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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
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(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *)
(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **)
Declare ML Module "rocq-runtime.plugins.ssreflect".
(** This file is the Gallina part of the ssreflect plugin implementation. Files that use the ssreflect plugin should always Require ssreflect and either Import ssreflect or Import ssreflect.SsrSyntax. Part of the contents of this file is technical and will only interest advanced developers; in addition the following are defined: #[#the str of v by f#]# == the Canonical s : str such that f s = v. #[#the str of v#]# == the Canonical s : str that coerces to v. argumentType c == the T such that c : forall x : T, P x. returnType c == the R such that c : T -> R. {type of c for s} == P s where c : forall x : T, P x. nonPropType == an interface for non-Prop Types: a nonPropType coerces to a Type, and only types that do _not_ have sort Prop are canonical nonPropType instances. This is useful for applied views (see mid-file comment). notProp T == the nonPropType instance for type T. 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.  #[#elaborate x#]# == triggers Coq elaboration to fill the holes of the term x The main use case is to trigger typeclass inference in the body of a ssreflect have := #[#elaborate body#]#.  Additionally we provide default intro pattern ltac views: - top of the stack actions: => /#[#apply#]#     := => hyp {}/hyp => /#[#swap#]#      := => x y; move: y x (also swap and preserves let bindings) => /#[#dup#]#       := => x; have copy := x; move: copy x (also copies and preserves let bindings) - calling rewrite from an intro pattern, use with parsimony: => /#[#1! rules#]#  := rewrite rules => /#[#! rules#]#   := rewrite !rules  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' // /= //= *)".
(**  Non ambiguous keyword to check if the SsrSyntax module is imported  **)
Reserved Notation "(* Use to test if 'SsrSyntax_is_Imported' *)".
Reserved Notation "<hidden n >" (at level 0, n at level 0,
format "<hidden  n >").
#[warning="-postfix-notation-not-level-1"]
Reserved Notation "T (* n *)" (at level 200, format "T  (* n *)").
End SsrSyntax.
Export SsrMatchingSyntax.
Export SsrSyntax.
(** Save primitive notation that will be overloaded. **)
Local Notation CoqGenericIf c vT vF := (if c then vT else vF) (only parsing).
Local Notation CoqGenericDependentIf c x R vT vF :=
(if c as x return R then vT else vF) (only parsing).
(** Reserve notation that introduced in this file. **)
Reserved Notation "'if' c 'then' vT 'else' vF" (at level 200,
c, vT, vF at level 200).
Reserved Notation "'if' c 'return' R 'then' vT 'else' vF" (at level 200,
c, R, vT, vF at level 200).
Reserved Notation "'if' c 'as' x 'return' R 'then' vT 'else' vF" (at level 200,
c, R, vT, vF at level 200, x name).
Reserved Notation "[ 'the' sT 'of' v 'by' f ]" (at level 0,
format "[ 'the'  sT  'of'  v  'by'  f ]").
Reserved Notation "[ 'the' sT 'of' v ]" (at level 0,
format "[ 'the'  sT  'of'  v ]").
Reserved Notation "{ 'type' 'of' c 'for' s }" (at level 0,
format "{ 'type'  'of'  c  'for'  s }").
Reserved Notation "=^~ r" (at level 100, format "=^~  r").
Reserved Notation "[ 'unlockable' 'of' C ]" (at level 0,
format "[ 'unlockable'  'of'  C ]").
Reserved Notation "[ 'unlockable' 'fun' C ]" (at level 0,
format "[ 'unlockable'  'fun'  C ]").
Reserved Notation "[ 'elaborate' x ]" (at level 0).
(** To define notations for tactic in intro patterns. When "=> /t" is parsed, "t:%ssripat" is actually interpreted. **)
Declare Scope ssripat_scope.
Delimit Scope ssripat_scope with ssripat.
(** 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 spurious trailing %%GEN_IF. **)
Declare Scope general_if_scope.
Delimit Scope general_if_scope with GEN_IF.
Notation "'if' c 'then' vT 'else' vF" :=
(CoqGenericIf c vT vF) (only parsing) : general_if_scope.
Notation "'if' c 'return' R 'then' vT 'else' vF" :=
(CoqGenericDependentIf c c R vT vF) (only parsing) : general_if_scope.
Notation "'if' c 'as' x 'return' R 'then' vT 'else' vF" :=
(CoqGenericDependentIf c x R vT vF) (only parsing) : general_if_scope.
(**  Force boolean interpretation of simple if expressions.  **)
Declare Scope boolean_if_scope.
Delimit Scope boolean_if_scope with BOOL_IF.
Notation "'if' c 'return' R 'then' vT 'else' vF" :=
(if c is true as c in bool return R then vT else vF) : boolean_if_scope.
Notation "'if' c 'then' vT 'else' vF" :=
(if c%bool is true as _ in bool return _ then vT else vF) : boolean_if_scope.
Notation "'if' c 'as' x 'return' R 'then' vT 'else' vF" :=
(if c%bool is true as x in bool return R then vT else vF) : 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.                                            **)
Declare Scope form_scope.
Delimit Scope form_scope with FORM.
Open Scope form_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.
Declare Scope ssr_scope.
Notation "<hidden n >" := (abstract _ n _) : ssr_scope.
Notation "T (* n *)" := (abstract T n abstract_key) : ssr_scope.
Open Scope ssr_scope.
Register abstract_lock as plugins.ssreflect.abstract_lock.
Register abstract_key as plugins.ssreflect.abstract_key.
Register abstract as plugins.ssreflect.abstract.
(**  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.
Variant put vT sT (v1 v2 : vT) (s : sT) : Prop := 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.
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 _))
(only parsing) : form_scope.
Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v (*coerce*) s s) _))
(only parsing) : form_scope.
(** The following are "format only" versions of the above notations. 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 "[ 'the' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _)
(only printing) : form_scope.
Notation "[ 'the' sT 'of' v ]" := (@get _ sT v _ _)
(only printing) : form_scope.
(** We would like to recognize Notation " #[# 'the' sT 'of' v : 'Type' #]#" := (@get Type sT v _ _) (at level 0, format " #[# 'the'  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 Structure "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) : 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.                                           **)
Variant phantom T (p : T) : Prop := Phantom.
Arguments phantom : clear implicits.
Arguments Phantom : clear implicits.
Variant phant (p : Type) : Prop := Phant.
(**  Internal tagging used by the implementation of the ssreflect elim.  **)
Definition protect_term (A : Type) (x : A) : A := x.
Register protect_term as plugins.ssreflect.protect_term.
(** The ssreflect idiom for a non-keyed pattern: - unkeyed t will 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 a and b, respectively, 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) : 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 compares 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).

(* Specifications: *)
Lemma master_key : unit. Proof. exact tt. Qed.
Definition locked A := let: tt := master_key in fun x : A => x.