Module mathcomp.order.order

From HB Require Import structures.
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq.
From mathcomp Require Import path fintype tuple bigop finset div prime finfun.
From mathcomp Require Import finset.
From mathcomp Require Export preorder.

                  Types equipped with order relations

NB: See CONTRIBUTING.md for an introduction to HB concepts and commands.

This files defines types equipped with order relations.

* How to use orders in MathComp?
Use one of the following modules implementing different theories (all
located in the module Order):
  Order.LTheory: partially ordered types and lattices excluding complement
                 and totality related theorems
  Order.CTheory: complemented lattices including Order.LTheory
  Order.TTheory: totally ordered types including Order.LTheory
   Order.Theory: ordered types including all of the above theory modules
To access the definitions, notations, and the theory from, say,
"Order.Xyz", insert "Import Order.Xyz." at the top of your scripts. You can
also "Import Order.Def." to enjoy shorter notations (e.g., min instead of
Order.min, nondecreasing instead of Order.nondecreasing, etc.).

In order to reason about abstract orders, notations are accessible by
opening the scope "order_scope" bound to the delimiting key "O"; however,
when dealing with another notation scope providing order notations for
a concrete instance (e.g., "ring_scope"), it is not recommended to open
"order_scope" at the same time.

* Control of inference (parsing) and printing
One characteristic of ordered types is that one carrier type may have
several orders. For example, natural numbers can be totally or partially
ordered by the less than or equal relation, the divisibility relation, and
their dual relations. Therefore, we need a way to control inference of
ordered type instances and printing of generic relations and operations on
ordered types. As a rule of thumb, we use the carrier type or its "alias"
(named copy) to control inference (using canonical structures), and use a
"display" to control the printing of notations.

Each generic interface and operation for ordered types has, as its first
argument, a "display" of type Order.disp_t. For example, the less than or
equal relation has type:
  Order.le : forall {d : Order.disp_t} {T : porderType d}, rel T,
where porderType d is the structure of partially ordered types with
display d. (@Order.le dvd_display _ m n) is printed as m %| n because
ordered type instances associated to the display dvd_display is intended
to represent natural numbers partially ordered by the divisibility
relation.

We stress that order structure inference can be triggered only from the
carrier type (or its alias), but not the display. For example, writing
m %| n for m and n of type nat does not trigger an inference of the
divisibility relation on natural numbers, which is associated to an alias
natdvd for nat; such an inference should be triggered through the use of
the corresponding alias, i.e., (m : natdvd) %| n. In other words, displays
are merely used to inform the user and the notation mechanism of what the
inference did; they are not additional input for the inference.

See below for various aliases and their associated displays.

NB: algebra/ssrnum.v provides the display ring_display to change the
scope of the usual notations to ring_scope.

Instantiating d with Disp tt tt or an unknown display will lead to a
default display for notations.

Alternative notation displays can be defined by :
1. declaring a new opaque definition of type unit. Using the idiom
   `Fact my_display : Order.disp_t. Proof. exact: Disp tt tt. Qed.`
2. using this symbol to tag canonical porderType structures using
   `HB.instance Definition _ := isPOrder.Build my_display my_type ...`,
3. declaring notations for the main operations of this library, by
   setting the first argument of the definition to the display, e.g.
   `Notation my_syndef_le x y := @Order.le my_display _ x y.` or
   `Notation "x <=< y" := @Order.lt my_display _ x y (at level ...).`
   Non overloaded notations will default to the default display.
We suggest the user to refer to the example of natdvd below as a guideline
example to add their own displays.

* Interfaces
We provide the following interfaces for types equipped with an order:

             porderType d == the type of partially ordered types
                             The HB class is called POrder.
            bPOrderType d == porderType with a bottom element (\bot)
                             The HB class is called BPOrder.
            tPOrderType d == porderType with a top element (\top)
                             The HB class is called TPOrder.
           tbPOrderType d == porderType with both a top and a bottom
                             The HB class is called TBPOrder.
    meetSemilatticeType d == the type of meet semilattices
                             The HB class is called MeetSemilattice.
   bMeetSemilatticeType d == meetSemilatticeType with a bottom element
                             The HB class is called BMeetSemilattice.
   tMeetSemilatticeType d == meetSemilatticeType with a top element
                             The HB class is called TMeetSemilattice.
  tbMeetSemilatticeType d == meetSemilatticeType with both a top and a
                             bottom
                             The HB class is called TBMeetSemilattice.
    joinSemilatticeType d == the type of join semilattices
                             The HB class is called JoinSemilattice.
   bJoinSemilatticeType d == joinSemilatticeType with a bottom element
                             The HB class is called BJoinSemilattice.
   tJoinSemilatticeType d == joinSemilatticeType with a top element
                             The HB class is called TJoinSemilattice.
  tbJoinSemilatticeType d == joinSemilatticeType with both a top and a
                             bottom
                             The HB class is called TBJoinSemilattice.
            latticeType d == the type of lattices
                             The HB class is called Lattice.
           bLatticeType d == latticeType with a bottom element
                             The HB class is called BLattice.
           tLatticeType d == latticeType with a top element
                             The HB class is called TLattice.
          tbLatticeType d == latticeType with both a top and a bottom
                             The HB class is called TBLattice.
       distrLatticeType d == the type of distributive lattices
                             The HB class is called DistrLattice.
      bDistrLatticeType d == distrLatticeType with a bottom element
                             The HB class is called BDistrLattice.
      tDistrLatticeType d == distrLatticeType with a top element
                             The HB class is called TDistrLattice.
     tbDistrLatticeType d == distrLatticeType with both a top and a bottom
                             The HB class is called TBDistrLattice.
              orderType d == the type of totally ordered types
                             The HB class is called Total.
             bOrderType d == orderType with a bottom element
                             The HB class is called BTotal.
             tOrderType d == orderType with a top element
                             The HB class is called TTotal.
            tbOrderType d == orderType with both a top and a bottom
                             The HB class is called TBTotal.
      cDistrLatticeType d == the type of relatively complemented
                             distributive lattices, where each interval
                             [a, b] is equipped with a complement operation
                             The HB class is called CDistrLattice.
     cbDistrLatticeType d == the type of sectionally complemented
                             distributive lattices, equipped with a bottom,
                             a relative complement operation, and a
                             difference operation, i.e., a complement
                             operation for each interval of the form
                             [\bot, b]
                             The HB class is called CBDistrLattice.
     ctDistrLatticeType d == the type of dually sectionally complemented
                             distributive lattices, equipped with a top,
                             a relative complement operation, and a
                             dual difference operation, i.e. a complement
                             operation for each interval of the form
                             [a, \top]
                             The HB class is called CTDistrLattice.
    ctbDistrLatticeType d == the type of complemented distributive
                             lattices, equipped with top, bottom,
                             difference, dual difference, and complement
                             The HB class is called CTBDistrLattice.
          finPOrderType d == the type of partially ordered finite types
                             The HB class is called FinPOrder.
         finBPOrderType d == finPOrderType with a bottom element
                             The HB class is called FinBPOrder.
         finTPOrderType d == finPOrderType with a top element
                             The HB class is called FinTPOrder.
        finTBPOrderType d == finPOrderType with both a top and a bottom
                             The HB class is called FinTBPOrder.
 finMeetSemilatticeType d == the type of finite meet semilattice types
                             The HB class is called FinMeetSemilattice.
finBMeetSemilatticeType d == finMeetSemilatticeType with a bottom element
                             Note that finTMeetSemilatticeType is just
                             finTBLatticeType.
                             The HB class is called FinBMeetSemilattice.
 finJoinSemilatticeType d == the type of finite join semilattice types
                             The HB class is called FinJoinSemilattice.
finTJoinSemilatticeType d == finJoinSemilatticeType with a top element
                             Note that finBJoinSemilatticeType is just
                             finTBLatticeType.
                             The HB class is called FinTJoinSemilattice.
         finLatticeType d == the type of finite lattices
                             The HB class is called FinLattice.
       finTBLatticeType d == the type of nonempty finite lattices
                             The HB class is called FinTBLattice.
    finDistrLatticeType d == the type of finite distributive lattices
                             The HB class is called FinDistrLattice.
  finTBDistrLatticeType d == the type of nonempty finite distributive
                             lattices
                             The HB class is called FinTBDistrLattice.
           finOrderType d == the type of totally ordered finite types
                             The HB class is called FinTotal.
         finTBOrderType d == the type of nonempty totally ordered finite
                             types
                             The HB class is called FinTBTotal.
   finCDistrLatticeType d == the type of finite relatively complemented
                             distributive lattices
                             The HB class is called FinCDistrLattice.
 finCTBDistrLatticeType d == the type of finite complemented distributive
                             lattices
                             The HB class is called FinCTBDistrLattice.

and their joins with subType:

         subPOrder d T P d' == join of porderType d' and subType
                               (P : pred T) such that val is monotonic
                               The HB class is called SubPOrder.
    meetSubLattice d T P d' == join of latticeType d' and subType
                               (P : pred T) such that val is monotonic and
                               a morphism for meet
                               The HB class is called MeetSubLattice.
    joinSubLattice d T P d' == join of latticeType d' and subType
                               (P : pred T) such that val is monotonic and
                               a morphism for join
                               The HB class is called JoinSubLattice.
        subLattice d T P d' == join of JoinSubLattice and MeetSubLattice
                               The HB class is called SubLattice.
   bJoinSubLattice d T P d' == join of JoinSubLattice and BLattice
                               such that val is a morphism for \bot
                               The HB class is called BJoinSubLattice.
   tMeetSubLattice d T P d' == join of MeetSubLattice and TLattice
                               such that val is a morphism for \top
                               The HB class is called TMeetSubLattice.
       bSubLattice d T P d' == join of SubLattice and BLattice
                               such that val is a morphism for \bot
                               The HB class is called BSubLattice.
       tSubLattice d T P d' == join of SubLattice and TLattice
                               such that val is a morphism for \top
                               The HB class is called BSubLattice.
          subOrder d T P d' == join of orderType d' and
                               subLatticeType d T P d'
                               The HB class is called SubOrder.
  subPOrderLattice d T P d' == join of SubPOrder and Lattice
                               The HB class is called SubPOrderLattice.
 subPOrderBLattice d T P d' == join of SubPOrder and BLattice
                               The HB class is called SubPOrderBLattice.
 subPOrderTLattice d T P d' == join of SubPOrder and TLattice
                               The HB class is called SubPOrderTLattice.
subPOrderTBLattice d T P d' == join of SubPOrder and TBLattice
                               The HB class is called SubPOrderTBLattice.
   meetSubBLattice d T P d' == join of MeetSubLattice and BLattice
                               The HB class is called MeetSubBLattice.
   meetSubTLattice d T P d' == join of MeetSubLattice and TLattice
                               The HB class is called MeetSubTLattice.
  meetSubTBLattice d T P d' == join of MeetSubLattice and TBLattice
                               The HB class is called MeetSubTBLattice.
   joinSubBLattice d T P d' == join of JoinSubLattice and BLattice
                               The HB class is called JoinSubBLattice.
   joinSubTLattice d T P d' == join of JoinSubLattice and TLattice
                               The HB class is called JoinSubTLattice.
  joinSubTBLattice d T P d' == join of JoinSubLattice and TBLattice
                               The HB class is called JoinSubTBLattice.
       subBLattice d T P d' == join of SubLattice and BLattice
                               The HB class is called SubBLattice.
       subTLattice d T P d' == join of SubLattice and TLattice
                               The HB class is called SubTLattice.
      subTBLattice d T P d' == join of SubLattice and TBLattice
                               The HB class is called SubTBLattice.
  bJoinSubTLattice d T P d' == join of BJoinSubLattice and TBLattice
                               The HB class is called BJoinSubTLattice.
  tMeetSubBLattice d T P d' == join of TMeetSubLattice and TBLattice
                               The HB class is called TMeetSubBLattice.
      bSubTLattice d T P d' == join of BSubLattice and TBLattice
                               The HB class is called BSubTLattice.
      tSubBLattice d T P d' == join of TSubLattice and TBLattice
                               The HB class is called TSubBLattice.
     tbSubBLattice d T P d' == join of BSubLattice and TSubLattice
                               The HB class is called TBSubLattice.

Morphisms between the above structures:

MeetLatticeMorphism.type d T d' T',
JoinLatticeMorphism.type d T d' T',
LatticeMorphism.type d T d' T' == nondecreasing function between two
                          lattices which are morphism for meet, join, and
                          meet/join respectively
BLatticeMorphism.type d T d' T' := {blmorphism T -> T'},
TLatticeMorphism.type d T d' T' := {tlmorphism T -> T'},
TBLatticeMorphism.type d T d' T' := {tblmorphism T -> T'}
                       == nondecreasing function between two lattices with
                          bottom/top which are morphism for bottom/top

Closedness predicates for the algebraic structures:

 meetLatticeClosed d T == predicate closed under meet on T : latticeType d
                          The HB class is MeetLatticeClosed.
 joinLatticeClosed d T == predicate closed under join on T : latticeType d
                          The HB class is JoinLatticeClosed.
     latticeClosed d T == predicate closed under meet and join
                          The HB class is JoinLatticeClosed.
    bLatticeClosed d T == predicate that contains bottom
                          The HB class is BLatticeClosed.
    tLatticeClosed d T == predicate that contains top
                          The HB class is TLatticeClosed.
   tbLatticeClosed d T == predicate that contains top and bottom
                          the HB class ie TBLatticeClosed.
bJoinLatticeClosed d T == predicate that contains bottom and is closed
                          under join
                          The HB class is BJoinLatticeClosed.
tMeetLatticeClosed d T == predicate that contains top and is closed under
                          meet
                          The HB class is TMeetLatticeClosed.

* Useful lemmas:
On orderType, leP, ltP, and ltgtP are the three main lemmas for case
analysis.
On porderType, one may use comparableP, comparable_leP, comparable_ltP,
and comparable_ltgtP, which are the four main lemmas for case analysis.

* Order relations and operations:
In general, an overloaded relation or operation on ordered types takes the
following arguments:
1. a display d of type Order.disp_t,
2. an instance T of the minimal structure it operates on, and
3. operands.
Here is the exhaustive list of all such operations together with their
default notation (defined in order_scope unless specified otherwise).

For T of type meetSemilatticeType d, and x, y of type T:
        x `&` y :=  @Order.meet d T x y
                ==  the meet of x and y
For T of type joinSemilatticeType d, and x, y of type T:
        x `|` y :=  @Order.join d T x y
                ==  the join of x and y

For T of type tMeetSemilatticeType d:
\meet_<range> e :=  \big[Order.meet / Order.top]_<range> e
                ==  iterated meet of a meet-semilattice with a top
                    the possible <range>s are documented in bigop.v
For T of type bJoinSemilatticeType d:
\join_<range> e :=  \big[Order.join / Order.bottom]_<range> e
                ==  iterated join of a join-semilattice with a bottom
                    the possible <range>s are documented in bigop.v

For T of type cDistrLatticeType d, and x, y, z of type T:
   rcompl x y z == the (relative) complement of z in [x, y]

For T of type cbDistrLatticeType d, and x, y of type T:
        x `\` y := @Order.diff d T x y
                == the (sectional) complement of y in [\bot, x],
                   i.e., rcompl \bot x y

For T of type ctDistrLatticeType d, and x, y of type T:
     codiff x y == the (dual sectional) complement of y in [x, \top],
                   i.e., rcompl x \top y

For T of type ctbDistrLatticeType d, and x of type T:
           ~` x := @Order.compl d T x
                == the complement of x in [\bot, \top],
                   i.e., rcompl \bot \top x

We provide aliases for various types and their displays:
           natdvd := nat (associated with display dvd_display)
                  == an alias for nat which is canonically ordered using
                     divisibility predicate dvdn
                     Notation %|, %<|, gcd, lcm are used instead of
                     <=, <, meet and join.
             T^d  := dual T,
                     where dual is a new definition for (fun T => T)
                     (associated with dual_display d where d is a display)
                  == an alias for T, such that if T is canonically
                     ordered, then T^d is canonically ordered with the
                     dual order, and displayed with an extra ^d in the
                     notation, i.e.,  <=^d, <^d, >=<^d, ><^d, `&`^d, `|`^d
                     are used and displayed instead of
                     <=, <, >=<, ><, `&`, `|`
    T *prod[d] T' := T * T'
                  == an alias for the cartesian product such that,
                     if T and T' are canonically ordered,
                     then T *prod[d] T' is canonically ordered in product
                     order, i.e.,
                     (x1, x2) <= (y1, y2) = (x1 <= y1) && (x2 <= y2),
                     and displayed in display d
          T *p T' := T *prod[prod_display d d'] T'
                     where d and d' are the displays of T and T',
                     respectively, and prod_display adds an extra ^p to
                     all notations
    T *lexi[d] T' := T * T'
                  == an alias for the cartesian product such that,
                     if T and T' are canonically ordered,
                     then T *lexi[d] T' is canonically ordered in
                     lexicographic order,
                     i.e., (x1, x2) <= (y1, y2) =
                             (x1 <= y1) && ((x1 >= y1) ==> (x2 <= y2))
                     and   (x1, x2) < (y1, y2) =
                             (x1 <= y1) && ((x1 >= y1) ==> (x2 < y2))
                     and displayed in display d
          T *l T' := T *lexi[lexi_display d d'] T'
                     where d and d' are the displays of T and T',
                     respectively, and lexi_display adds an extra ^l to
                     all notations
 seqprod_with d T := seq T
                  == an alias for seq, such that if T is canonically
                     ordered, then seqprod_with d T is canonically ordered
                     in product order, i.e.,
                     [:: x1, .., xn] <= [y1, .., yn] =
                       (x1 <= y1) && ... && (xn <= yn)
                     and displayed in display d
n.-tupleprod[d] T == same with n.tuple T
        seqprod T := seqprod_with (seqprod_display d) T
                     where d is the display of T, and seqprod_display adds
                     an extra ^sp to all notations
   n.-tupleprod T := n.-tuple[seqprod_display d] T
                     where d is the display of T
 seqlexi_with d T := seq T
                  == an alias for seq, such that if T is canonically
                     ordered, then seqprod_with d T is canonically ordered
                     in lexicographic order, i.e.,
                     [:: x1, .., xn] <= [y1, .., yn] =
                       (x1 <= x2) && ((x1 >= y1) ==> ((x2 <= y2) && ...))
                     and displayed in display d
n.-tuplelexi[d] T == same with n.tuple T
        seqlexi T := lexiprod_with (seqlexi_display d) T
                     where d is the display of T, and seqlexi_display adds
                     an extra ^sl to all notations
   n.-tuplelexi T := n.-tuple[seqlexi_display d] T
                     where d is the display of T
    {subset[d] T} := {set T}
                  == an alias for set which is canonically ordered by the
                     subset order and displayed in display d
       {subset T} := {subset[subset_display] T}

The following notations are provided to build substructures:
[SubChoice_isSubPOrder of U by <: with disp] ==
[SubChoice_isSubPOrder of U by <:] == porderType mixin for a subType
                         whose base type is a porderType
[SubPOrder_isSubLattice of U by <: with disp] ==
[SubPOrder_isSubLattice of U by <:] ==
[SubChoice_isSubLattice of U by <: with disp] ==
[SubChoice_isSubLattice of U by <:] == latticeType mixin for a subType
                         whose base type is a latticeType and whose
                         predicate is a latticeClosed
[SubPOrder_isBSubLattice of U by <: with disp] ==
[SubPOrder_isBSubLattice of U by <:] ==
[SubChoice_isBSubLattice of U by <: with disp] ==
[SubChoice_isBSubLattice of U by <:] == blatticeType mixin for a subType
                         whose base type is a blatticeType and whose
                         predicate is both a latticeClosed
                         and a bLatticeClosed
[SubPOrder_isTSubLattice of U by <: with disp] ==
[SubPOrder_isTSubLattice of U by <:] ==
[SubChoice_isTSubLattice of U by <: with disp] ==
[SubChoice_isTSubLattice of U by <:] == tlatticeType mixin for a subType
                         whose base type is a tlatticeType and whose
                         predicate is both a latticeClosed
                         and a tLatticeClosed
[SubPOrder_isTBSubLattice of U by <: with disp] ==
[SubPOrder_isTBSubLattice of U by <:] ==
[SubChoice_isTBSubLattice of U by <: with disp] ==
[SubChoice_isTBSubLattice of U by <:] == tblatticeType mixin for a subType
                         whose base type is a tblatticeType and whose
                         predicate is both a latticeClosed
                         and a tbLatticeClosed
[SubLattice_isSubOrder of U by <: with disp] ==
[SubLattice_isSubOrder of U by <:] ==
[SubPOrder_isSubOrder of U by <: with disp] ==
[SubPOrder_isSubOrder of U by <:] ==
[SubChoice_isSubOrder of U by <: with disp] ==
[SubChoice_isSubOrder of U by <:] == orderType mixin for a subType whose
                         base type is an orderType
  [POrder of U by <:] == porderType mixin for a subType whose base type is
                         a porderType
   [Order of U by <:] == orderType mixin for a subType whose base type is
                         an orderType

We provide expected instances of ordered types for bool, nat (for leq and
and dvdn), 'I_n, 'I_n.+1 (with a top and bottom), nat for dvdn,
T *prod[disp] T', T *lexi[disp] T', {t : T & T' x} (with lexicographic
ordering), seqprod_with d T (using product order), seqlexi_with d T
(with lexicographic ordering), n.-tupleprod[disp] (using product order),
n.-tuplelexi[d] T (with lexicographic ordering), on {subset[disp] T}
(using subset order) and all possible finite type instances.
(Use `HB.about type` to discover the instances on type.)

In order to get a canonical order on prod, seq, tuple or set, one may
import modules DefaultProdOrder or DefaultProdLexiOrder,
DefaultSeqProdOrder or DefaultSeqLexiOrder,
DefaultTupleProdOrder or DefaultTupleLexiOrder,
and DefaultSetSubsetOrder.

We also provide specialized versions of some theorems from path.v.

We provide Order.enum_val, Order.enum_rank, and Order.enum_rank_in, which
are monotonic variations of enum_val, enum_rank, and enum_rank_in
whenever the type is porderType, and their monotonicity is provided if
this order is total. The theory is in the module Order (Order.enum_valK,
Order.enum_rank_inK, etc) but Order.Enum can be imported to shorten these.

We provide an opaque monotonous bijection tagnat.sig / tagnat.rank between
the finite types {i : 'I_n & 'I_(p_ i)} and 'I_(\sum_i p_ i):
 tagnat.sig  : 'I_(\sum_i p_ i) -> {i : 'I_n & 'I_(p_ i)}
 tagnat.rank : {i : 'I_n & 'I_(p_ i)} -> 'I_(\sum_i p_ i)
 tagnat.sig1 : 'I_(\sum_i p_ i) -> 'I_n
 tagnat.sig2 : forall p : 'I_(\sum_i p_ i), 'I_(p_ (tagnat.sig1 p))
 tagnat.Rank : forall i, 'I_(p_ i) -> 'I_(\sum_i p_ i)

Acknowledgments: This file is based on prior work by D. Dreyer, G.
Gonthier, A. Nanevski, P-Y Strub, B. Ziliani

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Declare Scope order_scope.

Delimit Scope order_scope with O.
Local Open Scope order_scope.

Reserved Notation "A `&` B" (at level 48, left associativity).
Reserved Notation "A `|` B" (at level 52, left associativity).
Reserved Notation "A `\` B" (at level 50, left associativity).
Reserved Notation "~` A" (at level 35, right associativity).

