# Library ssreflect

(c) Copyright Microsoft Corporation and Inria. All rights reserved.

For bool_scope delimiter 'bool'.

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. The contents of this file is quite technical and should only interest advanced developers; features not covered by the Ssreflect reference manual, such as the Unlockable interface, phantom types, and the [the struct of T] construct, are covered by specific comments below.

Declare Ssr keywords: 'is' 'by' '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 camlp4 grammar will not factor properly.

Reserved Notation "(** x 'is' y 'by' z 'of' // /= //= *)" (at level 8).

Reserved Notation "(** 69 *)" (at level 69).

End SsrSyntax.

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 won't 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 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.

Open Scope form_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).

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 one of the 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 : struct_type to the inferred type of proj_val. Obviously the latter can be fixed by using an implicit 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). Special note for telescopes (structures with coercion-based inheritance): when using the "the" form to generate a substructure from a canonical hierarchy, on 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) : Type := 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.

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 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 option, we fake it by splitting the "the" keyword. We need to do this because the formatter will be thrown off by application collapsing, coercion insertion and beta reduction in the right hand sides of the above notations.

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 structures, i.e., in Canonical Structure 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 structure, as follows Canonical Structure 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 repack_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 ]" := (@repack_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 "repack" function, for telescopes, mixin classes, and join (multiple inheritance) classes; see fintype.v and ssralg.v for examples; they involve inferring the structure from instances of reflexivity or from phantoms (see below), rather than directly from the constructor as above.

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, the unit type with a phantom parameter). This can be used for type definitions that require some Structure on one of their parameters, to allow Coq to infer said structure rather that having to supply it explicitly or to resort to the "[is _ <: _]" notation, which interacts poorly with Notation. The definition of a (co)inductive type with a parameter p : p_type, that uses 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) : Type := 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") for the common case where p_type is Type.

CoInductive phantom (T : Type) (p : T) : Type := Phantom.

Implicit Arguments phantom [].

CoInductive phant (p : Type) : Type := Phant.

Internal tagging used by the implementation of the ssreflect elim.

We provide two strengths of term tagging : - (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 simpl tactic unless it is in a forcing context (e.g., in match foo t' with ... end, foo t' will be reduced if this allows match to be reduced. Note that nosimpl bar is simply notation for a term that beta-iota reduces to bar; hence unfold foo will replace foo by bar, and fold foo will replace bar by foo. A final warning: nosimpl only works if no reduction is apparent in t; in particular Definition foo x := nosimpl t. will usually not work. 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; it provides support for abstract data types, and selective rewriting. The equation t == (locked t) is provided as a lemma, but it should only be used for selective rewriting (see below). For ADTs, the unlock tactic should be used to remove locking. locked is also used as a placeholder for the implementation of flexible matching. Addendum: it appears that the use of a generic key confuses the term comparison heuristic of the Coq kernel, which thinks all locked terms have the same "head constant", and therefore immediately jumps to comparing their LAST argument. Furthermore, Coq still needs to delta expand a locked constant when comparing unequal terms, and, given the total absence of caching of comparisons, this causes an exponential blowup in comparisons that return false on terms that are built from many combinators, which is quite common in a modular development. As a stopgap, we use the module system to create opaque constants with an expansion lemma; this is pretty clumsy because design of the module language does not support such small-scale usage very well. See the definiiton of card and subset in fintype.v for examples of this. Of course the unlock tactic will not support the expansion of this new kind of opaque constants; to compensate for that we use "unlockable" canonical structures to store the expansion lemmas, which can then be retrieved by a generic "unlock" rewrite rule. Note that because of a technical limitation of the implementation of canonical projection in ssreflect 1.1, unlock must weaken the intensional equality between the constant and its definition to an extensional one.

To unlock opaque constants.

Structure unlockable (T : Type) (v : T) : Type :=

Unlockable {unlocked : T; _ : unlocked = v}.

Lemma unlock : forall aT rT (f : forall x : aT, rT x) (u : unlockable f) x,

unlocked u x = f x.

Unlockable {unlocked : T; _ : unlocked = v}.

Lemma unlock : forall aT rT (f : forall x : aT, rT x) (u : unlockable f) x,

unlocked u x = f x.

Definition locked_with key A := let: tt := key in fun x : A => x. Lemma unlock : forall key A, @locked_with key A = (fun x => x). Proof. case; split. Qed.

This should be Definition locked := locked_with master_key, but for compatibility with the ml4 code.

Definition locked A := let: tt := master_key in fun x : A => x.

Lemma lock : forall A x, x = locked x :> A.

Lemma lock : forall A x, x = locked x :> A.

Needed for locked predicates, in particular for eqType's.

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 ].

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 ].

The internal lemmas for the have tactics.

Definition ssr_have Plemma Pgoal (step : Plemma) rest : Pgoal := rest step.

Implicit Arguments ssr_have [Pgoal].

Definition ssr_suff Plemma Pgoal step (rest : Plemma) : Pgoal := step rest.

Implicit Arguments ssr_suff [Pgoal].

Definition ssr_wlog := ssr_suff.

Implicit Arguments ssr_wlog [Pgoal].

Internal N-ary congruence lemma 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 : forall n : nat,

let k B e := forall y : B, (e y y : Prop) in nary_congruence_statement n k.

View lemmas that don't use reflection.

Section ApplyIff.

Variables P Q : Prop.

Hypothesis eqPQ : P <-> Q.

Lemma iffLR : P -> Q. Qed.

Lemma iffRL : Q -> P. Qed.

Lemma iffLRn : ~P -> ~Q. Qed.

Lemma iffRLn : ~Q -> ~P. 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.