Library ssrbool

Require Import ssreflect.
Require Import ssrfun.
Require Export Bool.

 A theory of boolean predicates and operators. A large part of this file is 
 concerned with boolean reflection.                                         
 Definitions and notations:                                                 
               is_true b == the coercion of b : bool to Prop (:= b = true). 
                            This is just input and displayed as `b''.       
             reflect P b == the reflection inductive predicate, asserting   
                            that the logical proposition P : prop with the  
                            formula b : bool. Lemmas asserting reflect P b  
                            are often referred to as "views".               
  iffP, appP, sameP, rwP :: lemmas for direct manipulation of reflection    
                            views: iffP is used to prove reflection from    
                            logical equivalence, appP to compose views, and 
                            sameP and rwP to perform boolean and setoid     
                            rewriting.                                      
                   elimT :: coercion reflect >-> Funclass, which allows the 
                            direct application of `reflect' views to        
                            boolean assertions.                             
    contra, contraL, ... :: contraposition lemmas.                          
           altP my_viewP :: natural alternative for reflection; given       
                            lemma myvieP: 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                            been replaced by true and false, resp., with    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.   
           classically P == hP : P can be assumed when proving is_true b    
                         := forall b : bool, (P -> b) -> b.                 
                            This is equivalent to ~ (~ P) when P : Prop.    
                  a && b == the boolean conjunction of a and b.             
                  a || b == then 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.                             
 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 mixed. Explicit values of    
 pred T (i.e., lamdba terms) should always be used applicatively, while     
 values of collection types implementing the predType interface, such as    
 lists or sets should always be used as collective predicates; simpl_pred   
 predicates are the only type that can be used either way (however, the     
 x \in A notation will not simplify). We provide the following conversions  
             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 => T, with a cast on the argument.  
          [pred : T | P] == constant predicate P on type T.                 
          [pred x \in A] == [pred x | x \in A].                             
      [pred x \in A | E] == [pred x | (x \in A) && E].                      
           [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.                     
 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.      
 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 introduces the following suffix policy for lemma names:          
   A -- associativity, as in andbA : associative andb.                      
   C -- commutativity, as in andbC : commutative andb,                      
        or predicate complement, as in predC.                               
   D -- predicate difference, as in predD.                                  
   E -- elimination, as in negbEf : ~~ b = false -> b.                      
   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.                             
   K -- cancellation, as in negbK : involutive negb.                        
   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).                                   
   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.         


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, no associativity).
Reserved Notation "x \notin A" (at level 70, no associativity).
Reserved Notation "p1 =i p2" (at level 70, no associativity).

 We introduce a number of n-ary "list-style" notations, which 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; for similar reasons the separator cannot be omitted even     
 when last_arg can. The Notation declarations are complicated by the  
 separate treatments for fixed arities (binary for bool operators,    
 and all arities for Prop operators).                                 
   We also use the square brackets in comprehension-style notations   
 of the form                                                          
    [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 by an operator symbol or at least 
 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.

 The Coq library forgets to set argument scopes on bool ops. 


 An alternative to xorb that behaves somewhat better wrt simplification. 

Definition addb b := if b then negb else fun b' => b'.

 Bool operator notation; we need to redeclare && and || so they get the 
 correct argument scopes.                                               

Notation "~~ b" := (negb b) : bool_scope.
 Redundant for now; may be added if dependency on Bool is removed
Notation "b1 && b2" := (andb b1 b2) : bool_scope.
Notation "b1 || b2" := (orb b1 b2) : bool_scope.

Notation "b ==> c" := (implb b c) : bool_scope.
Notation "b1 (+) b2" := (addb b1 b2) : bool_scope.

 Coercion bool >-> Prop.                    

Coercion is_true b := b = true.


Ltac fold_prop := match goal with |- (?b = true) => change (is_true b) end.

Lemma prop_congr : forall b b' : bool, b = b' -> b = b' :> Prop.

Ltac prop_congr := apply: prop_congr.

 Lemmas for auto. 
 Shorter names. 
Definition isT := is_true_true.
Definition notF := not_false_is_true.

 Negation lemmas. 

 Note: in the general we take NEGATION as the standard form of a 
 false condition : hypotheses should be of the form ~~ b rather  
 than b = false or ~ b, as much as possible.                     