Reserved Notation "A `&^d` B" (at level 48, left associativity).
Reserved Notation "A `|^d` B" (at level 52, left associativity).
Reserved Notation "A `\^d` B" (at level 50, left associativity).
Reserved Notation "~^d` A" (at level 35, right associativity).

Reserved Notation "A `&^p` B" (at level 48, left associativity).
Reserved Notation "A `|^p` B" (at level 52, left associativity).
Reserved Notation "A `\^p` B" (at level 50, left associativity).
Reserved Notation "~^p` A" (at level 35, right associativity).

Reserved Notation "A `&^sp` B" (at level 48, left associativity).
Reserved Notation "A `|^sp` B" (at level 52, left associativity).
Reserved Notation "A `\^sp` B" (at level 50, left associativity).
Reserved Notation "~^sp` A" (at level 35, right associativity).

Reserved Notation "A `&^l` B" (at level 48, left associativity).
Reserved Notation "A `|^l` B" (at level 52, left associativity).
Reserved Notation "A `\^l` B" (at level 50, left associativity).
Reserved Notation "~^l` A" (at level 35, right associativity).

Reserved Notation "A `&^sl` B" (at level 48, left associativity).
Reserved Notation "A `|^sl` B" (at level 52, left associativity).
Reserved Notation "A `\^sl` B" (at level 50, left associativity).
Reserved Notation "~^sl` A" (at level 35, right associativity).

Reserved Notation "\gcd_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \gcd_ i '/ ' F ']'").
Reserved Notation "\gcd_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \gcd_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i <- r ) F"
  (F at level 41,
           format "'[' \gcd_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\gcd_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \gcd_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \gcd_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i | P ) F"
  (F at level 41,
           format "'[' \gcd_ ( i | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\gcd_ ( i : t ) F" (F at level 41).
Reserved Notation "\gcd_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \gcd_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i < n ) F"
  (F at level 41,
           format "'[' \gcd_ ( i < n ) F ']'").
Reserved Notation "\gcd_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \gcd_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \gcd_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\lcm_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \lcm_ i '/ ' F ']'").
Reserved Notation "\lcm_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \lcm_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i <- r ) F"
  (F at level 41,
           format "'[' \lcm_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\lcm_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \lcm_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \lcm_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i | P ) F"
  (F at level 41,
           format "'[' \lcm_ ( i | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\lcm_ ( i : t ) F" (F at level 41).
Reserved Notation "\lcm_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \lcm_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i < n ) F"
  (F at level 41,
           format "'[' \lcm_ ( i < n ) F ']'").
Reserved Notation "\lcm_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \lcm_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \lcm_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \meet_ i '/ ' F ']'").
Reserved Notation "\meet_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \meet_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i <- r ) F"
  (F at level 41,
           format "'[' \meet_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \meet_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \meet_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet_ ( i | P ) F"
  (F at level 41,
           format "'[' \meet_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\meet_ ( i : t ) F" (F at level 41).
Reserved Notation "\meet_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \meet_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i < n ) F"
  (F at level 41,
           format "'[' \meet_ ( i < n ) F ']'").
Reserved Notation "\meet_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \meet_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \meet_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \join_ i '/ ' F ']'").
Reserved Notation "\join_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \join_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i <- r ) F"
  (F at level 41,
           format "'[' \join_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \join_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \join_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join_ ( i | P ) F"
  (F at level 41,
           format "'[' \join_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\join_ ( i : t ) F" (F at level 41).
Reserved Notation "\join_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \join_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i < n ) F"
  (F at level 41,
           format "'[' \join_ ( i < n ) F ']'").
Reserved Notation "\join_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \join_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \join_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet^d_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \meet^d_ i '/ ' F ']'").
Reserved Notation "\meet^d_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \meet^d_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i <- r ) F"
  (F at level 41,
           format "'[' \meet^d_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \meet^d_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \meet^d_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i | P ) F"
  (F at level 41,
           format "'[' \meet^d_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\meet^d_ ( i : t ) F" (F at level 41).
Reserved Notation "\meet^d_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \meet^d_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i < n ) F"
  (F at level 41,
           format "'[' \meet^d_ ( i < n ) F ']'").
Reserved Notation "\meet^d_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \meet^d_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \meet^d_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join^d_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \join^d_ i '/ ' F ']'").
Reserved Notation "\join^d_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \join^d_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i <- r ) F"
  (F at level 41,
           format "'[' \join^d_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join^d_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \join^d_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \join^d_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i | P ) F"
  (F at level 41,
           format "'[' \join^d_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\join^d_ ( i : t ) F" (F at level 41).
Reserved Notation "\join^d_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \join^d_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i < n ) F"
  (F at level 41,
           format "'[' \join^d_ ( i < n ) F ']'").
Reserved Notation "\join^d_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \join^d_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \join^d_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet^p_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \meet^p_ i '/ ' F ']'").
Reserved Notation "\meet^p_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \meet^p_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i <- r ) F"
  (F at level 41,
           format "'[' \meet^p_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \meet^p_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \meet^p_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i | P ) F"
  (F at level 41,
           format "'[' \meet^p_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\meet^p_ ( i : t ) F" (F at level 41).
Reserved Notation "\meet^p_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \meet^p_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i < n ) F"
  (F at level 41,
           format "'[' \meet^p_ ( i < n ) F ']'").
Reserved Notation "\meet^p_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \meet^p_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \meet^p_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join^p_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \join^p_ i '/ ' F ']'").
Reserved Notation "\join^p_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \join^p_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i <- r ) F"
  (F at level 41,
           format "'[' \join^p_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join^p_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \join^p_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \join^p_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i | P ) F"
  (F at level 41,
           format "'[' \join^p_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\join^p_ ( i : t ) F" (F at level 41).
Reserved Notation "\join^p_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \join^p_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i < n ) F"
  (F at level 41,
           format "'[' \join^p_ ( i < n ) F ']'").
Reserved Notation "\join^p_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \join^p_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \join^p_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\min^p_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \min^p_ i '/ ' F ']'").
Reserved Notation "\min^p_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \min^p_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\min^p_ ( i <- r ) F"
  (F at level 41,
           format "'[' \min^p_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\min^p_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \min^p_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\min^p_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \min^p_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\min^p_ ( i | P ) F"
  (F at level 41,
           format "'[' \min^p_ ( i | P ) '/ ' F ']'").
Reserved Notation "\min^p_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\min^p_ ( i : t ) F" (F at level 41).
Reserved Notation "\min^p_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \min^p_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\min^p_ ( i < n ) F"
  (F at level 41,
           format "'[' \min^p_ ( i < n ) F ']'").
Reserved Notation "\min^p_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \min^p_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\min^p_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \min^p_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\max^p_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \max^p_ i '/ ' F ']'").
Reserved Notation "\max^p_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \max^p_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\max^p_ ( i <- r ) F"
  (F at level 41,
           format "'[' \max^p_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\max^p_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \max^p_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\max^p_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \max^p_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\max^p_ ( i | P ) F"
  (F at level 41,
           format "'[' \max^p_ ( i | P ) '/ ' F ']'").
Reserved Notation "\max^p_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\max^p_ ( i : t ) F" (F at level 41).
Reserved Notation "\max^p_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \max^p_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\max^p_ ( i < n ) F"
  (F at level 41,
           format "'[' \max^p_ ( i < n ) F ']'").
Reserved Notation "\max^p_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \max^p_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\max^p_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \max^p_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet^sp_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \meet^sp_ i '/ ' F ']'").
Reserved Notation "\meet^sp_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \meet^sp_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( i <- r ) F"
  (F at level 41,
           format "'[' \meet^sp_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \meet^sp_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \meet^sp_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( i | P ) F"
  (F at level 41,
           format "'[' \meet^sp_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\meet^sp_ ( i : t ) F" (F at level 41).
Reserved Notation "\meet^sp_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \meet^sp_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( i < n ) F"
  (F at level 41,
           format "'[' \meet^sp_ ( i < n ) F ']'").
Reserved Notation "\meet^sp_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \meet^sp_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet^sp_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \meet^sp_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join^sp_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \join^sp_ i '/ ' F ']'").
Reserved Notation "\join^sp_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \join^sp_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( i <- r ) F"
  (F at level 41,
           format "'[' \join^sp_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \join^sp_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \join^sp_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( i | P ) F"
  (F at level 41,
           format "'[' \join^sp_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\join^sp_ ( i : t ) F" (F at level 41).
Reserved Notation "\join^sp_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \join^sp_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( i < n ) F"
  (F at level 41,
           format "'[' \join^sp_ ( i < n ) F ']'").
Reserved Notation "\join^sp_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \join^sp_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join^sp_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \join^sp_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\min^sp_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \min^sp_ i '/ ' F ']'").
Reserved Notation "\min^sp_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \min^sp_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( i <- r ) F"
  (F at level 41,
           format "'[' \min^sp_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \min^sp_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \min^sp_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( i | P ) F"
  (F at level 41,
           format "'[' \min^sp_ ( i | P ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\min^sp_ ( i : t ) F" (F at level 41).
Reserved Notation "\min^sp_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \min^sp_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( i < n ) F"
  (F at level 41,
           format "'[' \min^sp_ ( i < n ) F ']'").
Reserved Notation "\min^sp_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \min^sp_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\min^sp_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \min^sp_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\max^sp_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \max^sp_ i '/ ' F ']'").
Reserved Notation "\max^sp_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \max^sp_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( i <- r ) F"
  (F at level 41,
           format "'[' \max^sp_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \max^sp_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \max^sp_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( i | P ) F"
  (F at level 41,
           format "'[' \max^sp_ ( i | P ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\max^sp_ ( i : t ) F" (F at level 41).
Reserved Notation "\max^sp_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \max^sp_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( i < n ) F"
  (F at level 41,
           format "'[' \max^sp_ ( i < n ) F ']'").
Reserved Notation "\max^sp_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \max^sp_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\max^sp_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \max^sp_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet^l_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \meet^l_ i '/ ' F ']'").
Reserved Notation "\meet^l_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \meet^l_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( i <- r ) F"
  (F at level 41,
           format "'[' \meet^l_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \meet^l_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \meet^l_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( i | P ) F"
  (F at level 41,
           format "'[' \meet^l_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\meet^l_ ( i : t ) F" (F at level 41).
Reserved Notation "\meet^l_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \meet^l_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( i < n ) F"
  (F at level 41,
           format "'[' \meet^l_ ( i < n ) F ']'").
Reserved Notation "\meet^l_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \meet^l_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet^l_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \meet^l_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join^l_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \join^l_ i '/ ' F ']'").
Reserved Notation "\join^l_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \join^l_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join^l_ ( i <- r ) F"
  (F at level 41,
           format "'[' \join^l_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join^l_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \join^l_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join^l_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \join^l_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join^l_ ( i | P ) F"
  (F at level 41,
           format "'[' \join^l_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join^l_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\join^l_ ( i : t ) F" (F at level 41).
Reserved Notation "\join^l_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \join^l_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join^l_ ( i < n ) F"
  (F at level 41,
           format "'[' \join^l_ ( i < n ) F ']'").
Reserved Notation "\join^l_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \join^l_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join^l_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \join^l_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\min^l_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \min^l_ i '/ ' F ']'").
Reserved Notation "\min^l_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \min^l_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\min^l_ ( i <- r ) F"
  (F at level 41,
           format "'[' \min^l_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\min^l_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \min^l_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\min^l_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \min^l_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\min^l_ ( i | P ) F"
  (F at level 41,
           format "'[' \min^l_ ( i | P ) '/ ' F ']'").
Reserved Notation "\min^l_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\min^l_ ( i : t ) F" (F at level 41).
Reserved Notation "\min^l_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \min^l_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\min^l_ ( i < n ) F"
  (F at level 41,
           format "'[' \min^l_ ( i < n ) F ']'").
Reserved Notation "\min^l_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \min^l_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\min^l_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \min^l_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\max^l_ i F"
  (at level 34, F at level 41, i at level 0,
           format "'[' \max^l_ i '/ ' F ']'").
Reserved Notation "\max^l_ ( i <- r | P ) F"
  (at level 34, F at level 41, i, r at level 60,
           format "'[' \max^l_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\max^l_ ( i <- r ) F"
  (F at level 41,
           format "'[' \max^l_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\max^l_ ( m <= i < n | P ) F"
  (F at level 41, i, n at level 60,
           format "'[' \max^l_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\max^l_ ( m <= i < n ) F"
  (F at level 41,
           format "'[' \max^l_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\max^l_ ( i | P ) F"
  (F at level 41,
           format "'[' \max^l_ ( i | P ) '/ ' F ']'").
Reserved Notation "\max^l_ ( i : t | P ) F" (F at level 41).
Reserved Notation "\max^l_ ( i : t ) F" (F at level 41).
Reserved Notation "\max^l_ ( i < n | P ) F"
  (F at level 41, n at level 60,
           format "'[' \max^l_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\max^l_ ( i < n ) F"
  (F at level 41,
           format "'[' \max^l_ ( i < n ) F ']'").
Reserved Notation "\max^l_ ( i 'in' A | P ) F"
  (F at level 41, A at level 60,
           format "'[' \max^l_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\max^l_ ( i 'in' A ) F"
  (F at level 41,
           format "'[' \max^l_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "'{' 'omorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'omorphism' U -> V }").
Reserved Notation "'{' 'mlmorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'mlmorphism' U -> V }").
Reserved Notation "'{' 'jlmorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'jlmorphism' U -> V }").
Reserved Notation "'{' 'lmorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'lmorphism' U -> V }").
Reserved Notation "'{' 'blmorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'blmorphism' U -> V }").
Reserved Notation "'{' 'tlmorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'tlmorphism' U -> V }").
Reserved Notation "'{' 'tblmorphism' U '->' V '}'"
  (U at level 98, V at level 99, format "{ 'tblmorphism' U -> V }").

Module Order.

Export Order.

#[key="T", primitive]
HB.mixin Record Preorder_isDuallyPOrder (d : disp_t) T & Preorder d T := {
  le_anti : antisymmetric (@le d T);
  ge_anti : antisymmetric (fun x y => @le d T y x);
}.

#[short(type="porderType")]
HB.structure Definition POrder (d : disp_t) :=
  { T of Preorder d T & Preorder_isDuallyPOrder d T }.

#[short(type="bPOrderType")]
HB.structure Definition BPOrder d := { T of hasBottom d T & POrder d T }.
#[short(type="tPOrderType")]
HB.structure Definition TPOrder d := { T of hasTop d T & POrder d T }.
#[short(type="tbPOrderType")]
HB.structure Definition TBPOrder d := { T of hasTop d T & BPOrder d T }.

Module POrderExports.
Arguments le_trans {d s} [_ _ _].
End POrderExports.
HB.export POrderExports.

#[key="T", primitive]
HB.mixin Record POrder_isMeetSemilattice d (T : Type) & POrder d T := {
  meet : T -> T -> T;
  lexI : forall x y z, (x <= meet y z) = (x <= y) && (x <= z);
}.

#[key="T", primitive]
HB.mixin Record POrder_isJoinSemilattice d T & POrder d T := {
  join : T -> T -> T;
  leUx : forall x y z, (join x y <= z) = (x <= z) && (y <= z);
}.

#[short(type="meetSemilatticeType")]
HB.structure Definition MeetSemilattice d :=
  { T of POrder d T & POrder_isMeetSemilattice d T }.

#[short(type="bMeetSemilatticeType")]
HB.structure Definition BMeetSemilattice d :=
  { T of MeetSemilattice d T & hasBottom d T }.

#[short(type="tMeetSemilatticeType")]
HB.structure Definition TMeetSemilattice d :=
  { T of MeetSemilattice d T & hasTop d T }.

#[short(type="tbMeetSemilatticeType")]
HB.structure Definition TBMeetSemilattice d :=
  { T of BMeetSemilattice d T & hasTop d T }.

#[short(type="joinSemilatticeType")]
HB.structure Definition JoinSemilattice d :=
  { T of POrder d T & POrder_isJoinSemilattice d T }.

#[short(type="bJoinSemilatticeType")]
HB.structure Definition BJoinSemilattice d :=
  { T of JoinSemilattice d T & hasBottom d T }.

#[short(type="tJoinSemilatticeType")]
HB.structure Definition TJoinSemilattice d :=
  { T of JoinSemilattice d T & hasTop d T }.

#[short(type="tbJoinSemilatticeType")]
HB.structure Definition TBJoinSemilattice d :=
  { T of BJoinSemilattice d T & hasTop d T }.

#[short(type="latticeType")]
HB.structure Definition Lattice d :=
  { T of JoinSemilattice d T & POrder_isMeetSemilattice d T }.

#[short(type="bLatticeType")]
HB.structure Definition BLattice d := { T of Lattice d T & hasBottom d T }.

#[short(type="tLatticeType")]
HB.structure Definition TLattice d := { T of Lattice d T & hasTop d T }.

#[short(type="tbLatticeType")]
HB.structure Definition TBLattice d := { T of BLattice d T & hasTop d T }.

Section LatticeDef.
Context {disp : disp_t} {T : latticeType disp}.

Variant lel_xor_gt (x y : T) :
  T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> Set :=
  | LelNotGt of x <= y : lel_xor_gt x y x x y y x x y y true false
  | GtlNotLe of y < x : lel_xor_gt x y y y x x y y x x false true.

Variant ltl_xor_ge (x y : T) :
  T -> T -> T -> T -> T -> T -> T -> T -> bool -> bool -> Set :=
  | LtlNotGe of x < y : ltl_xor_ge x y x x y y x x y y false true
  | GelNotLt of y <= x : ltl_xor_ge x y y y x x y y x x true false.

Variant comparel (x y : T) :
   T -> T -> T -> T -> T -> T -> T -> T ->
   bool -> bool -> bool -> bool -> bool -> bool -> Set :=
  | ComparelLt of x < y : comparel x y
    x x y y x x y y false false false true false true
  | ComparelGt of y < x : comparel x y
    y y x x y y x x false false true false true false
  | ComparelEq of x = y : comparel x y
    x x x x x x x x true true true true false false.

Variant incomparel (x y : T) :
  T -> T -> T -> T -> T -> T -> T -> T ->
  bool -> bool -> bool -> bool -> bool -> bool -> bool -> bool -> Set :=
  | InComparelLt of x < y : incomparel x y
    x x y y x x y y false false false true false true true true
  | InComparelGt of y < x : incomparel x y
    y y x x y y x x false false true false true false true true
  | InComparel of x >< y : incomparel x y
    x y y x (meet y x) (meet x y) (join y x) (join x y)
    false false false false false false false false
  | InComparelEq of x = y : incomparel x y
    x x x x x x x x true true true true false false true true.

End LatticeDef.

Module LatticeSyntax.

Notation "x `&` y" := (meet x y) : order_scope.
Notation "x `|` y" := (join x y) : order_scope.

End LatticeSyntax.
HB.export LatticeSyntax.

Module BLatticeSyntax.

Notation "\join_ ( i <- r | P ) F" :=
  (\big[@join _ _ / \bot]_(i <- r | P%B) F%O) : order_scope.
Notation "\join_ ( i <- r ) F" :=
  (\big[@join _ _ / \bot]_(i <- r) F%O) : order_scope.
Notation "\join_ ( i | P ) F" :=
  (\big[@join _ _ / \bot]_(i | P%B) F%O) : order_scope.
Notation "\join_ i F" :=
  (\big[@join _ _ / \bot]_i F%O) : order_scope.
Notation "\join_ ( i : I | P ) F" :=
  (\big[@join _ _ / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join_ ( i : I ) F" :=
  (\big[@join _ _ / \bot]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join_ ( m <= i < n | P ) F" :=
  (\big[@join _ _ / \bot]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\join_ ( m <= i < n ) F" :=
  (\big[@join _ _ / \bot]_(m <= i < n) F%O) : order_scope.
Notation "\join_ ( i < n | P ) F" :=
  (\big[@join _ _ / \bot]_(i < n | P%B) F%O) : order_scope.
Notation "\join_ ( i < n ) F" :=
  (\big[@join _ _ / \bot]_(i < n) F%O) : order_scope.
Notation "\join_ ( i 'in' A | P ) F" :=
  (\big[@join _ _ / \bot]_(i in A | P%B) F%O) : order_scope.
Notation "\join_ ( i 'in' A ) F" :=
  (\big[@join _ _ / \bot]_(i in A) F%O) : order_scope.

End BLatticeSyntax.
HB.export BLatticeSyntax.

Module TLatticeSyntax.

Notation "\meet_ ( i <- r | P ) F" :=
  (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet_ ( i <- r ) F" :=
  (\big[meet / \top]_(i <- r) F%O) : order_scope.
Notation "\meet_ ( i | P ) F" :=
  (\big[meet / \top]_(i | P%B) F%O) : order_scope.
Notation "\meet_ i F" :=
  (\big[meet / \top]_i F%O) : order_scope.
Notation "\meet_ ( i : I | P ) F" :=
  (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet_ ( i : I ) F" :=
  (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet_ ( m <= i < n | P ) F" :=
(\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\meet_ ( m <= i < n ) F" :=
(\big[meet / \top]_(m <= i < n) F%O) : order_scope.
Notation "\meet_ ( i < n | P ) F" :=
(\big[meet / \top]_(i < n | P%B) F%O) : order_scope.
Notation "\meet_ ( i < n ) F" :=
(\big[meet / \top]_(i < n) F%O) : order_scope.
Notation "\meet_ ( i 'in' A | P ) F" :=
(\big[meet / \top]_(i in A | P%B) F%O) : order_scope.
Notation "\meet_ ( i 'in' A ) F" :=
(\big[meet / \top]_(i in A) F%O) : order_scope.

End TLatticeSyntax.
HB.export TLatticeSyntax.

#[key="T", primitive]
HB.mixin Record Lattice_isDistributive d (T : Type) & Lattice d T := {
  meetUl : @left_distributive T T meet join;
  joinIl : @left_distributive T T join meet;
}.

#[short(type="distrLatticeType")]
HB.structure Definition DistrLattice d :=
  { T of Lattice_isDistributive d T & Lattice d T }.

#[short(type="bDistrLatticeType")]
HB.structure Definition BDistrLattice d :=
  { T of DistrLattice d T & hasBottom d T }.

#[short(type="tDistrLatticeType")]
HB.structure Definition TDistrLattice d :=
  { T of DistrLattice d T & hasTop d T }.

#[short(type="tbDistrLatticeType")]
HB.structure Definition TBDistrLattice d :=
  { T of BDistrLattice d T & hasTop d T }.

#[key="T", primitive]
HB.mixin Record DistrLattice_isTotal d T & DistrLattice d T :=
  { le_total : total (<=%O : rel T) }.

#[short(type="orderType")]
HB.structure Definition Total d :=
  { T of DistrLattice_isTotal d T & DistrLattice d T }.

#[short(type="bOrderType")]
HB.structure Definition BTotal d := { T of Total d T & hasBottom d T }.

#[short(type="tOrderType")]
HB.structure Definition TTotal d := { T of Total d T & hasTop d T }.

#[short(type="tbOrderType")]
HB.structure Definition TBTotal d := { T of BTotal d T & hasTop d T }.

#[key="T", primitive]
HB.mixin Record DistrLattice_hasRelativeComplement d T & DistrLattice d T := {
  rcompl : T -> T -> T -> T;
  rcomplPmeet : forall x y z, ((x `&` y) `|` z) `&` rcompl x y z = x `&` y;
  rcomplPjoin : forall x y z, ((y `|` x) `&` z) `|` rcompl x y z = y `|` x;
}.

