type phantom _ x -> phantom _ y. *) -(* *** In addition to their casual use in functional programming, identity *) -(* functions are often used to trigger static unification as part of the *) -(* construction of dependent Records and Structures. For example, if we need *) -(* a structure sT over a type T, we take as arguments T, sT, and a "dummy" *) -(* function T -> sort sT: *) -(* Definition foo T sT & T -> sort sT := ... *) -(* We can avoid specifying sT directly by calling foo (@id T), or specify *) -(* the call completely while still ensuring the consistency of T and sT, by *) -(* calling @foo T sT idfun. The phant_id type allows us to extend this trick *) -(* to non-Type canonical projections. It also allows us to sidestep *) -(* dependent type constraints when building explicit records, e.g., given *) -(* Record r := R {x; y : T(x)}. *) -(* if we need to build an r from a given y0 while inferring some x0, such *) -(* that y0 : T(x0), we pose *) -(* Definition mk_r .. y .. (x := ...) y' & phant_id y y' := R x y'. *) -(* Calling @mk_r .. y0 .. id will cause Coq to use y' := y0, while checking *) -(* the dependent type constraint y0 : T(x0). *) -(* *) -(* - Extensional equality for functions and relations (i.e. functions of two *) -(* arguments): *) -(* f1 =1 f2 == f1 x is equal to f2 x for all x. *) -(* f1 =1 f2 :> A == ... and f2 is explicitly typed. *) -(* f1 =2 f2 == f1 x y is equal to f2 x y for all x y. *) -(* f1 =2 f2 :> A == ... and f2 is explicitly typed. *) -(* *) -(* - Composition for total and partial functions: *) -(* f^~ y == function f with second argument specialised to y, *) -(* i.e., fun x => f x y *) -(* CAVEAT: conditional (non-maximal) implicit arguments *) -(* of f are NOT inserted in this context *) -(* @^~ x == application at x, i.e., fun f => f x *) -(* [eta f] == the explicit eta-expansion of f, i.e., fun x => f x *) -(* CAVEAT: conditional (non-maximal) implicit arguments *) -(* of f are NOT inserted in this context. *) -(* fun=> v := the constant function fun _ => v. *) -(* f1 \o f2 == composition of f1 and f2. *) -(* Note: (f1 \o f2) x simplifies to f1 (f2 x). *) -(* f1 \; f2 == categorical composition of f1 and f2. This expands to *) -(* to f2 \o f1 and (f1 \; f2) x simplifies to f2 (f1 x). *) -(* pcomp f1 f2 == composition of partial functions f1 and f2. *) -(* *) -(* *) -(* - Properties of functions: *) -(* injective f <-> f is injective. *) -(* cancel f g <-> g is a left inverse of f / f is a right inverse of g. *) -(* pcancel f g <-> g is a left inverse of f where g is partial. *) -(* ocancel f g <-> g is a left inverse of f where f is partial. *) -(* bijective f <-> f is bijective (has a left and right inverse). *) -(* involutive f <-> f is involutive. *) -(* *) -(* - Properties for operations. *) -(* left_id e op <-> e is a left identity for op (e op x = x). *) -(* right_id e op <-> e is a right identity for op (x op e = x). *) -(* left_inverse e inv op <-> inv is a left inverse for op wrt identity e, *) -(* i.e., (inv x) op x = e. *) -(* right_inverse e inv op <-> inv is a right inverse for op wrt identity e *) -(* i.e., x op (i x) = e. *) -(* self_inverse e op <-> each x is its own op-inverse (x op x = e). *) -(* idempotent op <-> op is idempotent for op (x op x = x). *) -(* associative op <-> op is associative, i.e., *) -(* x op (y op z) = (x op y) op z. *) -(* commutative op <-> op is commutative (x op y = y op x). *) -(* left_commutative op <-> op is left commutative, i.e., *) -(* x op (y op z) = y op (x op z). *) -(* right_commutative op <-> op is right commutative, i.e., *) -(* (x op y) op z = (x op z) op y. *) -(* left_zero z op <-> z is a left zero for op (z op x = z). *) -(* right_zero z op <-> z is a right zero for op (x op z = z). *) -(* left_distributive op1 