Lemma negbT : forall b, b = false -> ~~ b.

Lemma negbTE : forall b, ~~ b -> b = false.

Lemma negbF : forall b : bool, b -> ~~ b = false.

Lemma negbFE : forall b, ~~ b = false -> b.

Lemma negbK : involutive negb.

Lemma negbNE : forall b, ~~ ~~ b -> b.

Lemma negb_inj : injective negb.

Lemma negbLR : forall b c, b = ~~ c -> ~~ b = c.

Lemma negbRL : forall b c, ~~ b = c -> b = ~~ c.

Lemma contra : forall c b : bool, (c -> b) -> ~~ b -> ~~ c.
Definition contraNN := contra.

Lemma contraL : forall c b : bool, (c -> ~~ b) -> b -> ~~ c.
Definition contraTN := contraL.

Lemma contraR : forall c b : bool, (~~ c -> b) -> ~~ b -> c.
Definition contraNT := contraR.

Lemma contraLR : forall c b : bool, (~~ c -> ~~ b) -> b -> c.
Definition contraTT := contraLR.

Lemma contraT : forall b, (~~ b -> false) -> b.

Lemma wlog_neg : forall b, (~~ b -> b) -> b.

Lemma contraFT : forall c b : bool, (~~ c -> b) -> b = false -> c.

Lemma contraFN : forall c b : bool, (c -> b) -> b = false -> ~~ c.

Lemma contraTF : forall c b : bool, (c -> ~~ b) -> b -> c = false.

Lemma contraNF : forall c b : bool, (c -> b) -> ~~ b -> c = false.

Lemma contraFF : forall c b : bool, (c -> b) -> b = false -> c = false.

 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.


 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).

Lemma ifPn : if_spec (~~ b) b (if b then vT else vF).

Lemma if_same : (if b then vT else vT) = vT.

Lemma if_neg : (if ~~ b then vT else vF) = if b then vF else vT.

Lemma fun_if : f (if b then vT else vF) = if b then f vT else f vF.

Lemma if_arg : forall fT fF : A -> B,
  (if b then fT else fF) x = if b then fT x else fF x.

 Patch for a bug in ssreflect 8.1 that corrupts patterns where a 
 wildcard appears under a match. Usage:                          
   rewrite -ifE; set x := if_expr _ _ _.                         

Definition if_expr := if b then vT else vF.
Lemma ifE : (if b then vT else vF) = if_expr.

End BoolIf.

 The reflection predicate.                                          

Inductive reflect (P : Prop) : bool -> Set :=
  | ReflectT of P : reflect P true
  | ReflectF of ~ P : reflect P false.

 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.

Lemma introTF : (if c then P else ~ P) -> b = c.

Lemma elimNTF : ~~ b = c -> if c then ~ P else P.

Lemma elimTF : b = c -> if c then P else ~ P.

Lemma equivPif : (Q -> P) -> (P -> Q) -> if b then Q else ~ Q.

Lemma xorPif : Q \/ P -> ~ (Q /\ P) -> if b then ~ Q else Q.

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.

Lemma elimTFn : b = c -> if c then ~ P else P.

Lemma equivPifn : (Q -> P) -> (P -> Q) -> if b then ~ Q else Q.

Lemma xorPifn : Q \/ P -> ~ (Q /\ P) -> if b then Q else ~ Q.

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.

Lemma introF : ~ P -> b = false.

Lemma introN : ~ P -> ~~ b.

Lemma introNf : P -> ~~ b = false.

Lemma introTn : ~ P -> b'.

Lemma introFn : P -> b' = false.

Lemma elimT : b -> P.

Lemma elimF : b = false -> ~ P.

Lemma elimN : ~~ b -> ~P.

Lemma elimNf : ~~ b = false -> P.

Lemma elimTn : b' -> ~ P.

Lemma elimFn : b' = false -> P.

Lemma introP : (b -> Q) -> (~~ b -> ~ Q) -> reflect Q b.

Lemma iffP : (P -> Q) -> (Q -> P) -> reflect Q b.

Lemma appP : reflect Q b -> P -> Q.

Lemma sameP : reflect P c -> b = c.