#[short(type="cDistrLatticeType")]
HB.structure Definition CDistrLattice d :=
  { T of DistrLattice d T & DistrLattice_hasRelativeComplement d T }.

#[key="T", primitive]
HB.mixin Record CDistrLattice_hasSectionalComplement d T
         & CDistrLattice d T & hasBottom d T := {
  diff : T -> T -> T;
  diffErcompl : forall x y, diff x y = rcompl (\bot : T) x y;
}.

#[short(type="cbDistrLatticeType")]
HB.structure Definition CBDistrLattice d :=
  { T of CDistrLattice d T & hasBottom d T &
         CDistrLattice_hasSectionalComplement d T }.

#[key="T", primitive]
HB.mixin Record CDistrLattice_hasDualSectionalComplement d T
         & CDistrLattice d T & hasTop d T := {
  codiff : T -> T -> T;
  codiffErcompl : forall x y, codiff x y = rcompl x \top y;
}.

#[short(type="ctDistrLatticeType")]
HB.structure Definition CTDistrLattice d :=
  { T of CDistrLattice d T & hasTop d T &
         CDistrLattice_hasDualSectionalComplement d T }.

Module Import CBDistrLatticeSyntax.
Notation "x `\` y" := (diff x y) : order_scope.
End CBDistrLatticeSyntax.

#[key="T", primitive]
HB.mixin Record CDistrLattice_hasComplement d T &
         CTDistrLattice d T & CBDistrLattice d T := {
  compl : T -> T;
  complEdiff : forall x : T, compl x = (\top : T) `\` x;
  complEcodiff : forall x : T, compl x = codiff (\bot : T) x;
}.

#[short(type="ctbDistrLatticeType")]
HB.structure Definition CTBDistrLattice d :=
  { T of CBDistrLattice d T & CTDistrLattice d T &
         CDistrLattice_hasComplement d T }.

Module Import CTBDistrLatticeSyntax.
Notation "~` A" := (compl A) : order_scope.
End CTBDistrLatticeSyntax.

FINITE

DUAL

Proof.

by []. Qed.

HB.instance Definition _ d (T : meetSemilatticeType d) :=
POrder_isJoinSemilattice.Build (dual_display d) T^d (fun x y z => lexI z x y).

Lemma joinEdual d (T : meetSemilatticeType d) (x y : T) :
((x : T^d) `|^d` y) = (x `&` y).

Proof.

by []. Qed.

HB.saturate.

HB.instance Definition _ d (T : distrLatticeType d) :=
  Lattice_isDistributive.Build (dual_display d) T^d joinIl meetUl.

HB.instance Definition _ d (T : orderType d) :=
  DistrLattice_isTotal.Build (dual_display d) T^d (fun x y => le_total y x).

HB.saturate.

HB.instance Definition _ d (T : cDistrLatticeType d) :=
  DistrLattice_hasRelativeComplement.Build (dual_display d) T^d
    (fun x y => rcomplPjoin y x) (fun x y => rcomplPmeet y x).
HB.instance Definition _ d (T : ctDistrLatticeType d) :=
  CDistrLattice_hasSectionalComplement.Build (dual_display d) T^d codiffErcompl.
HB.instance Definition _ d (T : cbDistrLatticeType d) :=
  CDistrLattice_hasDualSectionalComplement.Build (dual_display d) T^d
    diffErcompl.
HB.instance Definition _ d (T : ctbDistrLatticeType d) :=
  CDistrLattice_hasComplement.Build (dual_display d) T^d
    complEcodiff complEdiff.

HB.saturate.

End DualOrder.
HB.export DualOrder.

THEORY

Module Import POrderTheory.

Include PreorderTheory.

Section POrderTheory.

Context {disp : disp_t} {T : porderType disp}.

Implicit Types (x y : T) (s : seq T).

Lemma le_anti: antisymmetric (<=%O : rel T).

Proof.

exact: le_anti. Qed.

Lemma ge_anti: antisymmetric (>=%O : rel T).

Proof.

by move=> x y /le_anti. Qed.

Lemma eq_le x y: (x == y) = (x <= y <= x).

Proof.

by apply/eqP/idP => [->|/le_anti]; rewrite ?lexx. Qed.

Lemma lt_def x y : (x < y) = (y != x) && (x <= y).

Proof.

rewrite andbC lt_le_def; case/boolP: (x <= y) => //= xy.
congr negb; apply/idP/eqP => [yx|->]; last exact/lexx.
by apply/le_anti; rewrite yx.
Qed.

Lemma lt_neqAle x y: (x < y) = (x != y) && (x <= y).

Proof.

by rewrite lt_def eq_sym. Qed.

Lemma le_eqVlt x y: (x <= y) = (x == y) || (x < y).

Proof.

by rewrite lt_neqAle; case: eqP => //= ->; rewrite lexx. Qed.

Definition lte_anti := (=^~ eq_le, @lt_asym disp T, @lt_le_asym disp T, @le_lt_asym disp T).

Lemma eq_geP {x y} : reflect (forall z, (z <= x) = (z <= y)) (x == y).

Proof.

by apply: (iffP idP) => [/eqP->//|/[dup]] /[!eq_le] -> <-; rewrite !lexx.
Qed.

Lemma eq_leP {x y} : reflect (forall z, (x <= z) = (y <= z)) (x == y).

Proof.

by apply: (iffP idP) => [/eqP->//|/[dup]] /[!eq_le] <- ->; rewrite !lexx.
Qed.

Lemma lt_sorted_uniq_le s : sorted <%O s = uniq s && sorted <=%O s.

Proof.

rewrite le_sorted_pairwise lt_sorted_pairwise uniq_pairwise -pairwise_relI.
by apply/eq_pairwise => ? ?; rewrite lt_neqAle.
Qed.

Lemma le_sorted_eq s1 s2 :
sorted <=%O s1 -> sorted <=%O s2 -> perm_eq s1 s2 -> s1 = s2.

Proof.

exact/sorted_eq/le_anti/le_trans. Qed.

Lemma count_lt_le_mem x s : (count (< x) s < count (<= x) s)%N = (x \in s).

Proof.

have := count_predUI (pred1 x) (< x) s.
have -> : count (predI (pred1 x) (< x)) s = 0%N.
rewrite (@eq_count _ _ pred0) ?count_pred0 // => y /=.
by rewrite lt_neqAle; case: eqP => //= ->; rewrite eqxx.
have /eq_count-> : [predU1 x & < x] =1 (<= x) by move=> y /=; rewrite le_eqVlt.
by rewrite addn0 => ->; rewrite -add1n leq_add2r -has_count has_pred1.
Qed.

Lemma comparable_ltgtP x y : x >=< y ->
compare x y (min y x) (min x y) (max y x) (max x y)
(y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y).

Proof.

rewrite /min /max />=<%O !le_eqVlt [y == x]eq_sym.
have := (eqVneq x y, (boolP (x < y), boolP (y < x))).
move=> [[->//|neq_xy /=] [[] xy [] //=]] ; do ?by rewrite ?ltxx; constructor.
by rewrite ltxx in xy.
by rewrite le_gtF // ltW.
Qed.

Lemma comparable_leP x y : x >=< y ->
le_xor_gt x y (min y x) (min x y) (max y x) (max x y) (x <= y) (y < x).

Proof.

by move=> /comparable_ltgtP [?|?|->]; constructor; rewrite // ltW. Qed.

Lemma comparable_ltP x y : x >=< y ->
lt_xor_ge x y (min y x) (min x y) (max y x) (max x y) (y <= x) (x < y).

Proof.

by move=> /comparable_ltgtP [?|?|->]; constructor; rewrite // ltW. Qed.

Lemma comparableP x y : incompare x y
  (min y x) (min x y) (max y x) (max x y)
  (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y)
  (y >=< x) (x >=< y).

Proof.

rewrite ![y >=< _]comparable_sym; have [c_xy|i_xy] := boolP (x >=< y).
by case: (comparable_ltgtP c_xy) => ?; constructor.
by rewrite /min /max ?incomparable_eqF ?incomparable_leF;
rewrite ?incomparable_ltF// 1?comparable_sym //; constructor.
Qed.

Lemma leifP x y C : reflect (x <= y ?= iff C) (if C then x == y else x < y).

Proof.

rewrite /leif le_eqVlt; apply: (iffP idP)=> [|[]].
by case: C => [/eqP->|lxy]; rewrite ?eqxx // lxy lt_eqF.
by move=> /orP[/eqP->|lxy] <-; rewrite ?eqxx // lt_eqF.
Qed.

Lemma leif_trans x1 x2 x3 C12 C23 :
x1 <= x2 ?= iff C12 -> x2 <= x3 ?= iff C23 -> x1 <= x3 ?= iff C12 && C23.

Proof.

move=> ltx12 ltx23; apply/leifP; rewrite -ltx12.
case eqx12: (x1 == x2).
by rewrite (eqP eqx12) lt_neqAle !ltx23 andbT; case C23.
by rewrite (@lt_le_trans _ _ x2) ?ltx23 // lt_neqAle eqx12 ltx12.
Qed.

Lemma leif_le x y : x <= y -> x <= y ?= iff (x >= y).

Proof.

by move=> lexy; split=> //; rewrite eq_le lexy. Qed.

Lemma leif_eq x y : x <= y -> x <= y ?= iff (x == y).

Proof.

by []. Qed.

Lemma ge_leif x y C : x <= y ?= iff C -> (y <= x) = C.

Proof.

by case=> le_xy; rewrite eq_le le_xy. Qed.

Lemma lt_leif x y C : x <= y ?= iff C -> (x < y) = ~~ C.

Proof.

by move=> le_xy; rewrite lt_neqAle !le_xy andbT. Qed.

Lemma ltNleif x y C : x <= y ?= iff ~~ C -> (x < y) = C.

Proof.

by move=> /lt_leif; rewrite negbK. Qed.

Lemma lteif_anti C1 C2 x y :
(x < y ?<= if C1) && (y < x ?<= if C2) = C1 && C2 && (x == y).

Proof.

by case: C1 C2 => [][]; rewrite lte_anti. Qed.

Lemma lteifN C x y : x < y ?<= if ~~ C -> ~~ (y < x ?<= if C).

Proof.

by case: C => /=; case: comparableP. Qed.

Lemma minEle x y : min x y = if x <= y then x else y.

Proof.

by case: comparableP. Qed.

Lemma maxEle x y : max x y = if x <= y then y else x.

Proof.

by case: comparableP. Qed.

Lemma comparable_minEgt x y : x >=< y -> min x y = if x > y then y else x.

Proof.

by case: comparableP. Qed.

Lemma comparable_maxEgt x y : x >=< y -> max x y = if x > y then x else y.

Proof.

by case: comparableP. Qed.

Lemma comparable_minEge x y : x >=< y -> min x y = if x >= y then y else x.

Proof.

by case: comparableP. Qed.

Lemma comparable_maxEge x y : x >=< y -> max x y = if x >= y then x else y.

Proof.

by case: comparableP. Qed.

Lemma min_l x y : x <= y -> min x y = x

Proof.

by case: comparableP. Qed.

Lemma min_r x y : y <= x -> min x y = y

Proof.

by case: comparableP. Qed.

Lemma max_l x y : y <= x -> max x y = x

Proof.

by case: comparableP. Qed.

Lemma max_r x y : x <= y -> max x y = y

Proof.

by case: comparableP. Qed.

Lemma eq_minl x y : (min x y == x) = (x <= y).

Proof.

by rewrite !(fun_if, if_arg) eqxx; case: comparableP. Qed.

Lemma eq_maxr x y : (max x y == y) = (x <= y).

Proof.

by rewrite !(fun_if, if_arg) eqxx; case: comparableP. Qed.

Lemma min_idPl x y : reflect (min x y = x) (x <= y).

Proof.

by rewrite -eq_minl; apply/eqP. Qed.

Lemma max_idPr x y : reflect (max x y = y) (x <= y).

Proof.

by rewrite -eq_maxr; apply/eqP. Qed.

Section Comparable2.
Context (z x y : T) (cmp_xy : x >=< y).

Lemma comparable_minC : min x y = min y x.

Proof.

by case: comparableP cmp_xy. Qed.

Lemma comparable_maxC : max x y = max y x.

Proof.

by case: comparableP cmp_xy. Qed.

Lemma comparable_eq_minr : (min x y == y) = (y <= x).

Proof.

by rewrite !(fun_if, if_arg) eqxx; case: comparableP cmp_xy. Qed.

Lemma comparable_eq_maxl : (max x y == x) = (y <= x).

Proof.

by rewrite !(fun_if, if_arg) eqxx; case: comparableP cmp_xy. Qed.

Lemma comparable_min_idPr : reflect (min x y = y) (y <= x).

Proof.

by rewrite -comparable_eq_minr; apply/eqP. Qed.

Lemma comparable_max_idPl : reflect (max x y = x) (y <= x).

Proof.

by rewrite -comparable_eq_maxl; apply/eqP. Qed.

Lemma comparable_lteifNE C :
x >=< y -> (x < y ?<= if ~~ C) = ~~ (y < x ?<= if C).

Proof.

by case: C => /=; case: comparableP. Qed.

End Comparable2.

Section Comparable3.
Context (x y z : T) (cmp_xy : x >=< y) (cmp_xz : x >=< z) (cmp_yz : y >=< z).
Let P := comparableP.

Lemma comparable_max_minl : max (min x y) z = min (max x z) (max y z).

Proof.

move: cmp_xy cmp_xz cmp_yz; rewrite !(fun_if, if_arg)/=.
move: (P x y) (P x z) (P y z).
move=> [xy|xy|xy|<-] [xz|xz|xz|<-] [yz|yz|yz|//->]//= _; rewrite ?ltxx//.
- by have := lt_trans xy (lt_trans yz xz); rewrite ltxx.
- by have := lt_trans xy (lt_trans xz yz); rewrite ltxx.
Qed.

End Comparable3.

Section Comparable4.
Context (x y z w : T) (cmp_xy : x >=< y) (cmp_zw : z >=< w).

Lemma comparable_le_min2 : x <= z -> y <= w ->
Order.min x y <= Order.min z w.

Proof.

move: cmp_xy cmp_zw => /comparable_leP[] xy /comparable_leP[] zw // xz yw.
- exact: le_trans xy yw.
- exact: le_trans (ltW xy) xz.
Qed.

Lemma comparable_le_max2 : x <= z -> y <= w ->
Order.max x y <= Order.max z w.

Proof.

move: cmp_xy cmp_zw => /comparable_leP[] xy /comparable_leP[] zw // xz yw.
- exact: le_trans yw (ltW zw).
- exact: le_trans xz zw.
Qed.

End Comparable4.

Lemma comparable_minAC x y z : x >=< y -> x >=< z -> y >=< z ->
min (min x y) z = min (min x z) y.

Proof.

move=> xy xz yz; rewrite -comparable_minA// [min y z]comparable_minC//.
by rewrite comparable_minA// 1?comparable_sym.
Qed.

Lemma comparable_maxAC x y z : x >=< y -> x >=< z -> y >=< z ->
max (max x y) z = max (max x z) y.

Proof.

move=> xy xz yz; rewrite -comparable_maxA// [max y z]comparable_maxC//.
by rewrite comparable_maxA// 1?comparable_sym.
Qed.

Lemma comparable_minCA x y z : x >=< y -> x >=< z -> y >=< z ->
min x (min y z) = min y (min x z).

Proof.

move=> xy xz yz; rewrite comparable_minA// [min x y]comparable_minC//.
by rewrite -comparable_minA// 1?comparable_sym.
Qed.

Lemma comparable_maxCA x y z : x >=< y -> x >=< z -> y >=< z ->
max x (max y z) = max y (max x z).

Proof.

move=> xy xz yz; rewrite comparable_maxA// [max x y]comparable_maxC//.
by rewrite -comparable_maxA// 1?comparable_sym.
Qed.

Lemma comparable_minACA x y z t :
x >=< y -> x >=< z -> x >=< t -> y >=< z -> y >=< t -> z >=< t ->
min (min x y) (min z t) = min (min x z) (min y t).

Proof.

move=> xy xz xt yz yt zt; rewrite comparable_minA// ?comparable_minl//.
rewrite [min _ z]comparable_minAC// -comparable_minA// ?comparable_minl//.
by rewrite inE comparable_sym.
Qed.

Lemma comparable_maxACA x y z t :
x >=< y -> x >=< z -> x >=< t -> y >=< z -> y >=< t -> z >=< t ->
max (max x y) (max z t) = max (max x z) (max y t).

Proof.

move=> xy xz xt yz yt zt; rewrite comparable_maxA// ?comparable_maxl//.
rewrite [max _ z]comparable_maxAC// -comparable_maxA// ?comparable_maxl//.
by rewrite inE comparable_sym.
Qed.

Lemma comparable_min_maxr x y z : x >=< y -> x >=< z -> y >=< z ->
min x (max y z) = max (min x y) (min x z).

Proof.

move=> xy xz yz; rewrite ![min x _]comparable_minC// ?comparable_maxr//.
by rewrite comparable_min_maxl// 1?comparable_sym.
Qed.

Lemma mono_in_leif (A : {pred T}) (f : T -> T) C :
{in A &, {mono f : x y / x <= y}} ->
{in A &, forall x y, (f x <= f y ?= iff C) = (x <= y ?= iff C)}.

Proof.

by move=> mf x y Ax Ay; rewrite /leif !eq_le !mf. Qed.

Lemma mono_leif (f : T -> T) C :
{mono f : x y / x <= y} ->
forall x y, (f x <= f y ?= iff C) = (x <= y ?= iff C).

Proof.

by move=> mf x y; rewrite /leif !eq_le !mf. Qed.

Lemma nmono_in_leif (A : {pred T}) (f : T -> T) C :
{in A &, {mono f : x y /~ x <= y}} ->
{in A &, forall x y, (f x <= f y ?= iff C) = (y <= x ?= iff C)}.

Proof.

by move=> mf x y Ax Ay; rewrite /leif !eq_le !mf. Qed.

Lemma nmono_leif (f : T -> T) C : {mono f : x y /~ x <= y} ->
forall x y, (f x <= f y ?= iff C) = (y <= x ?= iff C).

Proof.

by move=> mf x y; rewrite /leif !eq_le !mf. Qed.

Section bigminmax.
Context (I : Type) (r : seq I) (f : I -> T) (x0 x : T) (P : pred I).

Lemma bigmax_le : x0 <= x -> (forall i, P i -> f i <= x) ->
\big[max/x0]_(i <- r | P i) f i <= x.

Proof.

by move=> ? ?; elim/big_ind: _ => // *; rewrite maxEle; case: ifPn. Qed.

Lemma le_bigmin : x <= x0 -> (forall i, P i -> x <= f i) ->
x <= \big[min/x0]_(i <- r | P i) f i.

Proof.

by move=> ? ?; elim/big_ind: _ => // *; rewrite minEle; case: ifPn. Qed.

End bigminmax.

End POrderTheory.

#[global] Hint Resolve comparable_minr comparable_minl : core.
#[global] Hint Resolve comparable_maxr comparable_maxl : core.

Section ContraTheory.
Context {disp1 disp2 : disp_t} {T1 : porderType disp1} {T2 : porderType disp2}.
Implicit Types (x y : T1) (z t : T2) (b : bool) (m n : nat) (P : Prop).

Lemma contra_leT b x y : (~~ b -> x < y) -> (y <= x -> b).

Proof.

by case: comparableP; case: b. Qed.

Lemma contra_ltT b x y : (~~ b -> x <= y) -> (y < x -> b).

Proof.

by case: comparableP; case: b. Qed.

Lemma contra_leN b x y : (b -> x < y) -> (y <= x -> ~~ b).

Proof.

by case: comparableP; case: b. Qed.

Lemma contra_ltN b x y : (b -> x <= y) -> (y < x -> ~~ b).

Proof.

by case: comparableP; case: b. Qed.

Lemma contra_le_not P x y : (P -> x < y) -> (y <= x -> ~ P).

Proof.

by case: comparableP => // _ PF _ /PF. Qed.

Lemma contra_lt_not P x y : (P -> x <= y) -> (y < x -> ~ P).

Proof.

by case: comparableP => // _ PF _ /PF. Qed.

Lemma contra_leF b x y : (b -> x < y) -> (y <= x -> b = false).

Proof.

by case: comparableP; case: b => // _ /implyP. Qed.

Lemma contra_ltF b x y : (b -> x <= y) -> (y < x -> b = false).

Proof.

by case: comparableP; case: b => // _ /implyP. Qed.

Lemma contra_le_leq x y m n : ((n < m)%N -> y < x) -> (x <= y -> (m <= n)%N).

Proof.

by case: comparableP; case: ltngtP. Qed.

Lemma contra_le_ltn x y m n : ((n <= m)%N -> y < x) -> (x <= y -> (m < n)%N).

Proof.

by case: comparableP; case: ltngtP. Qed.

Lemma contra_lt_leq x y m n : ((n < m)%N -> y <= x) -> (x < y -> (m <= n)%N).

Proof.

by case: comparableP; case: ltngtP. Qed.

Lemma contra_lt_ltn x y m n : ((n <= m)%N -> y <= x) -> (x < y -> (m < n)%N).

Proof.

by case: comparableP; case: ltngtP. Qed.

End ContraTheory.

Section POrderMonotonyTheory.
Context {disp disp' : disp_t} {T : porderType disp} {T' : porderType disp'}.
Context (D D' : {pred T}) (f : T -> T').

Let leT_anti := @le_anti _ T.
Hint Resolve lexx lt_neqAle : core.

Let ge_antiT : antisymmetric (>=%O : rel T).

Proof.

by move=> ? ? /le_anti. Qed.

Lemma ltW_homo : {homo f : x y / x < y} -> {homo f : x y / x <= y}.

Proof.

exact: homoW. Qed.

Lemma ltW_nhomo : {homo f : x y /~ x < y} -> {homo f : x y /~ x <= y}.

Proof.

by apply: homoW=> // x y; rewrite eq_sym. Qed.

Lemma inj_homo_lt :
injective f -> {homo f : x y / x <= y} -> {homo f : x y / x < y}.

Proof.

exact: inj_homo. Qed.

Lemma inj_nhomo_lt :
injective f -> {homo f : x y /~ x <= y} -> {homo f : x y /~ x < y}.

Proof.

by apply: inj_homo=> // x y; rewrite eq_sym. Qed.

Lemma inc_inj : {mono f : x y / x <= y} -> injective f.

Proof.

exact: mono_inj. Qed.

Lemma dec_inj : {mono f : x y /~ x <= y} -> injective f.

Proof.

exact: mono_inj. Qed.

Lemma ltW_homo_in :
{in D & D', {homo f : x y / x < y}} -> {in D & D', {homo f : x y / x <= y}}.

Proof.

exact: homoW_in. Qed.

Lemma ltW_nhomo_in :
{in D & D', {homo f : x y /~ x < y}} -> {in D & D', {homo f : x y /~ x <= y}}.

Proof.

by apply: homoW_in=> // x y; rewrite eq_sym. Qed.

Lemma inj_homo_lt_in :
{in D & D', injective f} -> {in D & D', {homo f : x y / x <= y}} ->
{in D & D', {homo f : x y / x < y}}.

Proof.

exact: inj_homo_in. Qed.

Lemma inj_nhomo_lt_in :
{in D & D', injective f} -> {in D & D', {homo f : x y /~ x <= y}} ->
{in D & D', {homo f : x y /~ x < y}}.

Proof.

by apply: inj_homo_in=> // x y; rewrite eq_sym. Qed.

Lemma inc_inj_in : {in D &, {mono f : x y / x <= y}} ->
{in D &, injective f}.