op2 <-> op1 distributes over op2 to the left: *) -(* (x op2 y) op1 z = (x op1 z) op2 (y op1 z). *) -(* right_distributive op1 op2 <-> op distributes over add to the right: *) -(* x op1 (y op2 z) = (x op1 z) op2 (x op1 z). *) -(* interchange op1 op2 <-> op1 and op2 satisfy an interchange law: *) -(* (x op2 y) op1 (z op2 t) = (x op1 z) op2 (y op1 t). *) -(* Note that interchange op op is a commutativity property. *) -(* left_injective op <-> op is injective in its left argument: *) -(* x op y = z op y -> x = z. *) -(* right_injective op <-> op is injective in its right argument: *) -(* x op y = x op z -> y = z. *) -(* left_loop inv op <-> op, inv obey the inverse loop left axiom: *) -(* (inv x) op (x op y) = y for all x, y, i.e., *) -(* op (inv x) is always a left inverse of op x *) -(* rev_left_loop inv op <-> op, inv obey the inverse loop reverse left *) -(* axiom: x op ((inv x) op y) = y, for all x, y. *) -(* right_loop inv op <-> op, inv obey the inverse loop right axiom: *) -(* (x op y) op (inv y) = x for all x, y. *) -(* rev_right_loop inv op <-> op, inv obey the inverse loop reverse right *) -(* axiom: (x op y) op (inv y) = x for all x, y. *) -(* Note that familiar "cancellation" identities like x + y - y = x or *) -(* x - y + y = x are respectively instances of right_loop and rev_right_loop *) -(* The corresponding lemmas will use the K and NK/VK suffixes, respectively. *) -(* *) -(* - Morphisms for functions and relations: *) -(* {morph f : x / a >-> r} <-> f is a morphism with respect to functions *) -(* (fun x => a) and (fun x => r); if r == R[x], *) -(* this states that f a = R[f x] for all x. *) -(* {morph f : x / a} <-> f is a morphism with respect to the *) -(* function expression (fun x => a). This is *) -(* shorthand for {morph f : x / a >-> a}; note *) -(* that the two instances of a are often *) -(* interpreted at different types. *) -(* {morph f : x y / a >-> r} <-> f is a morphism with respect to functions *) -(* (fun x y => a) and (fun x y => r). *) -(* {morph f : x y / a} <-> f is a morphism with respect to the *) -(* function expression (fun x y => a). *) -(* {homo f : x / a >-> r} <-> f is a homomorphism with respect to the *) -(* predicates (fun x => a) and (fun x => r); *) -(* if r == R[x], this states that a -> R[f x] *) -(* for all x. *) -(* {homo f : x / a} <-> f is a homomorphism with respect to the *) -(* predicate expression (fun x => a). *) -(* {homo f : x y / a >-> r} <-> f is a homomorphism with respect to the *) -(* relations (fun x y => a) and (fun x y => r). *) -(* {homo f : x y / a} <-> f is a homomorphism with respect to the *) -(* relation expression (fun x y => a). *) -(* {mono f : x / a >-> r} <-> f is monotone with respect to projectors *) -(* (fun x => a) and (fun x => r); if r == R[x], *) -(* this states that R[f x] = a for all x. *) -(* {mono f : x / a} <-> f is monotone with respect to the projector *) -(* expression (fun x => a). *) -(* {mono f : x y / a >-> r} <-> f is monotone with respect to relators *) -(* (fun x y => a) and (fun x y => r). *) -(* {mono f : x y / a} <-> f is monotone with respect to the relator *) -(* expression (fun x y => a). *) -(* *) -(* The file also contains some basic lemmas for the above concepts. *) -(* Lemmas relative to cancellation laws use some abbreviated suffixes: *) -(* K - a cancellation rule like esymK : cancel (@esym T x y) (@esym T y x). *) -(* LR - a lemma moving an operation from the left hand side of a relation to *) -(* the right hand side, like canLR: cancel g f -> x = g y -> f x = y. *) -(* RL - a lemma moving an operation from the right to the left, e.g., canRL. *) -(* Beware that the LR and RL orientations refer to an "apply" (back chaining) *) -(* usage; when using the same lemmas with "have" or "move" (forward