Lemma decPcases : if b then P else ~ P.

Definition decP : {P} + {~ P}.

Lemma rwP : P <-> b.

Lemma rwP2 : reflect Q b -> (P <-> Q).

  Predicate family to reflect excluded middle in bool.
    This is the natural definition, but unfortunately it is unusable because
    matching for dependent type families in Coq is broken -- it tries to match
    indices in the prefix of the elimination predicate for the type as it is
    constructing it, which results in the wrong prefix on instances where one
    of the indices appear multible times, as in the boolP lemma below.
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.

Section BooleanAlternative.

 Workaround for the op-cited problem: split the Proposition and the boolean 
 formula into a predicate and argument, so that the index matching does not 
 capture (part of) the parameters. However, we can't do this in a generic   
 proposition, as usually the proposition and predicate have different       
 syntactic structure (e.g., while in eqP both = and == have the same final  
 argument, in andP if the last argument of && is b, the last argument of /\ 
 is (is_true b)), so the generic exmP theorem is restricted to propositions 
 in which the formula does not appear. We handle the identity case below    
 specially, using a Phantom to split the formula in Lemma orbNP.            
   Caveat: this kludge cannot possibly be made to work with atomic formulae 
 such as bool variables.                                                    
Variables (T : Type) (bP : T -> bool) (a : T).

CoInductive alt_spec (P : T -> Prop) : bool -> Type :=
  | AltTrue of P a : alt_spec P true
  | AltFalse of ~~ bP a : alt_spec P false.

Lemma altP : forall P, reflect P (bP a) -> alt_spec (fun _ : T => P) (bP a).

  This will become the official version when the dependent prefix capture
    problem gets fixed (see above)
Lemma boolP : exm_spec b1 b1 b1.
Proof. by case: b1; constructor. Qed.


Lemma boolP_proof : phantom bool (bP a) -> alt_spec [eta bP] (bP a).

End BooleanAlternative.

Notation boolP b := (boolP_proof (@Phantom bool b%B)).

 Classical reasoning becomes directly accessible for any bool subgoal.      
Definition classically P := forall b : bool, (P -> b) -> b.

Lemma classicP : forall P : Prop, classically P <-> ~ ~ P.

Lemma classic_bind : forall P Q,
  (P -> classically Q) -> (classically P -> classically Q).

Lemma classic_EM : forall P, classically ({P} + {~ P}).

Lemma classic_imply : forall P Q, (P -> classically Q) -> classically (P -> Q).

Lemma classic_pick : forall T P,
  classically ({x : T | P x} + (forall x, ~ P x)).

 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 ReflectConnectives.

Variable b1 b2 b3 b4 b5 : bool.

Lemma idP : reflect b1 b1.

Lemma idPn : reflect (~~ b1) (~~ b1).

Lemma negP : reflect (~ b1) (~~ b1).

Lemma negPn : reflect b1 (~~ ~~ b1).

Lemma negPf : reflect (b1 = false) (~~ b1).

Lemma andP : reflect (b1 /\ b2) (b1 && b2).

Lemma and3P : reflect [/\ b1, b2 & b3] [&& b1, b2 & b3].

Lemma and4P : reflect [/\ b1, b2, b3 & b4] [&& b1, b2, b3 & b4].

Lemma and5P : reflect [/\ b1, b2, b3, b4 & b5] [&& b1, b2, b3, b4 & b5].

Lemma orP : reflect (b1 \/ b2) (b1 || b2).

Lemma or3P : reflect [\/ b1, b2 | b3] [|| b1, b2 | b3].

Lemma or4P : reflect [\/ b1, b2, b3 | b4] [|| b1, b2, b3 | b4].

Lemma nandP : reflect (~~ b1 \/ ~~ b2) (~~ (b1 && b2)).

Lemma norP : reflect (~~ b1 /\ ~~ b2) (~~ (b1 || b2)).

Lemma implyP : reflect (b1 -> b2) (b1 ==> b2).

End ReflectConnectives.