Proof.

exact: mono_inj_in. Qed.

Lemma dec_inj_in :
{in D &, {mono f : x y /~ x <= y}} -> {in D &, injective f}.

Proof.

exact: mono_inj_in. Qed.

End POrderMonotonyTheory.

End POrderTheory.

Arguments leifP {disp T x y C}.
Arguments mono_in_leif [disp T A f C].
Arguments nmono_in_leif [disp T A f C].
Arguments mono_leif [disp T f C].
Arguments nmono_leif [disp T f C].
Arguments min_idPl {disp T x y}.
Arguments max_idPr {disp T x y}.
Arguments comparable_min_idPr {disp T x y _}.
Arguments comparable_max_idPl {disp T x y _}.

Module Import BPOrderTheory.

Export BPreorderTheory.

Section BPOrderTheory.
Context {disp : disp_t} {T : bPOrderType disp}.
Implicit Types (x y : T).

Lemma lex0 x : (x <= \bot) = (x == \bot).

Proof.

by rewrite le_eqVlt ltx0 orbF. Qed.

Lemma lt0x x : (\bot < x) = (x != \bot).

Proof.

by rewrite lt_def le0x andbT. Qed.

Variant eq0_xor_gt0 x : bool -> bool -> Set :=
Eq0NotPOs : x = \bot -> eq0_xor_gt0 x true false
| POsNotEq0 : \bot < x -> eq0_xor_gt0 x false true.

Lemma posxP x : eq0_xor_gt0 x (x == \bot) (\bot < x).

Proof.

by rewrite lt0x; have [] := eqVneq; constructor; rewrite ?lt0x. Qed.

End BPOrderTheory.
End BPOrderTheory.

Module Import TPOrderTheory.
Section TPOrderTheory.
Context {disp : disp_t} {T : tPOrderType disp}.
Implicit Types (x y : T).

Lemma le1x x : (\top <= x) = (x == \top)

Proof.

exact: (@lex0 _ T^d). Qed.

Lemma ltx1 x : (x < \top) = (x != \top)

Proof.

exact: (@lt0x _ T^d). Qed.

End TPOrderTheory.
End TPOrderTheory.

Module Import MeetTheory.
Section MeetTheory.
Context {disp : disp_t} {L : meetSemilatticeType disp}.
Implicit Types (x y : L).

Lemma lexI x y z : (x <= y `&` z) = (x <= y) && (x <= z).

Proof.

exact: lexI. Qed.

Lemma leIr x y : y `&` x <= x.

Proof.

by have:= le_refl (meet y x); rewrite lexI => /andP []. Qed.

Lemma leIl x y : x `&` y <= x.

Proof.

by have:= le_refl (meet x y); rewrite lexI => /andP []. Qed.

Lemma leIxl x y z : y <= x -> y `&` z <= x.

Proof.

exact/le_trans/leIl. Qed.

Lemma leIxr x y z : z <= x -> y `&` z <= x.

Proof.

exact/le_trans/leIr. Qed.

Lemma leIx2 x y z : (y <= x) || (z <= x) -> y `&` z <= x.

Proof.

by case/orP => [/leIxl|/leIxr]. Qed.

Lemma leEmeet x y : (x <= y) = (x `&` y == x).

Proof.

by rewrite eq_le lexI leIl lexx. Qed.

Lemma eq_meetl x y : (x `&` y == x) = (x <= y).

Proof.

by apply/esym/leEmeet. Qed.

Lemma eq_meetr x y : (x `&` y == y) = (y <= x).

Proof.

by rewrite eq_le lexI leIr lexx andbT. Qed.

Lemma meet_idPl {x y} : reflect (x `&` y = x) (x <= y).

Proof.

by rewrite -eq_meetl; apply/eqP. Qed.

Lemma meet_idPr {x y} : reflect (y `&` x = x) (x <= y).

Proof.

by rewrite -eq_meetr; apply/eqP. Qed.

Lemma meet_l x y : x <= y -> x `&` y = x

Proof.

exact/meet_idPl. Qed.

Lemma meet_r x y : y <= x -> x `&` y = y

Proof.

exact/meet_idPr. Qed.

Lemma leIidl x y : (x <= x `&` y) = (x <= y).

Proof.

by rewrite lexI lexx. Qed.

Lemma leIidr x y : (x <= y `&` x) = (x <= y).

Proof.

by rewrite lexI lexx andbT. Qed.

Lemma leI2 x y z t : x <= z -> y <= t -> x `&` y <= z `&` t.

Proof.

by move=> xz yt; rewrite lexI !leIx2 ?xz ?yt ?orbT //. Qed.

Lemma meetC : commutative (@meet _ L).

Proof.

by move=> x y; apply: le_anti; rewrite !lexI !leIr !leIl. Qed.

Lemma meetA : associative (@meet _ L).

Proof.

move=> x y z; apply: le_anti.
rewrite !lexI leIr leIl /= andbT -andbA.
rewrite ![_ `&` (_ `&` _) <= _]leIxr ?(leIr, leIl) //=.
by rewrite leIxl ?leIl // leIxl // leIr.
Qed.

HB.instance Definition _ := SemiGroup.isComLaw.Build L meet meetA meetC.

Lemma meetxx : idempotent_op (@meet _ L).

Proof.

by move=> x; apply/eqP; rewrite -leEmeet. Qed.

Lemma meetAC : right_commutative (@meet _ L).

Proof.

by move=> x y z; rewrite -!meetA [X in _ `&` X]meetC. Qed.

Lemma meetCA : left_commutative (@meet _ L).

Proof.

by move=> x y z; rewrite !meetA [X in X `&` _]meetC. Qed.

Lemma meetACA : interchange (@meet _ L) (@meet _ L).

Proof.

by move=> x y z t; rewrite !meetA [X in X `&` _]meetAC. Qed.

Lemma meetKI y x : x `&` (x `&` y) = x `&` y.

Proof.

by rewrite meetA meetxx. Qed.

Lemma meetIK y x : (x `&` y) `&` y = x `&` y.

Proof.

by rewrite -meetA meetxx. Qed.

Lemma meetKIC y x : x `&` (y `&` x) = x `&` y.

Proof.

by rewrite meetC meetIK meetC. Qed.

Lemma meetIKC y x : y `&` x `&` y = x `&` y.

Proof.

by rewrite meetAC meetC meetxx. Qed.

End MeetTheory.
End MeetTheory.

Arguments meet_idPl {disp L x y}.
Arguments meet_idPr {disp L x y}.

Module Import BMeetTheory.
Section BMeetTheory.
Context {disp : disp_t} {L : bMeetSemilatticeType disp}.

Lemma meet0x : left_zero \bot (@meet _ L).

Proof.

by move=> x; apply/eqP; rewrite -leEmeet. Qed.

Lemma meetx0 : right_zero \bot (@meet _ L).

Proof.

by move=> x; rewrite meetC meet0x. Qed.

HB.instance Definition _ := Monoid.isMulLaw.Build L \bot meet meet0x meetx0.

End BMeetTheory.
End BMeetTheory.

Module Import TMeetTheory.
Section TMeetTheory.
Context {disp : disp_t} {L : tMeetSemilatticeType disp}.
Implicit Types (I : finType) (T : eqType) (x y : L).

Lemma meetx1 : right_id \top (@meet _ L).

Proof.

by move=> x; apply/eqP; rewrite -leEmeet. Qed.

Lemma meet1x : left_id \top (@meet _ L).

Proof.

by move=> x; apply/eqP; rewrite meetC meetx1. Qed.

Lemma meet_eq1 x y : (x `&` y == \top) = (x == \top) && (y == \top).

Proof.

apply/idP/idP; last by move=> /andP[/eqP-> /eqP->]; rewrite meetx1.
by move=> /eqP xIy1; rewrite -!le1x -xIy1 leIl leIr.
Qed.

HB.instance Definition _ := Monoid.isMonoidLaw.Build L \top meet meet1x meetx1.

Lemma meets_inf_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) :
x \in r -> P x -> \meet_(i <- r | P i) F i <= F x.

Proof.

by move=> xr Px; rewrite (big_rem x) ?Px //= leIl. Qed.

Lemma meets_max_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) (u : L) :
x \in r -> P x -> F x <= u -> \meet_(x <- r | P x) F x <= u.

Proof.

by move=> ? ?; apply/le_trans/meets_inf_seq. Qed.

Lemma meets_inf I (j : I) (P : {pred I}) (F : I -> L) :
P j -> \meet_(i | P i) F i <= F j.

Proof.

exact: meets_inf_seq. Qed.

Lemma meets_max I (j : I) (u : L) (P : {pred I}) (F : I -> L) :
P j -> F j <= u -> \meet_(i | P i) F i <= u.

Proof.

exact: meets_max_seq. Qed.

Lemma meets_ge J (r : seq J) (P : {pred J}) (F : J -> L) (u : L) :
(forall x : J, P x -> u <= F x) -> u <= \meet_(x <- r | P x) F x.

Proof.

by move=> leFm; elim/big_rec: _ => // i x Px xu; rewrite lexI leFm. Qed.

Lemma meetsP_seq T (r : seq T) (P : {pred T}) (F : T -> L) (l : L) :
reflect (forall x : T, x \in r -> P x -> l <= F x)
(l <= \meet_(x <- r | P x) F x).

Proof.

apply: (iffP idP) => leFm => [x xr Px|].
exact/(le_trans leFm)/meets_inf_seq.
by rewrite big_seq_cond meets_ge// => x /andP[/leFm].
Qed.

Lemma meetsP I (l : L) (P : {pred I}) (F : I -> L) :
reflect (forall i : I, P i -> l <= F i) (l <= \meet_(i | P i) F i).

Proof.

by apply: (iffP (meetsP_seq _ _ _ _)) => H ? ?; apply: H. Qed.

Lemma le_meets I (A B : {set I}) (F : I -> L) :
A \subset B -> \meet_(i in B) F i <= \meet_(i in A) F i.

Proof.

by move=> /subsetP AB; apply/meetsP => i iA; apply/meets_inf/AB. Qed.

Lemma meets_setU I (A B : {set I}) (F : I -> L) :
\meet_(i in (A :|: B)) F i = \meet_(i in A) F i `&` \meet_(i in B) F i.

Proof.

rewrite -!big_enum; have /= <- := @big_cat _ _ meet.
apply/eq_big_idem; first exact: meetxx.
by move=> ?; rewrite mem_cat !fintype.mem_enum inE.
Qed.

Lemma meets_seq I (r : seq I) (F : I -> L) :
\meet_(i <- r) F i = \meet_(i in r) F i.

Proof.

by rewrite -big_enum; apply/eq_big_idem => ?; rewrite /= ?meetxx ?fintype.mem_enum.
Qed.

End TMeetTheory.
End TMeetTheory.

Module Import JoinTheory.
Section JoinTheory.
Context {disp : disp_t} {L : joinSemilatticeType disp}.
Implicit Types (x y : L).

Lemma leUx x y z : (x `|` y <= z) = (x <= z) && (y <= z).

Proof.

exact: leUx. Qed.

Lemma leUr x y : x <= y `|` x

Proof.

exact: (@leIr _ L^d). Qed.

Lemma leUl x y : x <= x `|` y

Proof.

exact: (@leIl _ L^d). Qed.

Lemma lexUl x y z : x <= y -> x <= y `|` z.

Proof.

exact: (@leIxl _ L^d). Qed.

Lemma lexUr x y z : x <= z -> x <= y `|` z.

Proof.

exact: (@leIxr _ L^d). Qed.

Lemma lexU2 x y z : (x <= y) || (x <= z) -> x <= y `|` z.

Proof.

exact: (@leIx2 _ L^d). Qed.

Lemma leEjoin x y : (x <= y) = (x `|` y == y).

Proof.

by rewrite [LHS](@leEmeet _ L^d) meetC. Qed.

Lemma eq_joinl x y : (x `|` y == x) = (y <= x).

Proof.

exact: (@eq_meetl _ L^d). Qed.

Lemma eq_joinr x y : (x `|` y == y) = (x <= y).

Proof.

exact: (@eq_meetr _ L^d). Qed.

Lemma join_idPl {x y} : reflect (y `|` x = y) (x <= y).

Proof.

exact: (@meet_idPl _ L^d). Qed.

Lemma join_idPr {x y} : reflect (x `|` y = y) (x <= y).

Proof.

exact: (@meet_idPr _ L^d). Qed.

Lemma join_l x y : y <= x -> x `|` y = x

Proof.

exact/join_idPl. Qed.

Lemma join_r x y : x <= y -> x `|` y = y

Proof.

exact/join_idPr. Qed.

Lemma leUidl x y : (x `|` y <= y) = (x <= y).

Proof.

exact: (@leIidr _ L^d). Qed.

Lemma leUidr x y : (y `|` x <= y) = (x <= y).

Proof.

exact: (@leIidl _ L^d). Qed.

Lemma leU2 x y z t : x <= z -> y <= t -> x `|` y <= z `|` t.

Proof.

exact: (@leI2 _ L^d). Qed.

Lemma joinC : commutative (@join _ L)

Proof.

exact: (@meetC _ L^d). Qed.

Lemma joinA : associative (@join _ L)

Proof.

exact: (@meetA _ L^d). Qed.

HB.instance Definition _ := SemiGroup.isComLaw.Build L join joinA joinC.

Lemma joinxx : idempotent_op (@join _ L).

Proof.

exact: (@meetxx _ L^d). Qed.

Lemma joinAC : right_commutative (@join _ L).

Proof.

exact: (@meetAC _ L^d). Qed.

Lemma joinCA : left_commutative (@join _ L).

Proof.

exact: (@meetCA _ L^d). Qed.

Lemma joinACA : interchange (@join _ L) (@join _ L).

Proof.

exact: (@meetACA _ L^d). Qed.

Lemma joinKU y x : x `|` (x `|` y) = x `|` y.

Proof.

exact: (@meetKI _ L^d). Qed.

Lemma joinUK y x : (x `|` y) `|` y = x `|` y.

Proof.

exact: (@meetIK _ L^d). Qed.

Lemma joinKUC y x : x `|` (y `|` x) = x `|` y.

Proof.

exact: (@meetKIC _ L^d). Qed.

Lemma joinUKC y x : y `|` x `|` y = x `|` y.

Proof.

exact: (@meetIKC _ L^d). Qed.

End JoinTheory.
End JoinTheory.

Arguments join_idPl {disp L x y}.
Arguments join_idPr {disp L x y}.

Module Import BJoinTheory.
Section BJoinTheory.
Context {disp : disp_t} {L : bJoinSemilatticeType disp}.
Implicit Types (I : finType) (T : eqType) (x y : L).

Lemma joinx0 : right_id \bot (@join _ L).

Proof.

exact: (@meetx1 _ L^d). Qed.

Lemma join0x : left_id \bot (@join _ L).

Proof.

exact: (@meet1x _ L^d). Qed.

Lemma join_eq0 x y : (x `|` y == \bot) = (x == \bot) && (y == \bot).

Proof.

exact: (@meet_eq1 _ L^d). Qed.

HB.instance Definition _ := Monoid.isMonoidLaw.Build L \bot join join0x joinx0.

Lemma joins_sup_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) :
x \in r -> P x -> F x <= \join_(i <- r | P i) F i.

Proof.

exact: (@meets_inf_seq _ L^d). Qed.

Lemma joins_min_seq T (r : seq T) (P : {pred T}) (F : T -> L) (x : T) (l : L) :
x \in r -> P x -> l <= F x -> l <= \join_(x <- r | P x) F x.

Proof.

exact: (@meets_max_seq _ L^d). Qed.

Lemma joins_sup I (j : I) (P : {pred I}) (F : I -> L) :
P j -> F j <= \join_(i | P i) F i.

Proof.

exact: (@meets_inf _ L^d). Qed.

Lemma joins_min I (j : I) (l : L) (P : {pred I}) (F : I -> L) :
P j -> l <= F j -> l <= \join_(i | P i) F i.

Proof.

exact: (@meets_max _ L^d). Qed.

Lemma joins_le J (r : seq J) (P : {pred J}) (F : J -> L) (u : L) :
(forall x : J, P x -> F x <= u) -> \join_(x <- r | P x) F x <= u.

Proof.

exact: (@meets_ge _ L^d). Qed.

Lemma joinsP_seq T (r : seq T) (P : {pred T}) (F : T -> L) (u : L) :
reflect (forall x : T, x \in r -> P x -> F x <= u)
(\join_(x <- r | P x) F x <= u).

Proof.

exact: (@meetsP_seq _ L^d). Qed.

Lemma joinsP I (u : L) (P : {pred I}) (F : I -> L) :
reflect (forall i : I, P i -> F i <= u) (\join_(i | P i) F i <= u).

Proof.

exact: (@meetsP _ L^d). Qed.

Lemma le_joins I (A B : {set I}) (F : I -> L) :
A \subset B -> \join_(i in A) F i <= \join_(i in B) F i.

Proof.

exact: (@le_meets _ L^d). Qed.

Lemma joins_setU I (A B : {set I}) (F : I -> L) :
\join_(i in (A :|: B)) F i = \join_(i in A) F i `|` \join_(i in B) F i.

Proof.

exact: (@meets_setU _ L^d). Qed.

Lemma joins_seq I (r : seq I) (F : I -> L) :
\join_(i <- r) F i = \join_(i in r) F i.

Proof.

exact: (@meets_seq _ L^d). Qed.

End BJoinTheory.
End BJoinTheory.

Module Import TJoinTheory.
Section TJoinTheory.
Context {disp : disp_t} {L : tJoinSemilatticeType disp}.

Lemma joinx1 : right_zero \top (@join _ L)

Proof.

exact: (@meetx0 _ L^d). Qed.

Lemma join1x : left_zero \top (@join _ L)

Proof.

exact: (@meet0x _ L^d). Qed.

HB.instance Definition _ := Monoid.isMulLaw.Build L \top join join1x joinx1.

End TJoinTheory.
End TJoinTheory.

Module Import LatticeTheory.
Section LatticeTheory.
Context {disp : disp_t} {L : latticeType disp}.
Implicit Types (x y : L).

Lemma meetUK x y : (x `&` y) `|` y = y

Proof.

exact/join_idPr/leIr. Qed.

Lemma meetUKC x y : (y `&` x) `|` y = y

Proof.

by rewrite meetC meetUK. Qed.

Lemma meetKUC y x : x `|` (y `&` x) = x

Proof.

by rewrite joinC meetUK. Qed.

Lemma meetKU y x : x `|` (x `&` y) = x

Proof.

by rewrite meetC meetKUC. Qed.

Lemma joinIK x y : (x `|` y) `&` y = y

Proof.

exact/meet_idPr/leUr. Qed.

Lemma joinIKC x y : (y `|` x) `&` y = y

Proof.

by rewrite joinC joinIK. Qed.

Lemma joinKIC y x : x `&` (y `|` x) = x

Proof.

by rewrite meetC joinIK. Qed.

Lemma joinKI y x : x `&` (x `|` y) = x

Proof.

by rewrite joinC joinKIC. Qed.

Lemma lcomparableP x y : incomparel x y
  (min y x) (min x y) (max y x) (max x y)
  (y `&` x) (x `&` y) (y `|` x) (x `|` y)
  (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y) (y >=< x) (x >=< y).

Proof.

by case: (comparableP x) => [hxy|hxy|hxy|->]; do 1?have hxy' := ltW hxy;
   rewrite ?(meetxx, joinxx);
   rewrite ?(meet_l hxy', meet_r hxy', join_l hxy', join_r hxy');
   constructor.
Qed.

Lemma lcomparable_ltgtP x y : x >=< y ->
  comparel x y (min y x) (min x y) (max y x) (max x y)
               (y `&` x) (x `&` y) (y `|` x) (x `|` y)
               (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y).

Proof.

by case: (lcomparableP x) => // *; constructor. Qed.

Lemma lcomparable_leP x y : x >=< y ->
lel_xor_gt x y (min y x) (min x y) (max y x) (max x y)
(y `&` x) (x `&` y) (y `|` x) (x `|` y) (x <= y) (y < x).

Proof.

by move/lcomparable_ltgtP => [/ltW xy|xy|->]; constructor. Qed.

Lemma lcomparable_ltP x y : x >=< y ->
ltl_xor_ge x y (min y x) (min x y) (max y x) (max x y)
(y `&` x) (x `&` y) (y `|` x) (x `|` y) (y <= x) (x < y).

Proof.

by move=> /lcomparable_ltgtP [xy|/ltW xy|->]; constructor. Qed.

End LatticeTheory.
End LatticeTheory.

Module Import DistrLatticeTheory.
Section DistrLatticeTheory.
Context {disp : disp_t} {L : distrLatticeType disp}.

Lemma meetUl : left_distributive (@meet _ L) (@join _ L).

Proof.

exact: meetUl. Qed.

Lemma meetUr : right_distributive (@meet _ L) (@join _ L).

Proof.

by move=> x y z; rewrite ![x `&` _]meetC meetUl. Qed.

Lemma joinIl : left_distributive (@join _ L) (@meet _ L).

Proof.

exact: joinIl. Qed.

Lemma joinIr : right_distributive (@join _ L) (@meet _ L).

Proof.

by move=> x y z; rewrite ![x `|` _]joinC joinIl. Qed.

#[warning="-HB.no-new-instance"]
HB.instance Definition _ := Monoid.isAddLaw.Build L meet join meetUl meetUr.
HB.instance Definition _ := Monoid.isAddLaw.Build L join meet joinIl joinIr.

End DistrLatticeTheory.
End DistrLatticeTheory.

Module Import BDistrLatticeTheory.
Section BDistrLatticeTheory.
Context {disp : disp_t} {L : bDistrLatticeType disp}.
Implicit Types (x y z : L).

Lemma leU2l_le y t x z : x `&` t = \bot -> x `|` y <= z `|` t -> x <= z.

Proof.

by move=> xIt0 /(leI2 (lexx x)); rewrite joinKI meetUr xIt0 joinx0 leIidl.
Qed.

Lemma leU2r_le y t x z : x `&` t = \bot -> y `|` x <= t `|` z -> x <= z.

Proof.

by rewrite joinC [_ `|` z]joinC => /leU2l_le H /H. Qed.

Lemma disjoint_lexUl z x y : x `&` z = \bot -> (x <= y `|` z) = (x <= y).

Proof.

move=> xz0; apply/idP/idP=> xy; last by rewrite lexU2 ?xy.
by apply: (@leU2l_le x z); rewrite ?joinxx.
Qed.

Lemma disjoint_lexUr z x y : x `&` z = \bot -> (x <= z `|` y) = (x <= y).

Proof.

by move=> xz0; rewrite joinC; rewrite disjoint_lexUl. Qed.

Lemma leU2E x y z t : x `&` t = \bot -> y `&` z = \bot ->
(x `|` y <= z `|` t) = (x <= z) && (y <= t).

Proof.

move=> dxt dyz; apply/idP/andP; last by case=> ? ?; exact: leU2.
by move=> lexyzt; rewrite (leU2l_le _ lexyzt) // (leU2r_le _ lexyzt).
Qed.

Lemma joins_disjoint (I : finType) (d : L) (P : {pred I}) (F : I -> L) :
(forall i : I, P i -> d `&` F i = \bot) -> d `&` \join_(i | P i) F i = \bot.

Proof.

move=> d_Fi_disj; have : \big[andb/true]_(i | P i) (d `&` F i == \bot).
rewrite big_all_cond; apply/allP => i _ /=.
by apply/implyP => /d_Fi_disj ->.
elim/big_rec2: _ => [|i y]; first by rewrite meetx0.
case; rewrite (andbF, andbT) // => Pi /(_ isT) dy /eqP dFi.
by rewrite meetUr dy dFi joinxx.
Qed.

