Library ssreflect

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

Module SsrSyntax.

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.

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  

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. 

Definition protect_term (A : Type) (x : A) : A := x.

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  
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.                   

Notation "'nosimpl' t" := (let: tt := tt in t) (at level 10, t at level 8).

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.

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.

Lemma master_key : unit. 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.

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 ].

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))

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.