Implicit Arguments idP [b1].
Implicit Arguments idPn [b1].
Implicit Arguments negP [b1].
Implicit Arguments negPn [b1].
Implicit Arguments negPf [b1].
Implicit Arguments andP [b1 b2].
Implicit Arguments and3P [b1 b2 b3].
Implicit Arguments and4P [b1 b2 b3 b4].
Implicit Arguments and5P [b1 b2 b3 b4 b5].
Implicit Arguments orP [b1 b2].
Implicit Arguments or3P [b1 b2 b3].
Implicit Arguments or4P [b1 b2 b3 b4].
Implicit Arguments nandP [b1 b2].
Implicit Arguments norP [b1 b2].
Implicit Arguments implyP [b1 b2].

 Shorter, more systematic names for the boolean connectives laws.       

Lemma andTb : left_id true andb.

Lemma andFb : left_zero false andb.

Lemma andbT : right_id true andb.

Lemma andbF : right_zero false andb.

Lemma andbb : idempotent andb.

Lemma andbC : commutative andb.

Lemma andbA : associative andb.

Lemma andbCA : left_commutative andb.

Lemma andbAC : right_commutative andb.

Lemma orTb : forall b, true || b.

Lemma orFb : left_id false orb.

Lemma orbT : forall b, b || true.

Lemma orbF : right_id false orb.

Lemma orbb : idempotent orb.

Lemma orbC : commutative orb.

Lemma orbA : associative orb.

Lemma orbCA : left_commutative orb.

Lemma orbAC : right_commutative orb.

Lemma andbN : forall b, b && ~~ b = false.

Lemma andNb : forall b, ~~ b && b = false.

Lemma orbN : forall b, b || ~~ b = true.

Lemma orNb : forall b, ~~ b || b = true.

Lemma andb_orl : left_distributive andb orb.

Lemma andb_orr : right_distributive andb orb.

Lemma orb_andl : left_distributive orb andb.

Lemma orb_andr : right_distributive orb andb.

Lemma andb_idl : forall a b : bool, (b -> a) -> a && b = b.
Lemma andb_idr : forall a b : bool, (a -> b) -> a && b = a.
Lemma andb_id2l : forall a b c : bool, (a -> b = c) -> a && b = a && c.
Lemma andb_id2r : forall a b c : bool, (b -> a = c) -> a && b = c && b.

Lemma orb_idl : forall a b : bool, (a -> b) -> a || b = b.
Lemma orbb_idr : forall a b : bool, (b -> a) -> a || b = a.
Lemma orb_id2l : forall a b c : bool, (~~ a -> b = c) -> a || b = a || c.
Lemma orb_id2r : forall a b c : bool, (~~ b -> a = c) -> a || b = c || b.

Lemma negb_and : forall b1 b2, ~~ (b1 && b2) = ~~ b1 || ~~ b2.

Lemma negb_or : forall b1 b2, ~~ (b1 || b2) = ~~ b1 && ~~ b2.

 Pseudo-cancellation -- i.e, absorbtion 

Lemma andbK : forall b1 b2, b1 && b2 || b1 = b1.

Lemma andKb : forall b1 b2, b1 || b2 && b1 = b1.

Lemma orbK : forall b1 b2, (b1 || b2) && b1 = b1.

Lemma orKb : forall b1 b2, b1 && (b2 || b1) = b1.

 Imply 

Lemma implybT : forall b, b ==> true.

Lemma implybF : forall b, (b ==> false) = ~~ b.

Lemma implyFb : forall b, false ==> b.

Lemma implyTb : forall b, (true ==> b) = b.

Lemma implybb : forall b, b ==> b.

Lemma negb_imply : forall b1 b2, ~~ (b1 ==> b2) = b1 && ~~ b2.

Lemma implybE : forall b1 b2, (b1 ==> b2) = ~~ b1 || b2.

Lemma implyNb : forall b1 b2, (~~ b1 ==> b2) = b1 || b2.

Lemma implybN : forall b1 b2, (b1 ==> ~~ b2) = (b2 ==> ~~ b1).

Lemma implybNN : forall b1 b2, (~~ b1 ==> ~~ b2) = b2 ==> b1.

Lemma implyb_idl : forall a b : bool, (~~ a -> b) -> (a ==> b) = b.
Lemma implyb_idr : forall a b : bool, (b -> ~~ a) -> (a ==> b) = ~~ a.
Lemma implyb_id2l : forall a b c : bool, (a -> b = c) -> (a ==> b) = (a ==> c).

 addition (xor) 

Lemma addFb : left_id false addb.