End BDistrLatticeTheory.
End BDistrLatticeTheory.

Module Import TDistrLatticeTheory.
Section TDistrLatticeTheory.
Context {disp : disp_t} {L : tDistrLatticeType disp}.
Implicit Types (x y : L).

Lemma leI2l_le y t x z : y `|` z = \top -> x `&` y <= z `&` t -> x <= z.

Proof.

by rewrite joinC; exact: (@leU2l_le _ L^d). Qed.

Lemma leI2r_le y t x z : y `|` z = \top -> y `&` x <= t `&` z -> x <= z.

Proof.

by rewrite joinC; exact: (@leU2r_le _ L^d). Qed.

Lemma cover_leIxl z x y : z `|` y = \top -> (x `&` z <= y) = (x <= y).

Proof.

by rewrite joinC; exact: (@disjoint_lexUl _ L^d). Qed.

Lemma cover_leIxr z x y : z `|` y = \top -> (z `&` x <= y) = (x <= y).

Proof.

by rewrite joinC; exact: (@disjoint_lexUr _ L^d). Qed.

Lemma leI2E x y z t : x `|` t = \top -> y `|` z = \top ->
(x `&` y <= z `&` t) = (x <= z) && (y <= t).

Proof.

by move=> ? ?; apply: (@leU2E _ L^d); rewrite meetC. Qed.

Lemma meets_total (I : finType) (d : L) (P : {pred I}) (F : I -> L) :
(forall i : I, P i -> d `|` F i = \top) -> d `|` \meet_(i | P i) F i = \top.

Proof.

exact: (@joins_disjoint _ L^d). Qed.

End TDistrLatticeTheory.
End TDistrLatticeTheory.

Module Import TotalTheory.
Section TotalTheory.
Context {disp : disp_t} {T : orderType disp}.
Implicit Types (x y z t : T) (s : seq T).

Definition le_total : total (<=%O : rel T) := le_total.
Hint Resolve le_total : core.

Lemma ge_total : total (>=%O : rel T).

Proof.

by move=> ? ?; apply: le_total. Qed.

Hint Resolve ge_total : core.

Lemma comparableT x y : x >=< y

Proof.

exact: le_total. Qed.

Hint Extern 0 (is_true (_ >=< _)%O) => solve [apply: comparableT] : core.

Lemma sort_le_sorted s : sorted <=%O (sort <=%O s).

Proof.

exact: sort_sorted. Qed.

Hint Resolve sort_le_sorted : core.

Lemma sort_lt_sorted s : sorted <%O (sort <=%O s) = uniq s.

Proof.

by rewrite lt_sorted_uniq_le sort_uniq sort_le_sorted andbT. Qed.

Lemma perm_sort_leP s1 s2 : reflect (sort <=%O s1 = sort <=%O s2) (perm_eq s1 s2).

Proof.

exact/perm_sortP/le_anti/le_trans/le_total. Qed.

Lemma filter_sort_le p s : filter p (sort <=%O s) = sort <=%O (filter p s).

Proof.

exact/filter_sort/le_trans/le_total. Qed.

Lemma mask_sort_le s (m : bitseq) :
{m_s : bitseq | mask m_s (sort <=%O s) = sort <=%O (mask m s)}.

Proof.

exact/mask_sort/le_trans/le_total. Qed.

Lemma sorted_mask_sort_le s (m : bitseq) :
sorted <=%O (mask m s) -> {m_s : bitseq | mask m_s (sort <=%O s) = mask m s}.

Proof.

exact/sorted_mask_sort/le_trans/le_total. Qed.

Lemma subseq_sort_le : {homo sort <=%O : s1 s2 / @subseq T s1 s2}.

Proof.

exact/subseq_sort/le_trans/le_total. Qed.

Lemma sorted_subseq_sort_le s1 s2 :
subseq s1 s2 -> sorted <=%O s1 -> subseq s1 (sort <=%O s2).

Proof.

exact/sorted_subseq_sort/le_trans/le_total. Qed.

Lemma mem2_sort_le s x y : x <= y -> mem2 s x y -> mem2 (sort <=%O s) x y.

Proof.

exact/mem2_sort/le_trans/le_total. Qed.

Lemma leNgt x y : (x <= y) = ~~ (y < x)

Proof.

exact: comparable_leNgt. Qed.

Lemma ltNge x y : (x < y) = ~~ (y <= x)

Proof.

exact: comparable_ltNge. Qed.

Definition ltgtP x y := LatticeTheory.lcomparable_ltgtP (comparableT x y).
Definition leP x y := LatticeTheory.lcomparable_leP (comparableT x y).
Definition ltP x y := LatticeTheory.lcomparable_ltP (comparableT x y).

Lemma wlog_le P :
(forall x y, P y x -> P x y) -> (forall x y, x <= y -> P x y) ->
forall x y, P x y.

Proof.

by move=> sP hP x y; case: (leP x y) => [| /ltW] /hP // /sP. Qed.

Lemma wlog_lt P :
    (forall x, P x x) ->
    (forall x y, (P y x -> P x y)) -> (forall x y, x < y -> P x y) ->
  forall x y, P x y.

Proof.

by move=> rP sP hP x y; case: (ltgtP x y) => [||->] // /hP // /sP. Qed.

Lemma neq_lt x y : (x != y) = (x < y) || (y < x)

Proof.

by case: ltgtP. Qed.

Lemma lt_total x y : x != y -> (x < y) || (y < x)

Proof.

by case: ltgtP. Qed.

Lemma eq_leLR x y z t :
(x <= y -> z <= t) -> (y < x -> t < z) -> (x <= y) = (z <= t).

Proof.

by rewrite !ltNge => ? /contraTT ?; apply/idP/idP. Qed.

Lemma eq_leRL x y z t :
(x <= y -> z <= t) -> (y < x -> t < z) -> (z <= t) = (x <= y).

Proof.

by move=> *; apply/esym/eq_leLR. Qed.

Lemma eq_ltLR x y z t :
(x < y -> z < t) -> (y <= x -> t <= z) -> (x < y) = (z < t).

Proof.

by rewrite !leNgt => ? /contraTT ?; apply/idP/idP. Qed.

Lemma eq_ltRL x y z t :
(x < y -> z < t) -> (y <= x -> t <= z) -> (z < t) = (x < y).

Proof.

by move=> *; apply/esym/eq_ltLR. Qed.

Lemma le_gtP {x y} : reflect (forall z, z < x -> z < y) (x <= y).

Proof.

by apply: (iffP idP) => [xy z /lt_le_trans|/(_ y)/[!ltNge]/contraTT]; apply.
Qed.

Lemma le_ltP {x y} : reflect (forall z, y < z -> x < z) (x <= y).

Proof.

by apply: (iffP idP) => [xy z /(le_lt_trans _)|/(_ x)/[!ltNge]/contraTT]; apply.
Qed.

Lemma eq_gtP {x y} : reflect (forall z, (z < x) = (z < y)) (x == y).

Proof.

by apply: (iffP eq_leP) => + k => /(_ k)/[!ltNge]/(congr1 negb); rewrite ?negbK.
Qed.

Lemma eq_ltP {x y} : reflect (forall z, (x < z) = (y < z)) (x == y).

Proof.

by apply: (iffP eq_geP) => + k => /(_ k)/[!ltNge]/(congr1 negb); rewrite ?negbK.
Qed.

Lemma meetEtotal x y : x `&` y = min x y

Proof.

by case: leP. Qed.

Lemma joinEtotal x y : x `|` y = max x y

Proof.

by case: leP. Qed.

Lemma minEgt x y : min x y = if x > y then y else x

Proof.

by case: ltP. Qed.

Lemma maxEgt x y : max x y = if x > y then x else y

Proof.

by case: ltP. Qed.

Lemma minEge x y : min x y = if x >= y then y else x

Proof.

by case: leP. Qed.

Lemma maxEge x y : max x y = if x >= y then x else y

Proof.

by case: leP. Qed.

Lemma minC : commutative (min : T -> T -> T).

Proof.

by move=> x y; apply: comparable_minC. Qed.

Lemma maxC : commutative (max : T -> T -> T).

Proof.

by move=> x y; apply: comparable_maxC. Qed.

Lemma minA : associative (min : T -> T -> T).

Proof.

by move=> x y z; apply: comparable_minA. Qed.

Lemma maxA : associative (max : T -> T -> T).

Proof.

by move=> x y z; apply: comparable_maxA. Qed.

Lemma minAC : right_commutative (min : T -> T -> T).

Proof.

by move=> x y z; apply: comparable_minAC. Qed.

Lemma maxAC : right_commutative (max : T -> T -> T).

Proof.

by move=> x y z; apply: comparable_maxAC. Qed.

Lemma minCA : left_commutative (min : T -> T -> T).

Proof.

by move=> x y z; apply: comparable_minCA. Qed.

Lemma maxCA : left_commutative (max : T -> T -> T).

Proof.

by move=> x y z; apply: comparable_maxCA. Qed.

Lemma minACA : interchange (min : T -> T -> T) min.

Proof.

by move=> x y z t; apply: comparable_minACA. Qed.

Lemma maxACA : interchange (max : T -> T -> T) max.

Proof.

by move=> x y z t; apply: comparable_maxACA. Qed.

Lemma eq_minr x y : (min x y == y) = (y <= x).

Proof.

exact: comparable_eq_minr. Qed.

Lemma eq_maxl x y : (max x y == x) = (y <= x).

Proof.

exact: comparable_eq_maxl. Qed.

Lemma min_idPr x y : reflect (min x y = y) (y <= x).

Proof.

exact: comparable_min_idPr. Qed.

Lemma max_idPl x y : reflect (max x y = x) (y <= x).

Proof.

exact: comparable_max_idPl. Qed.

Lemma le_min z x y : (z <= min x y) = (z <= x) && (z <= y).

Proof.

exact: comparable_le_min. Qed.

Lemma ge_min z x y : (min x y <= z) = (x <= z) || (y <= z).

Proof.

exact: comparable_ge_min. Qed.

Lemma lt_min z x y : (z < min x y) = (z < x) && (z < y).

Proof.

exact: comparable_lt_min. Qed.

Lemma gt_min z x y : (min x y < z) = (x < z) || (y < z).

Proof.

exact: comparable_gt_min. Qed.

Lemma le_max z x y : (z <= max x y) = (z <= x) || (z <= y).

Proof.

exact: comparable_le_max. Qed.

Lemma ge_max z x y : (max x y <= z) = (x <= z) && (y <= z).

Proof.

exact: comparable_ge_max. Qed.

Lemma lt_max z x y : (z < max x y) = (z < x) || (z < y).

Proof.

exact: comparable_lt_max. Qed.

Lemma gt_max z x y : (max x y < z) = (x < z) && (y < z).

Proof.

exact: comparable_gt_max. Qed.

Lemma minxK x y : max (min x y) y = y

Proof.

exact: comparable_minxK. Qed.

Lemma minKx x y : max x (min x y) = x

Proof.

exact: comparable_minKx. Qed.

Lemma maxxK x y : min (max x y) y = y

Proof.

exact: comparable_maxxK. Qed.

Lemma maxKx x y : min x (max x y) = x

Proof.

exact: comparable_maxKx. Qed.

Lemma max_minl : left_distributive (max : T -> T -> T) min.

Proof.

by move=> x y z; apply: comparable_max_minl. Qed.

Lemma min_maxl : left_distributive (min : T -> T -> T) max.

Proof.

by move=> x y z; apply: comparable_min_maxl. Qed.

Lemma max_minr : right_distributive (max : T -> T -> T) min.

Proof.

by move=> x y z; apply: comparable_max_minr. Qed.

Lemma min_maxr : right_distributive (min : T -> T -> T) max.

Proof.

by move=> x y z; apply: comparable_min_maxr. Qed.

HB.instance Definition _ := SemiGroup.isComLaw.Build T max maxA maxC.
HB.instance Definition _ := SemiGroup.isComLaw.Build T min minA minC.

Lemma leIx x y z : (meet y z <= x) = (y <= x) || (z <= x).

Proof.

by rewrite meetEtotal ge_min. Qed.

Lemma lexU x y z : (x <= join y z) = (x <= y) || (x <= z).

Proof.

by rewrite joinEtotal le_max. Qed.

Lemma ltxI x y z : (x < meet y z) = (x < y) && (x < z).

Proof.

by rewrite !ltNge leIx negb_or. Qed.

Lemma ltIx x y z : (meet y z < x) = (y < x) || (z < x).

Proof.

by rewrite !ltNge lexI negb_and. Qed.

Lemma ltxU x y z : (x < join y z) = (x < y) || (x < z).

Proof.

by rewrite !ltNge leUx negb_and. Qed.

Lemma ltUx x y z : (join y z < x) = (y < x) && (z < x).

Proof.

by rewrite !ltNge lexU negb_or. Qed.

Definition ltexI := (@lexI _ T, ltxI).
Definition lteIx := (leIx, ltIx).
Definition ltexU := (lexU, ltxU).
Definition lteUx := (@leUx _ T, ltUx).

Lemma le_min2 x y z t : x <= z -> y <= t -> Order.min x y <= Order.min z t.

Proof.

exact: comparable_le_min2. Qed.

Lemma le_max2 x y z t : x <= z -> y <= t -> Order.max x y <= Order.max z t.

Proof.

exact: comparable_le_max2. Qed.

Lemma lteifNE x y C : (x < y ?<= if ~~ C) = ~~ (y < x ?<= if C).

Proof.

by case: C => /=; case: leP. Qed.

Lemma lteif_minr z x y C :
(z < min x y ?<= if C) = (z < x ?<= if C) && (z < y ?<= if C).

Proof.

by case: C; rewrite /= (le_min, lt_min). Qed.

Lemma lteif_minl z x y C :
(min x y < z ?<= if C) = (x < z ?<= if C) || (y < z ?<= if C).

Proof.

by case: C; rewrite /= (ge_min, gt_min). Qed.

Lemma lteif_maxr z x y C :
(z < max x y ?<= if C) = (z < x ?<= if C) || (z < y ?<= if C).

Proof.

by case: C; rewrite /= (le_max, lt_max). Qed.

Lemma lteif_maxl z x y C :
(max x y < z ?<= if C) = (x < z ?<= if C) && (y < z ?<= if C).

Proof.

by case: C; rewrite /= (ge_max, gt_max). Qed.

Section ArgExtremum.
Context (I : finType) (i0 : I) (P : {pred I}) (F : I -> T) (Pi0 : P i0).

Lemma arg_minP: extremum_spec <=%O P F (arg_min i0 P F).

Proof.

by apply: extremumP => //; apply: le_trans. Qed.

Lemma arg_maxP: extremum_spec >=%O P F (arg_max i0 P F).

Proof.

by apply: extremumP => //; [apply: ge_refl | apply: ge_trans]. Qed.

End ArgExtremum.

Lemma count_le_gt x s : count (<= x) s = size s - count (> x) s.

Proof.

by rewrite -(count_predC (> x)) addKn; apply: eq_count => y; rewrite /= leNgt.
Qed.

Lemma count_lt_ge x s : count (< x) s = size s - count (>= x) s.

Proof.

by rewrite -(count_predC (>= x)) addKn; apply: eq_count => y; rewrite /= ltNge.
Qed.

Section bigminmax_Type.
Context (I : Type) (r : seq I) (x : T).
Implicit Types (P : pred I) (F : I -> T).

Lemma bigmin_mkcond P F : \big[min/x]_(i <- r | P i) F i =
\big[min/x]_(i <- r) (if P i then F i else x).

Proof.

by rewrite big_mkcond_idem //= minxx. Qed.

Lemma bigmax_mkcond P F :
\big[max/x]_(i <- r | P i) F i = \big[max/x]_(i <- r) if P i then F i else x.

Proof.

by rewrite big_mkcond_idem //= maxxx. Qed.

Lemma bigmin_mkcondl P Q F :
\big[min/x]_(i <- r | P i && Q i) F i
= \big[min/x]_(i <- r | Q i) if P i then F i else x.

Proof.

rewrite bigmin_mkcond [RHS]bigmin_mkcond.
by apply: eq_bigr => i _; case: P; case: Q.
Qed.

Lemma bigmin_mkcondr P Q F :
\big[min/x]_(i <- r | P i && Q i) F i
= \big[min/x]_(i <- r | P i) if Q i then F i else x.

Proof.

by under eq_bigl do rewrite andbC; apply: bigmin_mkcondl. Qed.

Lemma bigmax_mkcondl P Q F :
\big[max/x]_(i <- r | P i && Q i) F i
= \big[max/x]_(i <- r | Q i) if P i then F i else x.

Proof.

rewrite bigmax_mkcond [RHS]bigmax_mkcond.
by apply: eq_bigr => i _; case: P; case: Q.
Qed.

Lemma bigmax_mkcondr P Q F :
\big[max/x]_(i <- r | P i && Q i) F i
= \big[max/x]_(i <- r | P i) if Q i then F i else x.

Proof.

by under eq_bigl do rewrite andbC; apply: bigmax_mkcondl. Qed.

Lemma bigmin_split P F1 F2 :
\big[min/x]_(i <- r | P i) (min (F1 i) (F2 i)) =
min (\big[min/x]_(i <- r | P i) F1 i) (\big[min/x]_(i <- r | P i) F2 i).

Proof.

by rewrite big_split_idem //= minxx. Qed.

Lemma bigmax_split P F1 F2 :
\big[max/x]_(i <- r | P i) (max (F1 i) (F2 i)) =
max (\big[max/x]_(i <- r | P i) F1 i) (\big[max/x]_(i <- r | P i) F2 i).

Proof.

by rewrite big_split_idem //= maxxx. Qed.

Lemma bigmin_idl P F :
\big[min/x]_(i <- r | P i) F i = min x (\big[min/x]_(i <- r | P i) F i).

Proof.

by rewrite minC big_id_idem //= minxx. Qed.

Lemma bigmax_idl P F :
\big[max/x]_(i <- r | P i) F i = max x (\big[max/x]_(i <- r | P i) F i).

Proof.

by rewrite maxC big_id_idem //= maxxx. Qed.

Lemma bigmin_idr P F :
\big[min/x]_(i <- r | P i) F i = min (\big[min/x]_(i <- r | P i) F i) x.

Proof.

by rewrite [LHS]bigmin_idl minC. Qed.

Lemma bigmax_idr P F :
\big[max/x]_(i <- r | P i) F i = max (\big[max/x]_(i <- r | P i) F i) x.

Proof.

by rewrite [LHS]bigmax_idl maxC. Qed.

Lemma bigminID a P F : \big[min/x]_(i <- r | P i) F i =
min (\big[min/x]_(i <- r | P i && a i) F i)
(\big[min/x]_(i <- r | P i && ~~ a i) F i).

Proof.

by rewrite (bigID_idem _ _ a) //= minxx. Qed.

Lemma bigmaxID a P F : \big[max/x]_(i <- r | P i) F i =
max (\big[max/x]_(i <- r | P i && a i) F i)
(\big[max/x]_(i <- r | P i && ~~ a i) F i).

Proof.

by rewrite (bigID_idem _ _ a) //= maxxx. Qed.

End bigminmax_Type.

Let ge_min_id (x y : T) : x >= min x y

Proof.

by rewrite ge_min lexx. Qed.

Let le_max_id (x y : T) : x <= max x y

Proof.

by rewrite le_max lexx. Qed.

Lemma sub_bigmin [x0] I r (P P' : {pred I}) (F : I -> T) :
(forall i, P' i -> P i) ->
\big[min/x0]_(i <- r | P i) F i <= \big[min/x0]_(i <- r | P' i) F i.

Proof.

exact: (sub_le_big ge_refl). Qed.

Lemma sub_bigmax [x0] I r (P P' : {pred I}) (F : I -> T) :
(forall i, P i -> P' i) ->
\big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r | P' i) F i.

Proof.

exact: sub_le_big. Qed.

Local Notation "'{subset' x '<=' y '}'" :=
(sub_mem (mem x) (mem y)) (at level 0, x, y at level 1).

Lemma sub_bigmin_seq [x0] (I : eqType) r r' P (F : I -> T) : {subset r' <= r} ->
\big[min/x0]_(i <- r | P i) F i <= \big[min/x0]_(i <- r' | P i) F i.

Proof.

exact: (idem_sub_le_big ge_refl _ minxx). Qed.

Lemma sub_bigmax_seq [x0] (I : eqType) r r' P (F : I -> T) : {subset r <= r'} ->
\big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r' | P i) F i.

Proof.

exact: (idem_sub_le_big _ _ maxxx). Qed.

Lemma sub_bigmin_cond [x0] (I : eqType) r r' P P' (F : I -> T) :
{subset ([seq i <- r | P i]) <= ([seq i <- r' | P' i])} ->
\big[min/x0]_(i <- r' | P' i) F i <= \big[min/x0]_(i <- r | P i) F i.

Proof.

exact: (idem_sub_le_big_cond ge_refl _ minxx). Qed.

Lemma sub_bigmax_cond [x0] (I : eqType) r r' P P' (F : I -> T) :
{subset ([seq i <- r | P i]) <= ([seq i <- r' | P' i])} ->
\big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r' | P' i) F i.

Proof.

exact: (idem_sub_le_big_cond _ _ maxxx). Qed.

Lemma sub_in_bigmin [x0] [I : eqType] (r : seq I) (P P' : {pred I}) F :
{in r, forall i, P' i -> P i} ->
\big[min/x0]_(i <- r | P i) F i <= \big[min/x0]_(i <- r | P' i) F i.

Proof.

exact: (sub_in_le_big ge_refl). Qed.

Lemma sub_in_bigmax [x0] [I : eqType] (r : seq I) (P P' : {pred I}) F :
{in r, forall i, P i -> P' i} ->
\big[max/x0]_(i <- r | P i) F i <= \big[max/x0]_(i <- r | P' i) F i.

Proof.

exact: sub_in_le_big. Qed.

Lemma le_bigmin_nat [x0] n m n' m' P (F : nat -> T) :
(n <= n')%N -> (m' <= m)%N ->
\big[min/x0]_(n <= i < m | P i) F i <= \big[min/x0]_(n' <= i < m' | P i) F i.

Proof.

exact: (le_big_nat ge_refl). Qed.

Lemma le_bigmax_nat [x0] n m n' m' P (F : nat -> T) :
(n' <= n)%N -> (m <= m')%N ->
\big[max/x0]_(n <= i < m | P i) F i <= \big[max/x0]_(n' <= i < m' | P i) F i.

Proof.

exact: le_big_nat. Qed.

Lemma le_bigmin_nat_cond [x0] n m n' m' (P P' : pred nat) (F : nat -> T) :
(n <= n')%N -> (m' <= m)%N -> (forall i, (n' <= i < m')%N -> P' i -> P i) ->
\big[min/x0]_(n <= i < m | P i) F i <= \big[min/x0]_(n' <= i < m' | P' i) F i.

Proof.

exact: (le_big_nat_cond ge_refl). Qed.

Lemma le_bigmax_nat_cond [x0] n m n' m' (P P' : {pred nat}) (F : nat -> T) :
(n' <= n)%N -> (m <= m')%N -> (forall i, (n <= i < m)%N -> P i -> P' i) ->
\big[max/x0]_(n <= i < m | P i) F i <= \big[max/x0]_(n' <= i < m' | P' i) F i.

Proof.

exact: le_big_nat_cond. Qed.