chaining) *) -(* the directions will be reversed!. *) -(******************************************************************************) - -Set Implicit Arguments. -Unset Strict Implicit. -Unset Printing Implicit Defensive. - -Delimit Scope fun_scope with FUN. -Open Scope fun_scope. - -(* Notations for argument transpose *) -Notation "f ^~ y" := (fun x => f x y) - (at level 10, y at level 8, no associativity, format "f ^~ y") : fun_scope. -Notation "@^~ x" := (fun f => f x) - (at level 10, x at level 8, no associativity, format "@^~ x") : fun_scope. - -Delimit Scope pair_scope with PAIR. -Open Scope pair_scope. - -(* Notations for pair/conjunction projections *) -Notation "p .1" := (fst p) - (at level 2, left associativity, format "p .1") : pair_scope. -Notation "p .2" := (snd p) - (at level 2, left associativity, format "p .2") : pair_scope. - -Coercion pair_of_and P Q (PandQ : P /\ Q) := (proj1 PandQ, proj2 PandQ). - -Definition all_pair I T U (w : forall i : I, T i * U i) := - (fun i => (w i).1, fun i => (w i).2). - -(* Complements on the option type constructor, used below to *) -(* encode partial functions. *) - -Module Option. - -Definition apply aT rT (f : aT -> rT) x u := if u is Some y then f y else x. - -Definition default T := apply (fun x : T => x). - -Definition bind aT rT (f : aT -> option rT) := apply f None. - -Definition map aT rT (f : aT -> rT) := bind (fun x => Some (f x)). - -End Option. - -Notation oapp := Option.apply. -Notation odflt := Option.default. -Notation obind := Option.bind. -Notation omap := Option.map. -Notation some := (@Some _) (only parsing). - -(* Shorthand for some basic equality lemmas. *) - -Notation erefl := refl_equal. -Notation ecast i T e x := (let: erefl in _ = i := e return T in x). -Definition esym := sym_eq. -Definition nesym := sym_not_eq. -Definition etrans := trans_eq. -Definition congr1 := f_equal. -Definition congr2 := f_equal2. -(* Force at least one implicit when used as a view. *) -Prenex Implicits esym nesym. - -(* A predicate for singleton types. *) -Definition all_equal_to T (x0 : T) := forall x, unkeyed x = x0. - -Lemma unitE : all_equal_to tt. Proof. by case. Qed. - -(* A generic wrapper type *) - -Structure wrapped T := Wrap {unwrap : T}. -Canonical wrap T x := @Wrap T x. - -Prenex Implicits unwrap wrap Wrap. - -(* Syntax for defining auxiliary recursive function. *) -(* Usage: *) -(* Section FooDefinition. *) -(* Variables (g1 : T1) (g2 : T2). (globals) *) -(* Fixoint foo_auxiliary (a3 : T3) ... := *) -(* body, using [rec e3, ...] for recursive calls *) -(* where "[ 'rec' a3 , a4 , ... ]" := foo_auxiliary. *) -(* Definition foo x y .. := [rec e1, ...]. *) -(* + proofs about foo *) -(* End FooDefinition. *) - -Reserved Notation "[ 'rec' a0 ]" - (at level 0, format "[ 'rec' a0 ]"). -Reserved Notation "[ 'rec' a0 , a1 ]" - (at level 0, format "[ 'rec' a0 , a1 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 ]" - (at level 0, format "[ 'rec' a0 , a1 , a2 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 ]" - (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 ]" - (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]" - (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 ]" - (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]" - (at level 0, - format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]" - (at level 0, - format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]"). -Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]" - (at level 0, - format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]"). - -(* Definitions and notation for explicit functions with simplification, *) -(* i.e., which simpl and /= beta expand (this is complementary to nosimpl). *) - -Section SimplFun. - -Variables aT rT : Type. - -CoInductive