Lemma addbF : right_id false addb.

Lemma addbb : self_inverse false addb.

Lemma addbC : commutative addb.

Lemma addbA : associative addb.

Lemma addbCA : left_commutative addb.

Lemma addbAC : right_commutative addb.

Lemma andb_addl : left_distributive andb addb.

Lemma andb_addr : right_distributive andb addb.

Lemma addKb : left_loop id addb.

Lemma addbK : right_loop id addb.

Lemma addIb : left_injective addb.

Lemma addbI : right_injective addb.

Lemma addTb : forall b, true (+) b = ~~ b.

Lemma addbT : forall b, b (+) true = ~~ b.

Lemma addbN : forall b1 b2, b1 (+) ~~ b2 = ~~ (b1 (+) b2).
Lemma addNb : forall b1 b2, ~~ b1 (+) b2 = ~~ (b1 (+) b2).

Lemma addbP : forall b1 b2, b1 (+) b2 -> ~~ b1 = b2.

 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 specialize them  
 to some value 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 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 
 can't 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.v).          

   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    
 allows us to turn some specific Types (namely, any aT_:_predArgType)     
 into predicates, specifically, 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 just 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 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)             
 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.                            

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 SimplPred (p : pred T) : simpl_pred := SimplFun p.

Coercion pred_of_simpl (p : simpl_pred) : pred T := p : T -> bool.

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 : forall r1 r2, subrel r1 (relU r1 r2).

Lemma subrelUr : forall r1 r2, subrel r2 (relU r1 r2).

CoInductive mem_pred : Type := Mem of pred T.

Definition isMem pT topred mem := mem = (fun p : pT => Mem [eta topred p]).

Structure predType : Type := PredType {
  pred_sort :> Type;
  topred : pred_sort -> pred T;
  _ : {mem | isMem topred mem}
}.

Definition mkPredType pT toP := PredType (exist (@isMem pT toP) _ (erefl _)).

Canonical Structure predPredType := Eval hnf in @mkPredType (pred T) id.
Canonical Structure simplPredType := Eval hnf in mkPredType pred_of_simpl.

Coercion pred_of_mem mp : pred_sort predPredType :=
  let: Mem p := mp in [eta p].

Canonical Structure 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.

Implicit Arguments pred0 [T].
Implicit Arguments predT [T].

Notation "[ 'pred' : T | E ]" := (SimplPred (fun _ : T => E))
  (at level 0, format "[ 'pred' : T | E ]") : fun_scope.
Notation "[ 'pred' x | E ]" := (SimplPred (fun x => E))
  (at level 0, x ident, format "[ 'pred' x | E ]") : fun_scope.
Notation "[ 'pred' x : T | E ]" := (SimplPred (fun x : T => E))
  (at level 0, x ident, only parsing) : fun_scope.
Notation "[ 'rel' x y | E ]" := (SimplRel (fun x y => E))
  (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))
  (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 
 structure 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 structure should always resolve.                         

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.
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.


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.

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 "[ '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).

Lemma mem_topred : forall (p : pT), mem (topred p) = mem p.

Lemma topredE : forall x (p : pT), topred p x = (x \in p).

Lemma in_simpl : forall x (p : simpl_pred T), (x \in p) = p x.

Lemma simpl_predE : forall (p : pred T), [pred x | p x] =1 p.

Definition inE := (in_simpl, simpl_predE).
Lemma mem_simpl : forall (p : simpl_pred T), mem p = p :> pred T.

Definition memE := mem_simpl.
Lemma mem_mem : forall p : pT, (mem (mem p) = mem p) * (mem [mem p] = mem p).

End simpl_mem.

Section RelationProperties.

 Caveat: reflexive should not be used to state lemmas, since 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.

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.

End RelationProperties.

Lemma rev_trans : forall T (R : rel T),
  transitive R -> transitive (fun x y => R y x).

 Property localization 

Notation Local "{ 'all1' P }" := (forall x, P x : Prop) (at level 0).
Notation Local "{ 'all2' P }" := (forall x y, P x y : Prop) (at level 0).
Notation Local "{ 'all3' P }" := (forall x y z, P x y z: Prop) (at level 0).
Notation Local ph := (phantom _).