Lemma le_bigmin_ord [x0] n m (P : pred nat) (F : nat -> T) : (m <= n)%N ->
\big[min/x0]_(i < n | P i) F i <= \big[min/x0]_(i < m | P i) F i.

Proof.

exact: (le_big_ord ge_refl). Qed.

Lemma le_bigmax_ord [x0] n m (P : {pred nat}) (F : nat -> T) : (n <= m)%N ->
\big[max/x0]_(i < n | P i) F i <= \big[max/x0]_(i < m | P i) F i.

Proof.

exact: le_big_ord. Qed.

Lemma le_bigmin_ord_cond [x0] n m (P P' : pred nat) (F : nat -> T) :
(m <= n)%N -> (forall i : 'I_m, P' i -> P i) ->
\big[min/x0]_(i < n | P i) F i <= \big[min/x0]_(i < m | P' i) F i.

Proof.

exact: (le_big_ord_cond ge_refl). Qed.

Lemma le_bigmax_ord_cond [x0] n m (P P' : {pred nat}) (F : nat -> T) :
(n <= m)%N -> (forall i : 'I_n, P i -> P' i) ->
\big[max/x0]_(i < n | P i) F i <= \big[max/x0]_(i < m | P' i) F i.

Proof.

exact: le_big_ord_cond. Qed.

Lemma subset_bigmin [x0] [I : finType] [A A' P : {pred I}] (F : I -> T) :
A' \subset A ->
\big[min/x0]_(i in A | P i) F i <= \big[min/x0]_(i in A' | P i) F i.

Proof.

exact: (subset_le_big ge_refl). Qed.

Lemma subset_bigmax [x0] [I : finType] (A A' P : {pred I}) (F : I -> T) :
A \subset A' ->
\big[max/x0]_(i in A | P i) F i <= \big[max/x0]_(i in A' | P i) F i.

Proof.

exact: subset_le_big. Qed.

Lemma subset_bigmin_cond [x0] (I : finType) (A A' P P' : {pred I}) (F : I -> T) :
[set i in A' | P' i] \subset [set i in A | P i] ->
\big[min/x0]_(i in A | P i) F i <= \big[min/x0]_(i in A' | P' i) F i.

Proof.

exact: (subset_le_big_cond ge_refl). Qed.

Lemma subset_bigmax_cond [x0] (I : finType) (A A' P P' : {pred I}) (F : I -> T) :
[set i in A | P i] \subset [set i in A' | P' i] ->
\big[max/x0]_(i in A | P i) F i <= \big[max/x0]_(i in A' | P' i) F i.

Proof.

exact: subset_le_big_cond. Qed.

Section bigminmax_Type.
Context (I : Type) (r : seq I) (x : T).
Implicit Types (P : pred I) (F : I -> T).

Lemma bigmin_le_id P F : \big[min/x]_(i <- r | P i) F i <= x.

Proof.

by rewrite bigmin_idl. Qed.

Lemma bigmax_ge_id P F : \big[max/x]_(i <- r | P i) F i >= x.

Proof.

by rewrite bigmax_idl. Qed.

Lemma bigmin_eq_id P F :
(forall i, P i -> x <= F i) -> \big[min/x]_(i <- r | P i) F i = x.

Proof.

by move=> x_le; apply: le_anti; rewrite bigmin_le_id le_bigmin. Qed.

Lemma bigmax_eq_id P F :
(forall i, P i -> x >= F i) -> \big[max/x]_(i <- r | P i) F i = x.

Proof.

by move=> x_ge; apply: le_anti; rewrite bigmax_ge_id bigmax_le. Qed.

End bigminmax_Type.

Section bigminmax_eqType.
Context (I : eqType) (r : seq I) (x : T).
Implicit Types (P : pred I) (F : I -> T).

Lemma ge_bigmin_seq i0 P F :
i0 \in r -> P i0 -> \big[min/x]_(i <- r | P i) F i <= F i0.

Proof.

move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons.
move=> /predU1P[<-|i0t]; first by rewrite Pi0 ge_min// lexx.
by case: ifPn => Ph; [rewrite ge_min ih// orbT|rewrite ih].
Qed.

Lemma le_bigmax_seq i0 P F :
i0 \in r -> P i0 -> F i0 <= \big[max/x]_(i <- r | P i) F i.

Proof.

move=> + Pi0; elim: r => // h t ih; rewrite inE big_cons.
move=> /predU1P[<-|i0t]; first by rewrite Pi0 le_max// lexx.
by case: ifPn => Ph; [rewrite le_max ih// orbT|rewrite ih].
Qed.

Lemma bigmin_inf_seq i0 P t F :
i0 \in r -> P i0 -> F i0 <= t -> \big[min/x]_(i <- r | P i) F i <= t.

Proof.

by move=> ? ? ?; exact: le_trans (@ge_bigmin_seq i0 _ _ _ _) _. Qed.

Lemma bigmax_sup_seq i0 P t F :
i0 \in r -> P i0 -> t <= F i0 -> t <= \big[max/x]_(i <- r | P i) F i.

Proof.

by move=> ? ? ?; exact: le_trans (@le_bigmax_seq i0 _ _ _ _). Qed.

End bigminmax_eqType.

Section bigminmax_finType.
Context (I : finType) (x : T).
Implicit Types (P : pred I) (F : I -> T).

Lemma bigminD1 j P F : P j ->
\big[min/x]_(i | P i) F i = min (F j) (\big[min/x]_(i | P i && (i != j)) F i).

Proof.

by move/(bigD1 _) ->. Qed.

Lemma bigmaxD1 j P F : P j ->
\big[max/x]_(i | P i) F i = max (F j) (\big[max/x]_(i | P i && (i != j)) F i).

Proof.

by move/(bigD1 _) ->. Qed.

Lemma bigmin_le_cond j P F : P j -> \big[min/x]_(i | P i) F i <= F j.

Proof.

have := mem_index_enum j; rewrite unlock; elim: (index_enum I) => //= i l ih.
rewrite inE => /orP [/eqP-> ->|/ih leminlfi Pi]; first by rewrite ge_min lexx.
by case: ifPn => Pj; [rewrite ge_min leminlfi// orbC|exact: leminlfi].
Qed.

Lemma le_bigmax_cond j P F : P j -> F j <= \big[max/x]_(i | P i) F i.

Proof.

by move=> Pj; rewrite (bigmaxD1 _ Pj) le_max lexx. Qed.

Lemma bigmin_le j F : \big[min/x]_i F i <= F j.

Proof.

exact: bigmin_le_cond. Qed.

Lemma le_bigmax F j : F j <= \big[max/x]_i F i.

Proof.

exact: le_bigmax_cond. Qed.

Lemma bigmin_inf j P m F : P j -> F j <= m -> \big[min/x]_(i | P i) F i <= m.

Proof.

by move=> Pj ?; apply: le_trans (bigmin_le_cond _ Pj) _. Qed.

Lemma bigmax_sup j P m F : P j -> m <= F j -> m <= \big[max/x]_(i | P i) F i.

Proof.

by move=> Pj ?; apply: le_trans (le_bigmax_cond _ Pj). Qed.

Lemma bigmin_geP m P F :
reflect (m <= x /\ forall i, P i -> m <= F i)
(m <= \big[min/x]_(i | P i) F i).

Proof.

apply: (iffP idP) => [lemFi|[lemx lemPi]]; [split|exact: le_bigmin].
- by rewrite (le_trans lemFi)// bigmin_idl ge_min lexx.
- by move=> i Pi; rewrite (le_trans lemFi)// (bigminD1 _ Pi)// le_minl lexx.
Qed.

Lemma bigmax_leP m P F :
reflect (x <= m /\ forall i, P i -> F i <= m)
(\big[max/x]_(i | P i) F i <= m).

Proof.

apply: (iffP idP) => [|[? ?]]; last exact: bigmax_le.
rewrite bigmax_idl ge_max => /andP[-> leFm]; split=> // i Pi.
by apply: le_trans leFm; exact: le_bigmax_cond.
Qed.

Lemma bigmin_gtP m P F :
reflect (m < x /\ forall i, P i -> m < F i) (m < \big[min/x]_(i | P i) F i).

Proof.

apply: (iffP idP) => [lemFi|[lemx lemPi]]; [split|exact: lt_bigmin].
- by rewrite (lt_le_trans lemFi)// bigmin_idl ge_min lexx.
- by move=> i Pi; rewrite (lt_le_trans lemFi)// (bigminD1 _ Pi)// le_minl lexx.
Qed.

Lemma bigmax_ltP m P F :
reflect (x < m /\ forall i, P i -> F i < m) (\big[max/x]_(i | P i) F i < m).

Proof.

apply: (iffP idP) => [|[? ?]]; last exact: bigmax_lt.
rewrite bigmax_idl gt_max => /andP[-> ltFm]; split=> // i Pi.
by apply: le_lt_trans ltFm; exact: le_bigmax_cond.
Qed.

Lemma bigmin_eq_arg j P F : P j -> (forall i, P i -> F i <= x) ->
\big[min/x]_(i | P i) F i = F [arg min_(i < j | P i) F i].

Proof.

move=> Pi0; case: arg_minP => //= i Pi PF PFx.
apply/eqP; rewrite eq_le bigmin_le_cond //=.
by apply/bigmin_geP; split => //; exact: PFx.
Qed.

Lemma bigmax_eq_arg j P F : P j -> (forall i, P i -> x <= F i) ->
\big[max/x]_(i | P i) F i = F [arg max_(i > j | P i) F i].

Proof.

move=> Pi0; case: arg_maxP => //= i Pi PF PxF.
apply/eqP; rewrite eq_le le_bigmax_cond // andbT.
by apply/bigmax_leP; split => //; exact: PxF.
Qed.

Lemma eq_bigmin j P F : P j -> (forall i, P i -> F i <= x) ->
{i0 | i0 \in P & \big[min/x]_(i | P i) F i = F i0}.

Proof.

by move=> Pi0 Hx; rewrite (bigmin_eq_arg Pi0) //; eexists=> //; case: arg_minP.
Qed.

Lemma eq_bigmax j P F : P j -> (forall i, P i -> x <= F i) ->
{i0 | i0 \in P & \big[max/x]_(i | P i) F i = F i0}.

Proof.

by move=> Pi0 Hx; rewrite (bigmax_eq_arg Pi0) //; eexists=> //; case: arg_maxP.
Qed.

Lemma le_bigmin2 P F1 F2 : (forall i, P i -> F1 i <= F2 i) ->
\big[min/x]_(i | P i) F1 i <= \big[min/x]_(i | P i) F2 i.

Proof.

move=> FG; elim/big_ind2 : _ => // a b e f ba fe.
rewrite ge_min 2!le_min ba fe /= andbT.
move: (le_total a e) => /orP[/(le_trans ba)-> // | /(le_trans fe)->].
by rewrite orbT.
Qed.

Lemma le_bigmax2 P F1 F2 : (forall i, P i -> F1 i <= F2 i) ->
\big[max/x]_(i | P i) F1 i <= \big[max/x]_(i | P i) F2 i.

Proof.

move=> FG; elim/big_ind2 : _ => // a b e f ba fe.
rewrite le_max 2!ge_max ba fe /= andbT; have [//|/= af] := leP f a.
by rewrite (le_trans ba) // (le_trans _ fe) // ltW.
Qed.

Lemma bigmaxUl (A B : {set I}) F :
\big[max/x]_(i in A) F i <= \big[max/x]_(i in A :|: B) F i.

Proof.

by apply: sub_bigmax => t; rewrite in_setU => ->. Qed.

Lemma bigmaxUr (A B : {set I}) F :
\big[max/x]_(i in B) F i <= \big[max/x]_(i in A :|: B) F i.

Proof.

by under [leRHS]eq_bigl do rewrite setUC; apply: bigmaxUl. Qed.

Lemma bigminUl (A B : {set I}) F :
\big[min/x]_(i in A) F i >= \big[min/x]_(i in A :|: B) F i.

Proof.

by apply: sub_bigmin => t; rewrite in_setU => ->. Qed.

Lemma bigminUr (A B : {set I}) F :
\big[min/x]_(i in B) F i >= \big[min/x]_(i in A :|: B) F i.

Proof.

by under [leLHS]eq_bigl do rewrite setUC; apply: bigminUl. Qed.

Lemma bigmaxIl (A B : {set I}) F :
\big[max/x]_(i in A) F i >= \big[max/x]_(i in A :&: B) F i.

Proof.

by apply: sub_bigmax => t; rewrite in_setI => /andP[-> _]. Qed.

Lemma bigmaxIr (A B : {set I}) F :
\big[max/x]_(i in B) F i >= \big[max/x]_(i in A :&: B) F i.

Proof.

by under eq_bigl do rewrite setIC; apply: bigmaxIl. Qed.

Lemma bigminIl (A B : {set I}) F :
\big[min/x]_(i in A) F i <= \big[min/x]_(i in A :&: B) F i.

Proof.

by apply: sub_bigmin => t; rewrite in_setI => /andP[->_]. Qed.

Lemma bigminIr (A B : {set I}) F :
\big[min/x]_(i in B) F i <= \big[min/x]_(i in A :&: B) F i.

Proof.

by under [leRHS]eq_bigl do rewrite setIC; apply: bigminIl. Qed.

Lemma bigmaxD (A B : {set I}) F :
\big[max/x]_(i in B) F i >= \big[max/x]_(i in B :\: A) F i.

Proof.

by apply: sub_bigmax => t; rewrite in_setD => /andP[_->]. Qed.

Lemma bigminD (A B : {set I}) F :
\big[min/x]_(i in B) F i <= \big[min/x]_(i in B :\: A) F i.

Proof.

by apply: sub_bigmin => t; rewrite in_setD => /andP[_->]. Qed.

Lemma bigmaxU (A B : {set I}) F :
\big[max/x]_(i in A :|: B) F i
= max (\big[max/x]_(i in A) F i) (\big[max/x]_(i in B) F i).

Proof.

apply: le_anti; rewrite ge_max bigmaxUl bigmaxUr !andbT; apply/bigmax_leP.
split=> [|i /[!in_setU]/orP[iA|iB]]; first by rewrite le_max bigmax_ge_id.
- by rewrite le_max le_bigmax_cond.
- by rewrite le_max orbC le_bigmax_cond.
Qed.

Lemma bigminU (A B : {set I}) F :
\big[min/x]_(i in A :|: B) F i
= min (\big[min/x]_(i in A) F i) (\big[min/x]_(i in B) F i).

Proof.

apply: le_anti; rewrite le_min bigminUl bigminUr !andbT; apply/bigmin_geP.
split=> [|i /[!in_setU]/orP[iA|iB]]; first by rewrite ge_min bigmin_le_id.
- by rewrite ge_min bigmin_le_cond.
- by rewrite ge_min orbC bigmin_le_cond.
Qed.

Lemma bigmin_set1 j F : \big[min/x]_(i in [set j]) F i = min (F j) x.

Proof.

exact: big_set1E. Qed.

Lemma bigmax_set1 j F : \big[max/x]_(i in [set j]) F i = max (F j) x.

Proof.

exact: big_set1E. Qed.

End bigminmax_finType.

Lemma bigmin_imset [I J : finType] x [h : I -> J] [A : {set I}] (F : J -> T) :
\big[min/x]_(j in [set h x | x in A]) F j = \big[min/x]_(i in A) F (h i).

Proof.

by apply: big_imset_idem; apply: minxx. Qed.

Lemma bigmax_imset [I J : finType] x [h : I -> J] [A : {set I}] (F : J -> T) :
\big[max/x]_(j in [set h x | x in A]) F j = \big[max/x]_(i in A) F (h i).

Proof.

by apply: big_imset_idem; apply: maxxx. Qed.

End TotalTheory.

#[global] Hint Resolve le_total : core.
#[global] Hint Resolve ge_total : core.
#[global] Hint Extern 0 (is_true (_ >=< _)%O) => solve [apply: comparableT]
: core.
#[global] Hint Resolve sort_le_sorted : core.

Arguments min_idPr {disp T x y}.
Arguments max_idPl {disp T x y}.
Arguments bigmin_mkcond {disp T I r}.
Arguments bigmax_mkcond {disp T I r}.
Arguments bigminID {disp T I r}.
Arguments bigmaxID {disp T I r}.
Arguments bigminD1 {disp T I x} j.
Arguments bigmaxD1 {disp T I x} j.
Arguments bigmin_inf {disp T I x} j.
Arguments bigmax_sup {disp T I x} j.
Arguments bigmin_eq_arg {disp T I} x j.
Arguments bigmax_eq_arg {disp T I} x j.
Arguments eq_bigmin {disp T I x} j.
Arguments eq_bigmax {disp T I x} j.
Arguments ge_bigmin_seq {disp T I r} x i0 P.
Arguments le_bigmax_seq {disp T I r} x i0 P.
Arguments bigmin_inf_seq {disp T I r} x i0 P.
Arguments bigmax_sup_seq {disp T I r} x i0 P.

Module Import DualTotalTheory.
Section DualTotalTheory.
Context {disp : disp_t} {T : orderType disp}.
Implicit Type s : seq T.

Lemma sorted_filter_gt x s :
sorted <=%O s -> [seq y <- s | x < y] = drop (count (<= x) s) s.

Proof.

move=> s_sorted; rewrite count_le_gt -[LHS]revK -filter_rev.
rewrite (@sorted_filter_lt _ T^d); last by rewrite take_rev revK count_rev.
by rewrite rev_sorted.
Qed.

Lemma sorted_filter_ge x s :
sorted <=%O s -> [seq y <- s | x <= y] = drop (count (< x) s) s.

Proof.

move=> s_sorted; rewrite count_lt_ge -[LHS]revK -filter_rev.
rewrite (@sorted_filter_le _ T^d); last by rewrite take_rev revK count_rev.
by rewrite rev_sorted.
Qed.

Lemma nth_count_ge x x0 s i : sorted <=%O s ->
(count (< x) s <= i < size s)%N -> x <= nth x0 s i.

Proof.

move=> ss /andP[ige ilt]; rewrite -(subnKC ige) -nth_drop -sorted_filter_ge //.
apply/(all_nthP _ (filter_all _ _)).
by rewrite size_filter ltn_subLR // count_lt_ge subnK // count_size.
Qed.

Lemma nth_count_gt x x0 s i : sorted <=%O s ->
(count (<= x) s <= i < size s)%N -> x < nth x0 s i.

Proof.

move=> ss /andP[ige ilt]; rewrite -(subnKC ige) -nth_drop -sorted_filter_gt //.
apply/(all_nthP _ (filter_all _ _)).
by rewrite size_filter ltn_subLR // count_le_gt subnK // count_size.
Qed.

Lemma nth_count_eq x x0 s i : sorted <=%O s ->
(count (< x) s <= i < count (<= x) s)%N -> nth x0 s i = x.

Proof.

move=> ss /andP[ige ilt]; apply/le_anti.
by rewrite nth_count_le// nth_count_ge// ige (leq_trans ilt (count_size _ _)).
Qed.

End DualTotalTheory.
End DualTotalTheory.

Section ContraTheory.
Context {disp1 disp2 : disp_t} {T1 : porderType disp1} {T2 : orderType disp2}.
Implicit Types (x y : T1) (z t : T2) (b : bool) (m n : nat) (P : Prop).

Lemma contraTle b z t : (t < z -> ~~ b) -> (b -> z <= t).

Proof.

exact: comparable_contraTle. Qed.

Lemma contraTlt b z t : (t <= z -> ~~ b) -> (b -> z < t).

Proof.

exact: comparable_contraTlt. Qed.

Lemma contraPle P z t : (t < z -> ~ P) -> (P -> z <= t).

Proof.

exact: comparable_contraPle. Qed.

Lemma contraPlt P z t : (t <= z -> ~ P) -> (P -> z < t).

Proof.

exact: comparable_contraPlt. Qed.

Lemma contraNle b z t : (t < z -> b) -> (~~ b -> z <= t).

Proof.

exact: comparable_contraNle. Qed.

Lemma contraNlt b z t : (t <= z -> b) -> (~~ b -> z < t).

Proof.

exact: comparable_contraNlt. Qed.

Lemma contra_not_le P z t : (t < z -> P) -> (~ P -> z <= t).

Proof.

exact: comparable_contra_not_le. Qed.

Lemma contra_not_lt P z t : (t <= z -> P) -> (~ P -> z < t).

Proof.

exact: comparable_contra_not_lt. Qed.

Lemma contraFle b z t : (t < z -> b) -> (b = false -> z <= t).

Proof.

exact: comparable_contraFle. Qed.

Lemma contraFlt b z t : (t <= z -> b) -> (b = false -> z < t).

Proof.

exact: comparable_contraFlt. Qed.

Lemma contra_leq_le m n z t : (t < z -> (n < m)%N) -> ((m <= n)%N -> z <= t).

Proof.

exact: comparable_contra_leq_le. Qed.

Lemma contra_leq_lt m n z t : (t <= z -> (n < m)%N) -> ((m <= n)%N -> z < t).

Proof.

exact: comparable_contra_leq_lt. Qed.

Lemma contra_ltn_le m n z t : (t < z -> (n <= m)%N) -> ((m < n)%N -> z <= t).

Proof.

exact: comparable_contra_ltn_le. Qed.

Lemma contra_ltn_lt m n z t : (t <= z -> (n <= m)%N) -> ((m < n)%N -> z < t).

Proof.

exact: comparable_contra_ltn_lt. Qed.

Lemma contra_le x y z t : (t < z -> y < x) -> (x <= y -> z <= t).

Proof.

exact: comparable_contra_le. Qed.

Lemma contra_le_lt x y z t : (t <= z -> y < x) -> (x <= y -> z < t).

Proof.

exact: comparable_contra_le_lt. Qed.

Lemma contra_lt_le x y z t : (t < z -> y <= x) -> (x < y -> z <= t).

Proof.

exact: comparable_contra_lt_le. Qed.

Lemma contra_lt x y z t : (t <= z -> y <= x) -> (x < y -> z < t).

Proof.

exact: comparable_contra_lt. Qed.

End ContraTheory.

Section TotalMonotonyTheory.
Context {disp disp' : disp_t} {T : orderType disp} {T' : porderType disp'}.
Context (D : {pred T}) (f : T -> T').
Implicit Types (x y z : T) (u v w : T').

Let leT_anti := @le_anti _ T.
Let leT'_anti := @le_anti _ T'.
Let ltT_neqAle := @lt_neqAle _ T.
Let ltT'_neqAle := @lt_neqAle _ T'.
Let ltT_def := @lt_def _ T.
Let leT_total := @le_total _ T.

Lemma le_mono : {homo f : x y / x < y} -> {mono f : x y / x <= y}.

Proof.

exact: total_homo_mono. Qed.

Lemma le_nmono : {homo f : x y /~ x < y} -> {mono f : x y /~ x <= y}.

Proof.

by apply: total_homo_mono => // x y; rewrite eq_sym. Qed.

Lemma le_mono_in :
{in D &, {homo f : x y / x < y}} -> {in D &, {mono f : x y / x <= y}}.

Proof.

exact: total_homo_mono_in. Qed.

Lemma le_nmono_in :
{in D &, {homo f : x y /~ x < y}} -> {in D &, {mono f : x y /~ x <= y}}.

Proof.

by apply: total_homo_mono_in => // x y; rewrite eq_sym. Qed.

End TotalMonotonyTheory.

End TotalTheory.

Module Import CDistrLatticeTheory.
Section CDistrLatticeTheory.
Context {disp : disp_t} {L : cDistrLatticeType disp}.
Implicit Types (x y z : L).

Lemma rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof.

exact: rcomplPmeet. Qed.