simpl_fun := SimplFun of aT -> rT. - -Definition fun_of_simpl f := fun x => let: SimplFun lam := f in lam x. - -Coercion fun_of_simpl : simpl_fun >-> Funclass. - -End SimplFun. - -Notation "[ 'fun' : T => E ]" := (SimplFun (fun _ : T => E)) - (at level 0, - format "'[hv' [ 'fun' : T => '/ ' E ] ']'") : fun_scope. - -Notation "[ 'fun' x => E ]" := (SimplFun (fun x => E)) - (at level 0, x ident, - format "'[hv' [ 'fun' x => '/ ' E ] ']'") : fun_scope. - -Notation "[ 'fun' x : T => E ]" := (SimplFun (fun x : T => E)) - (at level 0, x ident, only parsing) : fun_scope. - -Notation "[ 'fun' x y => E ]" := (fun x => [fun y => E]) - (at level 0, x ident, y ident, - format "'[hv' [ 'fun' x y => '/ ' E ] ']'") : fun_scope. - -Notation "[ 'fun' x y : T => E ]" := (fun x : T => [fun y : T => E]) - (at level 0, x ident, y ident, only parsing) : fun_scope. - -Notation "[ 'fun' ( x : T ) y => E ]" := (fun x : T => [fun y => E]) - (at level 0, x ident, y ident, only parsing) : fun_scope. - -Notation "[ 'fun' x ( y : T ) => E ]" := (fun x => [fun y : T => E]) - (at level 0, x ident, y ident, only parsing) : fun_scope. - -Notation "[ 'fun' ( x : xT ) ( y : yT ) => E ]" := - (fun x : xT => [fun y : yT => E]) - (at level 0, x ident, y ident, only parsing) : fun_scope. - -(* For delta functions in eqtype.v. *) -Definition SimplFunDelta aT rT (f : aT -> aT -> rT) := [fun z => f z z]. - -(* Extensional equality, for unary and binary functions, including syntactic *) -(* sugar. *) - -Section ExtensionalEquality. - -Variables A B C : Type. - -Definition eqfun (f g : B -> A) : Prop := forall x, f x = g x. - -Definition eqrel (r s : C -> B -> A) : Prop := forall x y, r x y = s x y. - -Lemma frefl f : eqfun f f. Proof. by []. Qed. -Lemma fsym f g : eqfun f g -> eqfun g f. Proof. by move=> eq_fg x. Qed. - -Lemma ftrans f g h : eqfun f g -> eqfun g h -> eqfun f h. -Proof. by move=> eq_fg eq_gh x; rewrite eq_fg. Qed. - -Lemma rrefl r : eqrel r r. Proof. by []. Qed. - -End ExtensionalEquality. - -Typeclasses Opaque eqfun. -Typeclasses Opaque eqrel. - -Hint Resolve frefl rrefl. - -Notation "f1 =1 f2" := (eqfun f1 f2) - (at level 70, no associativity) : fun_scope. -Notation "f1 =1 f2 :> A" := (f1 =1 (f2 : A)) - (at level 70, f2 at next level, A at level 90) : fun_scope. -Notation "f1 =2 f2" := (eqrel f1 f2) - (at level 70, no associativity) : fun_scope. -Notation "f1 =2 f2 :> A" := (f1 =2 (f2 : A)) - (at level 70, f2 at next level, A at level 90) : fun_scope. - -Section Composition. - -Variables A B C : Type. - -Definition funcomp u (f : B -> A) (g : C -> B) x := let: tt := u in f (g x). -Definition catcomp u g f := funcomp u f g. -Local Notation comp := (funcomp tt). - -Definition pcomp (f : B -> option A) (g : C -> option B) x := obind f (g x). - -Lemma eq_comp f f' g g' : f =1 f' -> g =1 g' -> comp f g =1 comp f' g'. -Proof. by move=> eq_ff' eq_gg' x; rewrite /= eq_gg' eq_ff'. Qed. - -End Composition. - -Notation comp := (funcomp tt). -Notation "@ 'comp'" := (fun A B C => @funcomp A B C tt). -Notation "f1 \o f2" := (comp f1 f2) - (at level 50, format "f1 \o '/ ' f2") : fun_scope. -Notation "f1 \; f2" := (catcomp tt f1 f2) - (at level 60, right associativity, format "f1 \; '/ ' f2") : fun_scope. - -Notation "[ 'eta' f ]" := (fun x => f x) - (at level 0, format "[ 'eta' f ]") : fun_scope. - -Notation "'fun' => E" := (fun _ => E) (at level 200, only parsing) : fun_scope. - -Notation id := (fun x => x). -Notation "@ 'id' T" := (fun x : T => x) - (at level 10, T at level 8, only parsing) : fun_scope. - -Definition id_head T u x : T := let: tt := u in x. -Definition explicit_id_key := tt. -Notation idfun := (id_head tt). -Notation "@ 'idfun' T " := (@id_head T explicit_id_key) - (at level 10, T at level 8, format "@ 'idfun' T") : fun_scope. - -Definition