Section LocalProperties.

Variables T1 T2 T3 : Type.

Variables (d1 : mem_pred T1) (d2 : mem_pred T2) (d3 : mem_pred T3).
Notation Local ph := (phantom Prop).

Definition prop_for (x : T1) P & ph {all1 P} := P x.

Lemma forE : forall x P phP, @prop_for x P phP = P x.

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.

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     
 cause the expansion of the predicate definition, as Coq     
 needs to expose the quantifier to apply these lemmas. We    
 define some specialized variants to avoid this for some of  
 the ssrfun definitions.                                     

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}}.
Lemma in2W : {all2 P2} -> {in D1 & D2, {all2 P2}}.
Lemma in3W : {all3 P3} -> {in D1 & D2 & D3, {all3 P3}}.

Lemma in1T : {in T1, {all1 P1}} -> {all1 P1}.
Lemma in2T : {in T1 & T2, {all2 P2}} -> {all2 P2}.
Lemma in3T : {in T1 & T2 & T3, {all3 P3}} -> {all3 P3}.

Lemma sub_in1 : forall Ph : ph {all1 P1},
  prop_in1 d1' Ph -> prop_in1 d1 Ph.

Lemma sub_in11 : forall Ph : ph {all2 P2},
  prop_in11 d1' d2' Ph -> prop_in11 d1 d2 Ph.

Lemma sub_in111 : forall Ph : ph {all3 P3},
  prop_in111 d1' d2' d3' Ph -> prop_in111 d1 d2 d3 Ph.


Lemma on1W : allQ1 f -> {on D2, allQ1 f}.

Lemma on1lW : allQ1l f h -> {on D2, allQ1l f & h}.

Lemma on2W : allQ2 f -> {on D2 &, allQ2 f}.

Lemma on1T : {on T2, allQ1 f} -> allQ1 f.

Lemma on1lT : {on T2, allQ1l f & h} -> allQ1l f h.

Lemma on2T : {on T2 &, allQ2 f} -> allQ2 f.

Lemma subon1 : forall (Phf : ph (allQ1 f)) (Ph : ph (allQ1 f)),
  prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph.

Lemma subon1l : forall (Phf : ph (allQ1l f)) (Ph : ph (allQ1l f h)),
  prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph.

Lemma subon2 : forall (Phf : ph (allQ2 f)) (Ph : ph (allQ2 f)),
  prop_on2 d2' Phf Ph -> prop_on2 d2 Phf Ph.

Lemma can_in_inj : {in D1, cancel f g} -> {in D1 &, injective f}.

Lemma canLR_in : forall x y,
  {in D1, cancel f g} -> y \in D1 -> x = f y -> g x = y.

Lemma canRL_in : forall x y,
  {in D1, cancel f g} -> x \in D1 -> f x = y -> x = g y.

Lemma on_can_inj : {on D2, cancel f & g} -> {on D2 &, injective f}.

Lemma canLR_on : forall x y,
  {on D2, cancel f & g} -> f y \in D2 -> x = f y -> g x = y.

Lemma canRL_on : forall x y,
  {on D2, cancel f & g} -> f x \in D2 -> f x = y -> x = g y.

Lemma inW_bij : bijective f -> {in D1, bijective f}.

Lemma onW_bij : bijective f -> {on D2, bijective f}.

Lemma inT_bij : {in T1, bijective f} -> bijective f.

Lemma onT_bij : {on T2, bijective f} -> bijective f.

Lemma sub_in_bij : forall D1' : pred T1,
  {subset D1 <= D1'} -> {in D1', bijective f} -> {in D1, bijective f}.

Lemma subon_bij : forall D2' : pred T2,
 {subset D2 <= D2'} -> {on D2', bijective f} -> {on D2, bijective f}.

End LocalGlobal.

Lemma sub_in2 : forall T d d' (P : T -> T -> Prop),
  sub_mem d d' -> forall Ph : ph {all2 P}, prop_in2 d' Ph -> prop_in2 d Ph.

Lemma sub_in3 : forall 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.

Lemma sub_in12 : forall 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.

Lemma sub_in21 : forall 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.