Lemma rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.

Proof.

exact: rcomplPjoin. Qed.

Lemma rcomplKI x y z : x <= y -> (x `|` z) `&` rcompl x y z = x.

Proof.

by move=> lexy; have := rcomplPmeet x y z; rewrite (meet_l lexy). Qed.

Lemma rcomplKU x y z : x <= y -> (y `&` z) `|` rcompl x y z = y.

Proof.

by move=> lexy; have := rcomplPjoin x y z; rewrite (join_l lexy). Qed.

End CDistrLatticeTheory.
End CDistrLatticeTheory.

Module Import CBDistrLatticeTheory.
Section CBDistrLatticeTheory.
Context {disp : disp_t} {L : cbDistrLatticeType disp}.
Implicit Types (x y z : L).

Lemma diffErcompl x y : x `\` y = rcompl \bot x y.

Proof.

exact: diffErcompl. Qed.

Lemma diffKI x y : y `&` (x `\` y) = \bot.

Proof.

by have := rcomplKI y (le0x x); rewrite join0x diffErcompl. Qed.

Lemma diffIK x y : (x `\` y) `&` y = \bot.

Proof.

by rewrite meetC diffKI. Qed.

Lemma meetIB z x y : (z `&` y) `&` (x `\` y) = \bot.

Proof.

by rewrite -meetA diffKI meetx0. Qed.

Lemma meetBI z x y : (x `\` y) `&` (z `&` y) = \bot.

Proof.

by rewrite meetC meetIB. Qed.

Lemma joinIB y x : (x `&` y) `|` (x `\` y) = x.

Proof.

by rewrite diffErcompl rcomplKU. Qed.

Lemma joinBI y x : (x `\` y) `|` (x `&` y) = x.

Proof.

by rewrite joinC joinIB. Qed.

Lemma joinIBC y x : (y `&` x) `|` (x `\` y) = x.

Proof.

by rewrite meetC joinIB. Qed.

Lemma joinBIC y x : (x `\` y) `|` (y `&` x) = x.

Proof.

by rewrite meetC joinBI. Qed.

Lemma leBx x y : x `\` y <= x.

Proof.

by rewrite -[leRHS](joinIB y) leUr. Qed.

Hint Resolve leBx : core.

Lemma diffxx x : x `\` x = \bot.

Proof.

by have := diffKI x x; rewrite meet_r. Qed.

Lemma leBl z x y : x <= y -> x `\` z <= y `\` z.

Proof.

rewrite -[leLHS](joinIB z) -[leRHS](joinIB z).
by rewrite leU2E ?meetIB ?meetBI // => /andP [].
Qed.

Lemma diffKU y x : y `|` (x `\` y) = y `|` x.

Proof.

apply/eqP; rewrite eq_le leU2 //= leUx leUl.
by apply/meet_idPl; have := joinIB y x; rewrite joinIl join_l.
Qed.

Lemma diffUK y x : (x `\` y) `|` y = x `|` y.

Proof.

by rewrite joinC diffKU joinC. Qed.

Lemma leBKU y x : y <= x -> y `|` (x `\` y) = x.

Proof.

by move=> /join_r {2}<-; rewrite diffKU. Qed.

Lemma leBUK y x : y <= x -> (x `\` y) `|` y = x.

Proof.

by move=> leyx; rewrite joinC leBKU. Qed.

Lemma leBLR x y z : (x `\` y <= z) = (x <= y `|` z).

Proof.

apply/idP/idP; first by move=> /join_r <-; rewrite joinA diffKU joinAC leUr.
by rewrite -{1}[x](joinIB y) => /(leU2r_le (diffIK _ _)).
Qed.

Lemma diffUx x y z : (x `|` y) `\` z = (x `\` z) `|` (y `\` z).

Proof.

apply/eqP; rewrite eq_le leUx !leBl ?leUr ?leUl ?andbT //.
by rewrite leBLR joinA diffKU joinAC diffKU joinAC -joinA leUr.
Qed.

Lemma diff_eq0 x y : (x `\` y == \bot) = (x <= y).

Proof.

by rewrite -lex0 leBLR joinx0. Qed.

Lemma joinxB x y z : x `|` (y `\` z) = ((x `|` y) `\` z) `|` (x `&` z).

Proof.

by rewrite diffUx joinAC joinBI. Qed.

Lemma joinBx x y z : (y `\` z) `|` x = ((y `|` x) `\` z) `|` (z `&` x).

Proof.

by rewrite ![_ `|` x]joinC ![_ `&` x]meetC joinxB. Qed.

Lemma leBr z x y : x <= y -> z `\` y <= z `\` x.

Proof.

by move=> lexy; rewrite leBLR joinxB meet_r ?leBUK ?leUr ?lexUl. Qed.

Lemma leB2 x y z t : x <= z -> t <= y -> x `\` y <= z `\` t.

Proof.

by move=> /(@leBl t) ? /(@leBr x) /le_trans ->. Qed.

Lemma meet_eq0E_diff z x y : x <= z -> (x `&` y == \bot) = (x <= z `\` y).

Proof.

move=> xz; apply/idP/idP; last by move=> /meet_r <-; rewrite -meetA meetBI.
by move=> /eqP xIy_eq0; rewrite -[x](joinIB y) xIy_eq0 join0x leBl.
Qed.

Lemma leBRL x y z : (x <= z `\` y) = (x <= z) && (x `&` y == \bot).

Proof.

apply/idP/idP => [xyz|]; first by rewrite (@meet_eq0E_diff z) // (le_trans xyz).
by move=> /andP [?]; rewrite -meet_eq0E_diff.
Qed.

Lemma eq_diff x y z : (x `\` y == z) = (z <= x <= y `|` z) && (z `&` y == \bot).

Proof.

by rewrite eq_le leBLR leBRL andbCA andbA. Qed.

Lemma diffxU x y z : z `\` (x `|` y) = (z `\` x) `&` (z `\` y).

Proof.

apply/eqP; rewrite eq_le lexI !leBr ?leUl ?leUr //=.
rewrite leBRL leIx2 ?leBx //= meetUr meetAC diffIK -meetA diffIK.
by rewrite meet0x meetx0 joinx0.
Qed.

Lemma diffx0 x : x `\` \bot = x.

Proof.

by apply/eqP; rewrite eq_diff join0x meetx0 lexx eqxx. Qed.

Lemma diff0x x : \bot `\` x = \bot.

Proof.

by apply/eqP; rewrite eq_diff joinx0 meet0x lexx eqxx le0x. Qed.

Lemma diffIx x y z : (x `&` y) `\` z = (x `\` z) `&` (y `\` z).

Proof.

apply/eqP; rewrite eq_diff joinIr ?leI2 ?diffKU ?leUr ?leBx //=.
by rewrite -meetA diffIK meetx0.
Qed.

Lemma meetxB x y z : x `&` (y `\` z) = (x `&` y) `\` z.

Proof.

by rewrite diffIx -{1}[x](joinBI z) meetUl meetIB joinx0. Qed.

Lemma meetBx x y z : (x `\` y) `&` z = (x `&` z) `\` y.

Proof.

by rewrite ![_ `&` z]meetC meetxB. Qed.

Lemma diffxI x y z : x `\` (y `&` z) = (x `\` y) `|` (x `\` z).

Proof.

apply/eqP; rewrite eq_diff leUx !leBx //= joinIl joinA joinCA !diffKU.
rewrite joinCA -joinA [_ `|` x]joinC ![x `|` _]join_l //.
by rewrite -joinIl leUr /= meetUl {1}[_ `&` z]meetC ?meetBI joinx0.
Qed.

Lemma diffBx x y z : (x `\` y) `\` z = x `\` (y `|` z).

Proof.

apply/eqP; rewrite eq_diff leBr ?leUl //=.
by rewrite diffxU joinIr diffKU -joinIr meet_l ?leUr //= -meetA diffIK meetx0.
Qed.

Lemma diffxB x y z : x `\` (y `\` z) = (x `\` y) `|` (x `&` z).

Proof.

rewrite -[y in RHS](joinIB z) diffxU joinIl diffxI -joinA joinBI join_r //.
by rewrite joinBx meetKU meetA meetAC diffIK meet0x joinx0 meet_r.
Qed.

Lemma joinBK x y : (y `|` x) `\` x = (y `\` x).

Proof.

by rewrite diffUx diffxx joinx0. Qed.

Lemma joinBKC x y : (x `|` y) `\` x = (y `\` x).

Proof.

by rewrite diffUx diffxx join0x. Qed.

Lemma disj_le x y : x `&` y == \bot -> (x <= y) = (x == \bot).

Proof.

by rewrite [x == \bot]eq_sym -eq_meetl => /eqP ->. Qed.

Lemma disj_leC x y : y `&` x == \bot -> (x <= y) = (x == \bot).

Proof.

by rewrite meetC => /disj_le. Qed.

Lemma disj_diffl x y : x `&` y == \bot -> x `\` y = x.

Proof.

by move=> dxy; apply/eqP; rewrite eq_diff dxy lexx leUr. Qed.

Lemma disj_diffr x y : x `&` y == \bot -> y `\` x = y.

Proof.

by rewrite meetC => /disj_diffl. Qed.

Lemma lt0B x y : x < y -> \bot < y `\` x.

Proof.

by move=> ?; rewrite lt_leAnge le0x leBLR joinx0 /= lt_geF. Qed.

End CBDistrLatticeTheory.
End CBDistrLatticeTheory.

Module Import CTDistrLatticeTheory.
Section CTDistrLatticeTheory.
Context {disp : disp_t} {L : ctDistrLatticeType disp}.
Implicit Types (x y z : L).

Lemma codiffErcompl x y : codiff x y = rcompl x \top y.

Proof.

exact: codiffErcompl. Qed.

End CTDistrLatticeTheory.
End CTDistrLatticeTheory.

Module Import CTBDistrLatticeTheory.
Section CTBDistrLatticeTheory.
Context {disp : disp_t} {L : ctbDistrLatticeType disp}.
Implicit Types (x y z : L).

Lemma complEdiff x : ~` x = \top `\` x

Proof.

exact: complEdiff. Qed.

#[deprecated(since="mathcomp 2.3.0", use=complEdiff)]
Notation complE := complEdiff.

Lemma complEcodiff x : ~` x = codiff \bot x

Proof.

exact: complEcodiff. Qed.

Lemma complErcompl x : ~` x = rcompl \bot \top x.

Proof.

by rewrite complEdiff diffErcompl. Qed.

Lemma diff1x x : \top `\` x = ~` x.

Proof.

exact/esym/complEdiff. Qed.

Lemma diffE x y : x `\` y = x `&` ~` y.

Proof.

by rewrite complEdiff meetxB meetx1. Qed.

Lemma complK : involutive (@compl _ L).

Proof.

by move=> x; rewrite !complEdiff diffxB diffxx meet1x join0x. Qed.

Lemma compl_inj : injective (@compl _ L).

Proof.

exact/inv_inj/complK. Qed.

Lemma disj_leC x y : (x `&` y == \bot) = (x <= ~` y).

Proof.

by rewrite -diff_eq0 diffE complK. Qed.

Lemma leCx x y : (~` x <= y) = (~` y <= x).

Proof.

by rewrite !complEdiff !leBLR joinC. Qed.

Lemma lexC x y : (x <= ~` y) = (y <= ~` x).

Proof.

by rewrite -[x in LHS]complK leCx complK. Qed.

Lemma leC x y : (~` x <= ~` y) = (y <= x).

Proof.

by rewrite leCx complK. Qed.

Lemma complU x y : ~` (x `|` y) = ~` x `&` ~` y.

Proof.

by rewrite !complEdiff diffxU. Qed.

Lemma complI x y : ~` (x `&` y) = ~` x `|` ~` y.

Proof.

by rewrite !complEdiff diffxI. Qed.

Lemma joinxC x : x `|` ~` x = \top.

Proof.

by rewrite complEdiff diffKU joinx1. Qed.

Lemma joinCx x : ~` x `|` x = \top.

Proof.

by rewrite joinC joinxC. Qed.

Lemma meetxC x : x `&` ~` x = \bot.

Proof.

by rewrite complEdiff diffKI. Qed.

Lemma meetCx x : ~` x `&` x = \bot.

Proof.

by rewrite meetC meetxC. Qed.

Lemma compl1 : ~` \top = \bot :> L.

Proof.

by rewrite complEdiff diffxx. Qed.

Lemma compl0 : ~` \bot = \top :> L.

Proof.

by rewrite -compl1 complK. Qed.

Lemma complB x y : ~` (x `\` y) = ~` x `|` y.

Proof.

by rewrite diffE complI complK. Qed.

Lemma leBC x y : x `\` y <= ~` y.

Proof.

by rewrite leBLR joinxC lex1. Qed.

Lemma compl_joins (J : Type) (r : seq J) (P : {pred J}) (F : J -> L) :
~` (\join_(j <- r | P j) F j) = \meet_(j <- r | P j) ~` F j.

Proof.

by elim/big_rec2: _=> [|i x y ? <-]; rewrite ?compl0 ?complU. Qed.

Lemma compl_meets (J : Type) (r : seq J) (P : {pred J}) (F : J -> L) :
~` (\meet_(j <- r | P j) F j) = \join_(j <- r | P j) ~` F j.

Proof.

by elim/big_rec2: _=> [|i x y ? <-]; rewrite ?compl1 ?complI. Qed.

End CTBDistrLatticeTheory.
End CTBDistrLatticeTheory.

FACTORIES

HB.factory Record Preorder_isPOrder (d : disp_t) T & Preorder d T := {
le_anti : antisymmetric (@le d T);
}.

HB.builders Context (d : disp_t) T & Preorder_isPOrder d T.

Let ge_anti : antisymmetric (fun x y => @le d T y x).

Proof.

by move=> x y; rewrite andbC; apply: le_anti. Qed.

HB.instance Definition _ := Preorder_isDuallyPOrder.Build d T le_anti ge_anti.

HB.end.

HB.factory Record isPOrder (d : disp_t) T & Choice T := {
  le : rel T;
  lt : rel T;
  lt_def : forall x y, lt x y = (y != x) && (le x y);
  le_refl : reflexive le;
  le_anti : antisymmetric le;
  le_trans : transitive le;
}.

HB.builders Context (d : disp_t) T & isPOrder d T.

Let lt_le_def x y : lt x y = le x y && ~~ le y x.

Proof.

rewrite lt_def andbC; case /boolP: (le x y) => //= xy.
have [->|/negP xyE /=] := eqVneq y x; first by rewrite le_refl.
by apply/esym/negP => yx; apply/xyE/eqP/le_anti; rewrite yx.
Qed.

HB.instance Definition _ := isPreorder.Build d T lt_le_def le_refl le_trans.
HB.instance Definition _ := Preorder_isPOrder.Build d T le_anti.

HB.end.

HB.factory Record Le_isPOrder (d : disp_t) T & Choice T := {
  le : rel T;
  le_refl : reflexive le;
  le_anti : antisymmetric le;
  le_trans : transitive le;
}.

HB.builders Context (d : disp_t) T & Le_isPOrder d T.
HB.instance Definition _ := @Le_isPreorder.Build d T le le_refl le_trans.
HB.instance Definition _ := @Preorder_isPOrder.Build d T le_anti.
HB.end.

HB.factory Record LtLe_isPOrder (d : disp_t) T & Choice T := {
  le : rel T;
  lt : rel T;
  le_def : forall x y, le x y = (x == y) || lt x y;
  lt_irr : irreflexive lt;
  lt_trans : transitive lt;
}.

HB.builders Context (d : disp_t) T & LtLe_isPOrder d T.

HB.instance Definition _ := @LtLe_isPreorder.Build d T le lt le_def lt_irr lt_trans.

Let le_anti : antisymmetric le.

Proof.

move=> x y; rewrite !le_def [y == _]eq_sym.
have [//|neq_xy/=] := eqVneq x y => /andP[xy yx].
by have := lt_trans xy yx; rewrite lt_irr.
Qed.

HB.instance Definition _ := @Preorder_isPOrder.Build d T le_anti.

HB.end.

HB.factory Record Lt_isPOrder (d : disp_t) T & Choice T := {
  lt : rel T;
  lt_irr : irreflexive lt;
  lt_trans : transitive lt;
}.

HB.builders Context d T & Lt_isPOrder d T.
#[warning="-HB.no-new-instance"]
HB.instance Definition _ := @LtLe_isPOrder.Build d T
  _ lt (fun _ _ => erefl) lt_irr lt_trans.
HB.end.

HB.factory Record POrder_Meet_isSemilattice d T & POrder d T := {
  meet : T -> T -> T;
  meetC : commutative meet;
  meetA : associative meet;
  leEmeet : forall x y, (x <= y) = (meet x y == x);
}.

HB.builders Context d T & POrder_Meet_isSemilattice d T.

Fact meetxx : idempotent_op meet.

Proof.

by move=> x; apply/eqP; rewrite -leEmeet. Qed.

Fact lexI x y z : (x <= meet y z) = (x <= y) && (x <= z).

Proof.

rewrite !leEmeet; apply/eqP/andP => [<-|[/eqP<- /eqP<-]].
split; apply/eqP; last by rewrite meetA -meetA meetxx.
by rewrite -!meetA (meetC z) (meetA y) meetxx.
by rewrite -!meetA (meetC z) -meetA (meetA y) !meetxx.
Qed.

HB.instance Definition _ := @POrder_isMeetSemilattice.Build d T meet lexI.

HB.end.

HB.factory Record POrder_Join_isSemilattice d T & POrder d T := {
  join : T -> T -> T;
  joinC : commutative join;
  joinA : associative join;
  leEjoin : forall x y, (y <= x) = (join x y == x);
}.

HB.builders Context d T & POrder_Join_isSemilattice d T.

Fact joinxx : idempotent_op join.

Proof.

by move=> x; apply/eqP; rewrite -leEjoin. Qed.

Fact leUx x y z : (join x y <= z) = (x <= z) && (y <= z).
rewrite !leEjoin; apply/eqP/andP => [<-|[/eqP<- /eqP<-]].
  split; apply/eqP; last by rewrite joinA -joinA joinxx.
  by rewrite -joinA (joinC _ x) (joinA x) joinxx.
by rewrite -!joinA (joinC y) -joinA (joinA x) !joinxx.

HB.instance Definition _ := @POrder_isJoinSemilattice.Build d T join leUx.

HB.end.

HB.factory Record POrder_MeetJoin_isLattice d T & POrder d T := {
  meet : T -> T -> T;
  join : T -> T -> T;
  meetP : forall x y z, (x <= meet y z) = (x <= y) && (x <= z);
  joinP : forall x y z, (join x y <= z) = (x <= z) && (y <= z);
}.

HB.builders Context d T & POrder_MeetJoin_isLattice d T.
HB.instance Definition _ := @POrder_isMeetSemilattice.Build d T meet meetP.
HB.instance Definition _ := @POrder_isJoinSemilattice.Build d T join joinP.
HB.end.

HB.factory Record POrder_isLattice d T & POrder d T := {
  meet : T -> T -> T;
  join : T -> T -> T;
  meetC : commutative meet;
  joinC : commutative join;
  meetA : associative meet;
  joinA : associative join;
  joinKI : forall y x, meet x (join x y) = x;
  meetKU : forall y x, join x (meet x y) = x;
  leEmeet : forall x y, (x <= y) = (meet x y == x);
}.

HB.builders Context d T & POrder_isLattice d T.

Fact leEjoin x y : (y <= x) = (join x y == x).

Proof.

rewrite leEmeet; apply/eqP/eqP => <-.
by rewrite meetC meetKU.
by rewrite joinC joinKI.
Qed.

Fact meetxx : idempotent_op meet.

Proof.

by move=> x; apply/eqP; rewrite -leEmeet. Qed.

Fact lexI x y z : (x <= meet y z) = (x <= y) && (x <= z).

Proof.

Fact joinxx : idempotent_op join.

Proof.

by move=> x; apply/eqP; rewrite -leEjoin. Qed.

Proof.

by move=> x y z; rewrite ![x `&` _]meetC meetUl. Qed.

Let joinIl : left_distributive (@join _ T) (@meet _ T).

Proof.

by move=> x y z; rewrite meetUr joinIK meetUl -joinA meetUKC. Qed.

HB.instance Definition _ := Lattice_isDistributive.Build d T meetUl joinIl.

HB.end.

HB.factory Record POrder_Meet_isDistrLattice d T & POrder d T := {
  meet : T -> T -> T;
  join : T -> T -> T;
  meetC : commutative meet;
  joinC : commutative join;
  meetA : associative meet;
  joinA : associative join;
  joinKI : forall y x, meet x (join x y) = x;
  meetKU : forall y x, join x (meet x y) = x;
  leEmeet : forall x y, (x <= y) = (meet x y == x);
  meetUl : left_distributive meet join;
}.

HB.builders Context d T & POrder_Meet_isDistrLattice d T.

HB.instance Definition _ := @POrder_isLattice.Build d T
  meet join meetC joinC meetA joinA joinKI meetKU leEmeet.
HB.instance Definition _ :=
  Lattice_Meet_isDistrLattice.Build d T meetUl.

HB.end.

HB.factory Record isMeetJoinDistrLattice (d : disp_t) T & Choice T := {
  le : rel T;
  lt : rel T;
  meet : T -> T -> T;
  join : T -> T -> T;
  le_def : forall x y : T, le x y = (meet x y == x);
  lt_def : forall x y : T, lt x y = (y != x) && le x y;
  meetC : commutative meet;
  joinC : commutative join;
  meetA : associative meet;
  joinA : associative join;
  joinKI : forall y x : T, meet x (join x y) = x;
  meetKU : forall y x : T, join x (meet x y) = x;
  meetUl : left_distributive meet join;
  meetxx : idempotent_op meet;
}.

HB.builders Context d T & isMeetJoinDistrLattice d T.

Fact le_refl : reflexive le

Proof.

by move=> x; rewrite le_def meetxx. Qed.

Fact le_anti : antisymmetric le.

Proof.

by move=> x y; rewrite !le_def meetC => /andP [] /eqP {2}<- /eqP ->. Qed.

Fact le_trans : transitive le.

Proof.

move=> y x z; rewrite !le_def => /eqP lexy /eqP leyz; apply/eqP.
by rewrite -[in LHS]lexy -meetA leyz lexy.
Qed.

Fact lt_le_def x y : lt x y = (le x y) && ~~ (le y x).

Proof.

rewrite lt_def andbC; case/boolP: (le x y) => //= xy.
congr negb; apply/eqP/idP => [->|yx]; first exact/le_refl.
by apply/le_anti/andP; split.
Qed.

HB.instance Definition _ := isPreorder.Build d T
  lt_le_def le_refl le_trans.
HB.instance Definition _ := Preorder_isPOrder.Build d T le_anti.

HB.instance Definition _ := @POrder_Meet_isDistrLattice.Build d T
  meet join meetC joinC meetA joinA joinKI meetKU le_def meetUl.

HB.end.

HB.factory Record BDistrLattice_hasSectionalComplement d T
    & BDistrLattice d T := {
  diff : T -> T -> T;
  diffKI : forall x y, y `&` diff x y = \bot;
  joinIB : forall x y, (x `&` y) `|` diff x y = x;
}.

Module hasRelativeComplement.
#[deprecated(since="mathcomp 2.3.0",
  use=BDistrLattice_hasSectionalComplement.Build)]
Notation Build d T :=
  (BDistrLattice_hasSectionalComplement.Build d T) (only parsing).
End hasRelativeComplement.