phant_id T1 T2 v1 v2 := phantom T1 v1 -> phantom T2 v2. - -(* Strong sigma types. *) - -Section Tag. - -Variables (I : Type) (i : I) (T_ U_ : I -> Type). - -Definition tag := projT1. -Definition tagged : forall w, T_(tag w) := @projT2 I [eta T_]. -Definition Tagged x := @existT I [eta T_] i x. - -Definition tag2 (w : @sigT2 I T_ U_) := let: existT2 _ _ i _ _ := w in i. -Definition tagged2 w : T_(tag2 w) := let: existT2 _ _ _ x _ := w in x. -Definition tagged2' w : U_(tag2 w) := let: existT2 _ _ _ _ y := w in y. -Definition Tagged2 x y := @existT2 I [eta T_] [eta U_] i x y. - -End Tag. - -Arguments Tagged [I i]. -Arguments Tagged2 [I i]. -Prenex Implicits tag tagged Tagged tag2 tagged2 tagged2' Tagged2. - -Coercion tag_of_tag2 I T_ U_ (w : @sigT2 I T_ U_) := - Tagged (fun i => T_ i * U_ i)%type (tagged2 w, tagged2' w). - -Lemma all_tag I T U : - (forall x : I, {y : T x & U x y}) -> - {f : forall x, T x & forall x, U x (f x)}. -Proof. by move=> fP; exists (fun x => tag (fP x)) => x; case: (fP x). Qed. - -Lemma all_tag2 I T U V : - (forall i : I, {y : T i & U i y & V i y}) -> - {f : forall i, T i & forall i, U i (f i) & forall i, V i (f i)}. -Proof. by case/all_tag=> f /all_pair[]; exists f. Qed. - -(* Refinement types. *) - -(* Prenex Implicits and renaming. *) -Notation sval := (@proj1_sig _ _). -Notation "@ 'sval'" := (@proj1_sig) (at level 10, format "@ 'sval'"). - -Section Sig. - -Variables (T : Type) (P Q : T -> Prop). - -Lemma svalP (u : sig P) : P (sval u). Proof. by case: u. Qed. - -Definition s2val (u : sig2 P Q) := let: exist2 _ _ x _ _ := u in x. - -Lemma s2valP u : P (s2val u). Proof. by case: u. Qed. - -Lemma s2valP' u : Q (s2val u). Proof. by case: u. Qed. - -End Sig. - -Prenex Implicits svalP s2val s2valP s2valP'. - -Coercion tag_of_sig I P (u : @sig I P) := Tagged P (svalP u). - -Coercion sig_of_sig2 I P Q (u : @sig2 I P Q) := - exist (fun i => P i /\ Q i) (s2val u) (conj (s2valP u) (s2valP' u)). - -Lemma all_sig I T P : - (forall x : I, {y : T x | P x y}) -> - {f : forall x, T x | forall x, P x (f x)}. -Proof. by case/all_tag=> f; exists f. Qed. - -Lemma all_sig2 I T P Q : - (forall x : I, {y : T x | P x y & Q x y}) -> - {f : forall x, T x | forall x, P x (f x) & forall x, Q x (f x)}. -Proof. by case/all_sig=> f /all_pair[]; exists f. Qed. - -Section Morphism. - -Variables (aT rT sT : Type) (f : aT -> rT). - -(* Morphism property for unary and binary functions *) -Definition morphism_1 aF rF := forall x, f (aF x) = rF (f x). -Definition morphism_2 aOp rOp := forall x y, f (aOp x y) = rOp (f x) (f y). - -(* Homomorphism property for unary and binary relations *) -Definition homomorphism_1 (aP rP : _ -> Prop) := forall x, aP x -> rP (f x). -Definition homomorphism_2 (aR rR : _ -> _ -> Prop) := - forall x y, aR x y -> rR (f x) (f y). - -(* Stability property for unary and binary relations *) -Definition monomorphism_1 (aP rP : _ -> sT) := forall x, rP (f x) = aP x. -Definition monomorphism_2 (aR rR : _ -> _ -> sT) := - forall x y, rR (f x) (f y) = aR x y. - -End Morphism. - -Notation "{ 'morph' f : x / a >-> r }" := - (morphism_1 f (fun x => a) (fun x => r)) - (at level 0, f at level 99, x ident, - format "{ 'morph' f : x / a >-> r }") : type_scope. - -Notation "{ 'morph' f : x / a }" := - (morphism_1 f (fun x => a) (fun x => a)) - (at level 0, f at level 99, x ident, - format "{ 'morph' f : x / a }") : type_scope. - -Notation "{ 'morph' f : x y / a >-> r }" := - (morphism_2 f (fun x y => a) (fun x y => r)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'morph' f : x y / a >-> r }") : type_scope. - -Notation "{ 'morph' f : x y / a }" := - (morphism_2 f (fun x y => a) (fun x y => a)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'morph' f : x y / a }") : type_scope. - -Notation "{ 'homo' f : x / a >-> r }" := - (homomorphism_1 f (fun x => a) (fun x => r)) - (at level 0, f at level 99, x ident, - format "{ 'homo' f : x / a >-> r }") : type_scope. - -Notation "{ 'homo' f : x / a }" := - (homomorphism_1 f (fun x => a) (fun x => a)) - (at level 0, f at level 99, x ident, - format "{ 'homo' f : x / a }") : type_scope. - -Notation "{ 'homo' f : x y / a >-> r }" := - (homomorphism_2 f (fun x y => a) (fun x y => r)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'homo' f : x y / a >-> r }") : type_scope. - -Notation "{ 'homo' f : x y / a }" := - (homomorphism_2 f (fun x y => a) (fun x y => a)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'homo' f : x y / a }") : type_scope. - -Notation "{ 'homo' f : x y /~ a }" := - (homomorphism_2 f (fun y x => a) (fun x y => a)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'homo' f : x y /~ a }") : type_scope. - -Notation "{ 'mono' f : x / a >-> r }" := - (monomorphism_1 f (fun x => a) (fun x => r)) - (at level 0, f at level 99, x ident, - format "{ 'mono' f : x / a >-> r }") : type_scope. - -Notation "{ 'mono' f : x / a }" := - (monomorphism_1 f (fun x => a) (fun x => a)) - (at level 0, f at level 99, x ident, - format "{ 'mono' f : x / a }") : type_scope. - -Notation "{ 'mono' f : x y / a >-> r }" := - (monomorphism_2 f (fun x y => a) (fun x y => r)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'mono' f : x y / a >-> r }") : type_scope. - -Notation "{ 'mono' f : x y / a }" := - (monomorphism_2 f (fun x y => a) (fun x y => a)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'mono' f : x y / a }") : type_scope. - -Notation "{ 'mono' f : x y /~ a }" := - (monomorphism_2 f (fun y x => a) (fun x y => a)) - (at level 0, f at level 99, x ident, y ident, - format "{ 'mono' f : x y /~ a }") : type_scope. - -(* In an intuitionistic setting, we have two degrees of injectivity. The *) -(* weaker one gives only simplification, and the strong one provides a left *) -(* inverse (we show in `fintype' that they coincide for finite types). *) -(* We also define an intermediate version where the left inverse is only a *) -(* partial function. *) - -Section Injections. - -(* rT must come first so we can use @ to mitigate the Coq 1st order *) -(* unification bug (e..g., Coq can't infer rT from a "cancel" lemma). *) -Variables (rT aT : Type) (f : aT -> rT). - -Definition injective := forall x1 x2, f x1 = f x2 -> x1 = x2. - -Definition cancel g := forall x, g (f x) = x. - -Definition pcancel g := forall x, g (f x) = Some x. - -Definition ocancel (g : aT -> option rT) h := forall x, oapp h x (g x) = x. - -Lemma can_pcan g : cancel g -> pcancel (fun y => Some (g y)). -Proof. by move=> fK x; congr (Some _). Qed. - -Lemma pcan_inj g : pcancel g -> injective. -Proof. by move=> fK x y /(congr1 g); rewrite !fK => [[]]. Qed. - -Lemma can_inj g : cancel g -> injective. -Proof. by move/can_pcan; apply: pcan_inj. Qed. - -Lemma canLR g x y : cancel g -> x = f y -> g x = y. -Proof. by move=> fK ->. Qed. - -Lemma canRL g x y : cancel g -> f x = y -> x = g y. -Proof. by move=> fK <-. Qed. - -End Injections. - -Lemma Some_inj {T} : injective (@Some T). Proof. by move=> x y []. Qed. - -(* Force implicits to use as a view. *) -Prenex Implicits Some_inj. - -(* cancellation lemmas for dependent type casts. *) -Lemma esymK T x y : cancel (@esym T x y) (@esym T y x). -Proof. by case: y /. Qed. - -Lemma etrans_id T x y (eqxy : x = y :> T) : etrans (erefl x) eqxy = eqxy. -Proof. by case: y / eqxy. Qed. - -Section InjectionsTheory. - -Variables (A B C : Type) (f g : B -> A) (h : C -> B). - -Lemma inj_id : injective (@id A). -Proof. by []. Qed. - -Lemma inj_can_sym f' : cancel f f' -> injective f' -> cancel f' f. -Proof. by move=> fK injf' x; apply: injf'. Qed. - -Lemma inj_comp : injective f -> injective h -> injective (f \o h). -Proof. by move=> injf injh x y /injf; apply: injh. Qed. - -Lemma can_comp f' h' : cancel f f' -> cancel h h' -> cancel (f \o h) (h' \o f'). -Proof. by move=> fK hK x; rewrite /= fK hK. Qed. - -Lemma pcan_pcomp f' h' : - pcancel f f' -> pcancel h h' -> pcancel (f \o h) (pcomp h' f'). -Proof. by move=> fK hK x; rewrite /pcomp fK /= hK. Qed. - -Lemma eq_inj : injective f -> f =1 g -> injective g. -Proof. by move=> injf eqfg x y; rewrite -2!eqfg; apply: injf. Qed. - -Lemma eq_can f' g' : cancel f f' -> f =1 g -> f' =1 g' -> cancel g g'. -Proof. by move=> fK eqfg eqfg' x; rewrite -eqfg -eqfg'. Qed. - -Lemma inj_can_eq f' : cancel f f' -> injective f' -> cancel g f' -> f =1 g. -Proof. by move=> fK injf' gK x; apply: injf'; rewrite fK. Qed. - -End InjectionsTheory. - -Section Bijections. - -Variables (A B : Type) (f : B -> A). - -CoInductive bijective : Prop := Bijective g of cancel f g & cancel g f. - -Hypothesis bijf : bijective. - -Lemma bij_inj : injective f. -Proof. by case: bijf => g fK _; apply: can_inj fK. Qed. - -Lemma bij_can_sym f' : cancel f' f <-> cancel f f'. -Proof. -split=> fK; first exact: inj_can_sym fK bij_inj. -by case: bijf => h _ hK x; rewrite -[x]hK fK. -Qed. - -Lemma bij_can_eq f' f'' : cancel f f' -> cancel f f'' -> f' =1 f''. -Proof. -by move=> fK fK'; apply: (inj_can_eq _ bij_inj); apply/bij_can_sym. -Qed. - -End Bijections. - -Section BijectionsTheory. - -Variables (A B C : Type) (f : B -> A) (h : C -> B). - -Lemma eq_bij : bijective f -> forall g, f =1 g -> bijective g. -Proof. by case=> f' fK f'K g eqfg; exists f'; eapply eq_can; eauto. Qed. - -Lemma bij_comp : bijective f -> bijective h -> bijective (f \o h). -Proof. -by move=> [f' fK f'K] [h' hK h'K]; exists (h' \o f'); apply: can_comp; auto. -Qed. - -Lemma bij_can_bij : bijective f -> forall f', cancel f f' -> bijective f'. -Proof. by move=> bijf; exists f; first by apply/(bij_can_sym bijf). Qed. - -End BijectionsTheory. - -Section Involutions. - -Variables (A : Type) (f : A -> A). - -Definition involutive := cancel f f. - -Hypothesis Hf : involutive. - -Lemma inv_inj : injective f. Proof. exact: can_inj Hf. Qed. -Lemma inv_bij : bijective f. Proof. by exists f. Qed. - -End Involutions. - -Section OperationProperties. - -Variables S T R : Type. - -Section SopTisR. -Implicit Type op : S -> T -> R. -Definition left_inverse e inv op := forall x, op (inv x) x = e. -Definition right_inverse e inv op := forall x, op x (inv x) = e. -Definition left_injective op := forall x, injective (op^~ x). -Definition right_injective op := forall y, injective (op y). -End SopTisR. - - -Section SopTisS. -Implicit Type op : S -> T -> S. -Definition right_id e op := forall x, op x e = x. -Definition left_zero z op := forall x, op z x = z. -Definition right_commutative op := forall x y z, op (op x y) z = op (op x z) y. -Definition left_distributive op add := - forall x y z, op (add x y) z = add (op x z) (op y z). -Definition right_loop inv op := forall y, cancel (op^~ y) (op^~ (inv y)). -Definition rev_right_loop inv op := forall y, cancel (op^~ (inv y)) (op^~ y). -End SopTisS. - -Section SopTisT. -Implicit Type op : S -> T -> T. -Definition left_id e op := forall x, op e x = x. -Definition right_zero z op := forall x, op x z = z. -Definition left_commutative op := forall x y z, op x (op y z) = op y (op x z). -Definition right_distributive op add := - forall x y z, op x (add y z) = add (op x y) (op x z). -Definition left_loop inv op := forall x, cancel (op x) (op (inv x)). -Definition rev_left_loop inv op := forall x, cancel (op (inv x)) (op x). -End SopTisT. - -Section SopSisT. -Implicit Type op : S -> S -> T. -Definition self_inverse e op := forall x, op x x = e. -Definition commutative op := forall x y, op x y = op y x. -End SopSisT. - -Section SopSisS. -Implicit Type op : S -> S -> S. -Definition idempotent op := forall x, op x x = x. -Definition associative op := forall x y z, op x (op y z) = op (op x y) z. -Definition interchange op1 op2 := - forall x y z t, op1 (op2 x y) (op2 z t) = op2 (op1 x z) (op1 y t). -End SopSisS. - -End OperationProperties. - - - - - - - - - - --- a/test-suite/bugs/closed/2800.v +++ /dev/null @@ -1,19 +0,0 @@ -Goal False. - -intuition - match goal with - | |- _ => idtac " foo" - end. - - lazymatch goal with _ => idtac end. - match goal with _ => idtac end. - unshelve lazymatch goal with _ => idtac end. - unshelve match goal with _ => idtac end. - unshelve (let x := I in idtac). -Abort. - -Require Import ssreflect. - -Goal True. -match goal with _ => idtac end => //. -Qed. --- a/test-suite/bugs/closed/5692.v +++ /dev/null @@ -1,88 +0,0 @@ -Set Primitive Projections. -Require Import ZArith ssreflect. - -Module Test1. - -Structure semigroup := SemiGroup { - sg_car :> Type; - sg_op : sg_car -> sg_car -> sg_car; -}. - -Structure monoid := Monoid { - monoid_car :> Type; - monoid_op : monoid_car -> monoid_car -> monoid_car; - monoid_unit : monoid_car; -}. - -Coercion monoid_sg (X : monoid) : semigroup := - SemiGroup (monoid_car X) (monoid_op X). -Canonical Structure monoid_sg. - -Parameter X : monoid. -Parameter x y : X. - -Check (sg_op _ x y). - -End Test1. - -Module Test2. - -Structure semigroup := SemiGroup { - sg_car :> Type; - sg_op : sg_car -> sg_car -> sg_car; -}. - -Structure monoid := Monoid { - monoid_car :> Type; - monoid_op : monoid_car -> monoid_car -> monoid_car; - monoid_unit : monoid_car; - monoid_left_id x : monoid_op monoid_unit x = x; -}. - -Coercion monoid_sg (X : monoid) : semigroup := - SemiGroup (monoid_car X) (monoid_op X). -Canonical Structure monoid_sg. - -Canonical Structure nat_sg := SemiGroup nat plus. -Canonical Structure nat_monoid := Monoid nat plus 0 plus_O_n. - -Lemma foo (x : nat) : 0 + x = x. -Proof. -apply monoid_left_id. -Qed. - -End Test2. - -Module Test3. - -Structure semigroup := SemiGroup { - sg_car :> Type; - sg_op : sg_car -> sg_car -> sg_car; -}. - -Structure group := Something { - group_car :> Type; - group_op : group_car -> group_car -> group_car; - group_neg : group_car -> group_car; - group_neg_op' x y : group_neg (group_op x y) = group_op (group_neg x) (group_neg y) -}. - -Coercion group_sg (X : group) : semigroup := - SemiGroup (group_car X) (group_op X). -Canonical Structure group_sg. - -Axiom group_neg_op : forall (X : group) (x y : X), - group_neg X (sg_op (group_sg X) x y) = sg_op (group_sg X) (group_neg X x) (group_neg X y). - -Canonical Structure Z_sg := SemiGroup Z Z.add . -Canonical Structure Z_group := Something Z Z.add Z.opp Z.opp_add_distr. - -Lemma foo (x y : Z) : - sg_op Z_sg (group_neg Z_group x) (group_neg Z_group y) = - group_neg Z_group (sg_op Z_sg x y). -Proof. - rewrite -group_neg_op. - reflexivity. -Qed. - -End Test3. --- a/test-suite/bugs/closed/6634.v +++ /dev/null @@ -1,6 +0,0 @@ -From Coq Require Import ssreflect. - -Lemma normalizeP (p : tt = tt) : p = p. -Proof. -Fail move: {2} tt p. -Abort. --- a/test-suite/bugs/closed/6910.v +++ /dev/null @@ -1,5 +0,0 @@ -From Coq Require Import ssreflect ssrfun. - -(* We should be able to use Some_inj as a view: *) -Lemma foo (x y : nat) : Some x = Some y -> x = y. -Proof. by move/Some_inj. Qed. --- a/test-suite/output/ssr_clear.v +++ /dev/null @@ -1,6 +0,0 @@ -Require Import ssreflect. - -Example foo : True -> True. -Proof. -Fail move=> {NO_SUCH_NAME}. -Abort. --- a/test-suite/success/ssr_delayed_clear_rename.v +++ /dev/null @@ -1,5 +0,0 @@ -Require Import ssreflect. -Example foo (t t1 t2 : True) : True /\ True -> True -> True. -Proof. -move=>[{t1 t2 t} t1 t2] t. -Abort.