#[deprecated(since="mathcomp 2.3.0", use=BDistrLattice_hasSectionalComplement)]
Notation hasRelativeComplement d T :=
  (BDistrLattice_hasSectionalComplement d T) (only parsing).

HB.builders Context d T & BDistrLattice_hasSectionalComplement d T.

Definition rcompl x y z := (x `&` y) `|` diff (y `|` x) z.

Fact rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof.

by rewrite meetUr joinIKC meetUl diffKI joinx0 meetKU. Qed.

Fact rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.

Proof.

by rewrite joinCA joinIB joinA meetUK joinC. Qed.

HB.instance Definition _ :=
@DistrLattice_hasRelativeComplement.Build d T rcompl rcomplPmeet rcomplPjoin.

Fact diffErcompl x y : diff x y = rcompl \bot x y.

Proof.

by rewrite /rcompl meet0x join0x joinx0. Qed.

HB.instance Definition _ :=
  @CDistrLattice_hasSectionalComplement.Build d T diff diffErcompl.

HB.end.

HB.factory Record TDistrLattice_hasDualSectionalComplement d T
    & TDistrLattice d T := {
  codiff : T -> T -> T;
  codiffKU : forall x y, y `|` codiff x y = \top;
  meetUB : forall x y, (x `|` y) `&` codiff x y = x;
}.

HB.builders Context d T & TDistrLattice_hasDualSectionalComplement d T.

Definition rcompl x y z := (y `|` x) `&` codiff (x `&` y) z.

Fact rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof.

by rewrite meetCA meetUB meetA joinIK. Qed.

Fact rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.

Proof.

by rewrite joinIr meetUKC joinIl codiffKU meetx1 joinKI. Qed.

HB.instance Definition _ :=
@DistrLattice_hasRelativeComplement.Build d T rcompl rcomplPmeet rcomplPjoin.

Fact codiffErcompl x y : codiff x y = rcompl x \top y.

Proof.

by rewrite /rcompl join1x meet1x meetx1. Qed.

HB.instance Definition _ :=
  @CDistrLattice_hasDualSectionalComplement.Build d T codiff codiffErcompl.

HB.end.

HB.factory Record CBDistrLattice_hasComplement d T
    & CBDistrLattice d T & hasTop d T := {
  compl : T -> T;
  complEdiff : forall x, compl x = (\top : T) `\` x;
}.

Module hasComplement.
#[deprecated(since="mathcomp 2.3.0", use=CBDistrLattice_hasComplement.Build)]
Notation Build d T := (CBDistrLattice_hasComplement.Build d T) (only parsing).
End hasComplement.

#[deprecated(since="mathcomp 2.3.0", use=CBDistrLattice_hasComplement)]
Notation hasComplement d T := (CBDistrLattice_hasComplement d T) (only parsing).

HB.builders Context d T & CBDistrLattice_hasComplement d T.

HB.instance Definition _ := @CDistrLattice_hasDualSectionalComplement.Build d T
  (fun x y => rcompl x \top y) (fun _ _ => erefl).

Fact complEcodiff (x : T) : compl x = codiff (\bot : T) x.

Proof.

by rewrite complEdiff diffErcompl. Qed.

HB.instance Definition _ :=
  @CDistrLattice_hasComplement.Build d T compl complEdiff complEcodiff.

HB.end.

HB.factory Record CTDistrLattice_hasComplement d T
    & CTDistrLattice d T & hasBottom d T := {
  compl : T -> T;
  complEcodiff : forall x, compl x = codiff (\bot : T) x;
}.

HB.builders Context d T & CTDistrLattice_hasComplement d T.

HB.instance Definition _ := @CDistrLattice_hasSectionalComplement.Build d T
  (fun x y => rcompl (\bot : T) x y) (fun _ _ => erefl).

Fact complEdiff (x : T) : compl x = (\top : T) `\` x.

Proof.

by rewrite complEcodiff codiffErcompl. Qed.

HB.instance Definition _ :=
  @CDistrLattice_hasComplement.Build d T compl complEdiff complEcodiff.

HB.end.

HB.factory Record TBDistrLattice_hasComplement d T & TBDistrLattice d T := {
  compl : T -> T;
  joinxC : forall x, x `|` compl x = \top;
  meetxC : forall x, x `&` compl x = \bot;
}.

HB.builders Context d T & TBDistrLattice_hasComplement d T.

Definition diff x y := x `&` compl y.
Definition codiff x y := x `|` compl y.
Definition rcompl x y z := (x `&` y) `|` diff (y `|` x) z.

Fact diffKI x y : y `&` diff x y = \bot.

Proof.

by rewrite meetCA meetxC meetx0. Qed.

Fact joinIB x y : (x `&` y) `|` diff x y = x.

Proof.

by rewrite -meetUr joinxC meetx1. Qed.

HB.instance Definition _ :=
@BDistrLattice_hasSectionalComplement.Build d T diff diffKI joinIB.

Fact codiffErcompl x y : codiff x y = rcompl x \top y.

Proof.

by rewrite /rcompl /diff join1x meetx1 meet1x. Qed.

HB.instance Definition _ :=
@CDistrLattice_hasDualSectionalComplement.Build d T codiff codiffErcompl.

Fact complEdiff x : compl x = diff \top x

Proof.

exact/esym/meet1x. Qed.

Fact complEcodiff x : compl x = codiff \bot x

Proof.

exact/esym/join0x. Qed.

HB.instance Definition _ :=
@CDistrLattice_hasComplement.Build d T compl complEdiff complEcodiff.

HB.end.

HB.factory Record Lattice_isTotal d T & Lattice d T := {
le_total : total (<=%O : rel T)
}.

HB.builders Context d T & Lattice_isTotal d T.

Fact meetUl : @left_distributive T T meet join.

Proof.

pose leP x y := lcomparable_leP (le_total x y); move=> x y z; apply/esym.
by case: (leP x y) (leP x z) (leP y z) => [|/ltW] xy [|/ltW] xz [|/ltW] yz;
(apply/join_idPl || apply/join_idPr) => //; apply: le_trans xy.
Qed.

HB.instance Definition _ := Lattice_Meet_isDistrLattice.Build d T meetUl.
HB.instance Definition _ := DistrLattice_isTotal.Build d T le_total.

HB.end.

HB.factory Record POrder_isTotal d T & POrder d T := {
  le_total : total (<=%O : rel T) }.

HB.builders Context d T & POrder_isTotal d T.

Implicit Types (x y z : T).

Let comparableT x y : x >=< y := le_total x y.

Fact ltgtP x y :
  compare x y (min y x) (min x y) (max y x) (max x y)
              (y == x) (x == y) (x >= y) (x <= y) (x > y) (x < y).

Proof.

exact: comparable_ltgtP. Qed.

Fact leP x y : le_xor_gt x y
(min y x) (min x y) (max y x) (max x y) (x <= y) (y < x).

Proof.

exact: comparable_leP. Qed.

Definition meet := @min _ T.
Definition join := @max _ T.

Fact meetC : commutative meet.

Proof.

by move=> x y; rewrite /meet; have [] := ltgtP. Qed.

Fact joinC : commutative join.

Proof.

by move=> x y; rewrite /join; have [] := ltgtP. Qed.

Fact meetA : associative meet.

Proof.

move=> x y z; rewrite /meet /min !(fun_if, if_arg).
case: (leP z y) (leP y x) (leP z x) => [] zy [] yx [] zx//=.
by have := le_lt_trans (le_trans zy yx) zx; rewrite ltxx.
by apply/eqP; rewrite eq_le zx ltW// (lt_trans yx).
Qed.

Fact joinA : associative join.

Proof.

Fact joinKI y x : meet x (join x y) = x.

Proof.

rewrite /meet /join /min /max !(fun_if, if_arg).
by have []// := ltgtP x y; rewrite ltxx.
Qed.

Fact meetKU y x : join x (meet x y) = x.

Proof.

rewrite /meet /join /min /max !(fun_if, if_arg).
by have []// := ltgtP x y; rewrite ltxx.
Qed.

Fact leEmeet x y : (x <= y) = (meet x y == x).

Proof.

by rewrite /meet; case: leP => ?; rewrite ?eqxx ?lt_eqF. Qed.

HB.instance Definition _ := @POrder_isLattice.Build d T
  meet join meetC joinC meetA joinA joinKI meetKU leEmeet.
HB.instance Definition _ :=
  Lattice_isTotal.Build d T comparableT.
HB.end.

HB.factory Record isOrder (d : disp_t) T & Choice T := {
  le : rel T;
  lt : rel T;
  meet : T -> T -> T;
  join : T -> T -> T;
  lt_def : forall x y, lt x y = (y != x) && le x y;
  meet_def : forall x y, meet x y = if lt x y then x else y;
  join_def : forall x y, join x y = if lt x y then y else x;
  le_anti : antisymmetric le;
  le_trans : transitive le;
  le_total : total le;
}.

HB.builders Context d T & isOrder d T.

Fact le_refl : reflexive le.

Proof.

by move=> x; case: (le x x) (le_total x x). Qed.

Fact lt_le_def x y : lt x y = (le x y) && ~~ (le y x).

Proof.

rewrite lt_def andbC; case/boolP: (le x y) => //= xy.
congr negb; apply/eqP/idP => [->|yx]; first exact/le_refl.
by apply/le_anti/andP; split.
Qed.

HB.instance Definition _ := isPreorder.Build d T
lt_le_def le_refl le_trans.
HB.instance Definition _ := Preorder_isPOrder.Build d T le_anti.

Section GeneratedOrder.

Local Definition T' := T.
HB.instance Definition _ := POrder.on T'.
HB.instance Definition _ := POrder_isTotal.Build d T' le_total.
Implicit Types (x y z : T').

Fact meetE x y : meet x y = x `&` y

Proof.

by rewrite meet_def. Qed.

Fact joinE x y : join x y = x `|` y

Proof.

by rewrite join_def. Qed.

Fact meetC : commutative meet.

Proof.

by move=> *; rewrite !meetE meetC. Qed.

Fact joinC : commutative join.

Proof.

by move=> *; rewrite !joinE joinC. Qed.

Fact meetA : associative meet.

Proof.

by move=> *; rewrite !meetE meetA. Qed.

Fact joinA : associative join.

Proof.

by move=> *; rewrite !joinE joinA. Qed.

Fact joinKI y x : meet x (join x y) = x.

Proof.

by rewrite meetE joinE joinKI. Qed.

Fact meetKU y x : join x (meet x y) = x.

Proof.

by rewrite meetE joinE meetKU. Qed.

Fact meetUl : left_distributive meet join.

Proof.

by move=> *; rewrite !meetE !joinE meetUl. Qed.

Fact meetxx : idempotent_op meet.

Proof.

by move=> *; rewrite meetE meetxx. Qed.

Fact le_def x y : (x <= y) = (meet x y == x).

Proof.

by rewrite meetE (eq_meetl x y). Qed.

End GeneratedOrder.

HB.instance Definition _ := @POrder_Meet_isDistrLattice.Build d T
  meet join meetC joinC meetA joinA joinKI meetKU le_def meetUl.
HB.instance Definition _ := DistrLattice_isTotal.Build d T le_total.

HB.end.

HB.factory Record LtOrder (d : disp_t) T & Choice T := {
  le : rel T;
  lt : rel T;
  meet : T -> T -> T;
  join : T -> T -> T;
  le_def : forall x y, le x y = (x == y) || lt x y;
  meet_def : forall x y, meet x y = if lt x y then x else y;
  join_def : forall x y, join x y = if lt x y then y else x;
  lt_irr : irreflexive lt;
  lt_trans : transitive lt;
  lt_total : forall x y, x != y -> lt x y || lt y x;
}.

HB.builders Context d T & LtOrder d T.

Fact lt_def x y : lt x y = (y != x) && le x y.

Proof.

by rewrite le_def; case: eqVneq => //= ->; rewrite lt_irr. Qed.

Fact meet_def_le x y : meet x y = if lt x y then x else y.

Proof.

by rewrite meet_def lt_def; case: eqP. Qed.

Fact join_def_le x y : join x y = if lt x y then y else x.

Proof.

by rewrite join_def lt_def; case: eqP. Qed.

Fact le_anti : antisymmetric le.

Proof.

move=> x y; rewrite !le_def; case: eqVneq => //= _ /andP [] hxy.
by move/(lt_trans hxy); rewrite lt_irr.
Qed.

Fact le_trans : transitive le.

Proof.

move=> y x z; rewrite !le_def; case: eqVneq => [->|_] //=.
by case: eqVneq => [-> ->|_ hxy /(lt_trans hxy) ->]; rewrite orbT.
Qed.

Fact le_total : total le.

Proof.

by move=> x y; rewrite !le_def; case: eqVneq => //; exact: lt_total. Qed.

HB.instance Definition _ :=
  isOrder.Build d T lt_def meet_def_le join_def_le le_anti le_trans le_total.
HB.end.

HB.factory Record MonoTotal disp T & POrder disp T := {
  disp' : disp_t;
  T' : orderType disp';
  f : T -> T';
  f_mono : {mono f : x y / x <= y}
}.
HB.builders Context disp T & MonoTotal disp T.
Fact totalT : total (<=%O : rel T).

Proof.

by move=> x y; rewrite -!f_mono le_total. Qed.

HB.instance Definition _ := POrder_isTotal.Build disp T totalT.
HB.end.

Module CancelPartial.
Import PreCancelPartial.
Section CancelPartial.
Variables (disp : disp_t) (T : choiceType).
Variables (disp' : disp_t) (T' : porderType disp') (f : T -> T').

Section Pcan.
Variables (f' : T' -> option T) (f_can : pcancel f f').

Fact anti : antisymmetric (le f).

Proof.

by move=> ? ? /le_anti; apply: pcan_inj. Qed.

Fact lt_def x y :
lt f x y = (y != x) && le f x y.

Proof.

by rewrite /lt lt_def (inj_eq (pcan_inj f_can)). Qed.

Definition Pcan := isPOrder.Build disp (Choice.Pack (Choice.class T))
  lt_def (@refl T disp' T' f) anti (@trans T disp' T' f).

End Pcan.

Definition Can f' (f_can : cancel f f') := Pcan (can_pcan f_can).

End CancelPartial.
End CancelPartial.

Notation PCanIsPartial := CancelPartial.Pcan.
Notation CanIsPartial := CancelPartial.Can.

#[export]
HB.instance Definition _ (disp : disp_t) (T : choiceType)
  (disp' : disp_t) (T' : porderType disp') (f : T -> T')
  (f' : T' -> option T) (f_can : pcancel f f') :=
  Preorder_isPOrder.Build disp (pcan_type f_can) (CancelPartial.anti f_can).

#[export]
HB.instance Definition _ (disp : disp_t) (T : choiceType)
  (disp' : disp_t) (T' : porderType disp') (f : T -> T') (f' : T' -> T)
  (f_can : cancel f f') :=
  Preorder_isPOrder.Build disp (can_type f_can)
    (CancelPartial.anti (can_pcan f_can)).

Section CancelTotal.
Variables (disp : disp_t) (T : choiceType).
Variables (disp' : disp_t) (T' : orderType disp') (f : T -> T').

Section PCan.

Variables (f' : T' -> option T) (f_can : pcancel f f').

#[local]
HB.instance Definition _ :=
   MonoTotal.Build disp (pcan_type f_can) (fun _ _ => erefl).

Definition PCanIsTotal : DistrLattice_isTotal _ (pcan_type f_can) :=
  Total.on (pcan_type f_can).

End PCan.

Section Can.

Variables (f' : T' -> T) (f_can : cancel f f').

#[local]
HB.instance Definition _ :=
   MonoTotal.Build disp (can_type f_can) (fun _ _ => erefl).

Definition CanIsTotal : DistrLattice_isTotal _ (can_type f_can) :=
  Total.on (can_type f_can).

End Can.
End CancelTotal.

HB.factory Record IsoBottom disp T & Order.POrder disp T := {
  disp' : Order.disp_t;
  T' : bPOrderType disp';
  f : T -> T';
  f' : T' -> T;
  f_can : cancel f f';
  f'_can : cancel f' f;
  f_mono : {mono f : x y / x <= y}%O;
}.

HB.builders Context disp T & IsoBottom disp T.

Definition bottom := f' \bot%O.
Lemma isobottom x : (bottom <= x)%O.

Proof.

by rewrite /bottom -f_mono f'_can Order.le0x. Qed.

HB.instance Definition _ := Order.hasBottom.Build _ T isobottom.
HB.end.

HB.factory Record IsoTop disp T & Order.POrder disp T := {
  disp' : Order.disp_t;
  T' : tPOrderType disp';
  f : T -> T';
  f' : T' -> T;
  f_can : cancel f f';
  f'_can : cancel f' f;
  f_mono : {mono f : x y / x <= y}%O;
}.

HB.builders Context disp T & IsoTop disp T.

Definition top := f' \top%O.
Lemma isotop x : (x <= top)%O.

Proof.

by rewrite /top -f_mono f'_can Order.lex1. Qed.

HB.instance Definition _ := Order.hasTop.Build _ T isotop.
HB.end.

HB.factory Record IsoLattice disp T & POrder disp T := {
  disp' : disp_t;
  T' : latticeType disp';
  f : T -> T';
  f' : T' -> T;
  f_can : cancel f f';
  f'_can : cancel f' f;
  f_mono : {mono f : x y / x <= y};
}.

HB.builders Context disp T & IsoLattice disp T.

Definition meet (x y : T) := f' (meet (f x) (f y)).
Definition join (x y : T) := f' (join (f x) (f y)).

Fact meetC : commutative meet

Proof.

by move=> x y; rewrite /meet meetC. Qed.

Fact joinC : commutative join

Proof.

by move=> x y; rewrite /join joinC. Qed.

Fact meetA : associative meet.

Proof.

by move=> y x z; rewrite /meet !f'_can meetA. Qed.

Fact joinA : associative join.

Proof.

by move=> y x z; rewrite /join !f'_can joinA. Qed.

Fact joinKI y x : meet x (join x y) = x.

Proof.

by rewrite /meet /join f'_can joinKI f_can. Qed.

Fact meetKI y x : join x (meet x y) = x.

Proof.

by rewrite /join /meet f'_can meetKU f_can. Qed.

Fact meet_eql x y : (x <= y) = (meet x y == x).

Proof.

by rewrite /meet -(can_eq f_can) f'_can eq_meetl f_mono. Qed.

HB.instance Definition _ := POrder_isLattice.Build _ T
  meetC joinC meetA joinA joinKI meetKI meet_eql.

HB.end.

HB.factory Record IsoDistrLattice disp T & POrder disp T := {
  disp' : disp_t;
  T' : distrLatticeType disp';
  f : T -> T';
  f' : T' -> T;
  f_can : cancel f f';
  f'_can : cancel f' f;
  f_mono : {mono f : x y / x <= y};
}.

HB.builders Context disp T & IsoDistrLattice disp T.

HB.instance Definition _ := IsoLattice.Build _ T f_can f'_can f_mono.

Fact meetUl : left_distributive (meet : T -> T -> T) join.

Proof.

by move=> x y z; rewrite /meet /join /= !f'_can meetUl. Qed.

HB.instance Definition _ := Lattice_Meet_isDistrLattice.Build _ T meetUl.

HB.end.

Export OrderMorphismTheory.

Lemma omorph_lt (d : disp_t) (T : porderType d) (d' : disp_t) (T' : porderType d')
(f : {omorphism T -> T'}) : injective f -> {homo f : x y / x < y}.

Proof.

by move/inj_homo_lt; apply; apply: omorph_le. Qed.

Proof.

exact: omorphI_subproof. Qed.

Lemma omorphU (f : {jlmorphism T -> T'}) : {morph f : x y / x `|` y}.

Proof.

exact: omorphU_subproof. Qed.

End Properties.

Section IdCompFun.

Variables (d : disp_t) (T : latticeType d) (d' : disp_t) (T' : latticeType d').
Variables (d'' : disp_t) (T'' : latticeType d'').

Section MeetCompFun.

Variables (f : {mlmorphism T' -> T''}) (g : {mlmorphism T -> T'}).

Fact idfun_is_meet_morphism : meet_morphism (@idfun T)

Proof.

by []. Qed.

#[export]
HB.instance Definition _ := isMeetLatticeMorphism.Build d T d T idfun
idfun_is_meet_morphism.

Fact comp_is_meet_morphism : meet_morphism (f \o g).

Proof.

by move=> x y; rewrite /= !omorphI. Qed.

#[export]
HB.instance Definition _ := isMeetLatticeMorphism.Build d T d'' T'' (f \o g)
comp_is_meet_morphism.

End MeetCompFun.

Section JoinCompFun.

Variables (f : {jlmorphism T' -> T''}) (g : {jlmorphism T -> T'}).

Fact idfun_is_join_morphism : join_morphism (@idfun T)

Proof.

by []. Qed.

#[export]
HB.instance Definition _ := isJoinLatticeMorphism.Build d T d T idfun
idfun_is_join_morphism.

Fact comp_is_join_morphism : join_morphism (f \o g).

Proof.

by move=> x y; rewrite /= !omorphU. Qed.

#[export]
HB.instance Definition _ := isJoinLatticeMorphism.Build d T d'' T'' (f \o g)
  comp_is_join_morphism.

End JoinCompFun.

End IdCompFun.

End LatticeMorphismTheory.
End LatticeMorphismTheory.

HB.mixin Record isBLatticeMorphism d (T : bLatticeType d)
    d' (T' : bLatticeType d') (apply : T -> T') := {
  omorph0_subproof : apply \bot = \bot;
}.

HB.mixin Record isTLatticeMorphism d (T : tLatticeType d)
    d' (T' : tLatticeType d') (apply : T -> T') := {
  omorph1_subproof : apply \top = \top;
}.

HB.structure Definition BLatticeMorphism d (T : bLatticeType d)
    d' (T' : bLatticeType d') := {f of isBLatticeMorphism d T d' T' f}.

HB.structure Definition TLatticeMorphism d (T : tLatticeType d)
    d' (T' : tLatticeType d') := {f of isTLatticeMorphism d T d' T' f}.

HB.structure Definition TBLatticeMorphism d (T : tbLatticeType d)
    d' (T' : tbLatticeType d') :=
  {f of @BLatticeMorphism d T d' T' f & @TLatticeMorphism d T d' T' f}.

Module TBLatticeMorphismExports.
Notation "{ 'blmorphism' T -> T' }" :=
  (@BLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation "{ 'tlmorphism' T -> T' }" :=
  (@TLatticeMorphism.type _ T%type _ T'%type) : type_scope.
Notation "{ 'tblmorphism' T -> T' }" :=
  (@TBLatticeMorphism.type _ T%type _ T'%type) : type_scope.
End TBLatticeMorphismExports.
HB.export TBLatticeMorphismExports.

Module Import BLatticeMorphismTheory.
Section BLatticeMorphismTheory.

Section Properties.

Variables (d : disp_t) (T : bLatticeType d).
Variables (d' : disp_t) (T' : bLatticeType d').
Variables (f : {blmorphism T -> T'}).

Lemma omorph0 : f \bot = \bot.

Proof.

exact: omorph0_subproof. Qed.

End Properties.

Section IdCompFun.

Variables (d : disp_t) (T : bLatticeType d).
Variables (d' : disp_t) (T' : bLatticeType d').
Variables (d'' : disp_t) (T'' : bLatticeType d'').
Variables (f : {blmorphism T' -> T''}) (g : {blmorphism T -> T'}).

Fact idfun_is_bottom_morphism : (@idfun T) \bot = \bot

Proof.

by []. Qed.

#[export]
HB.instance Definition _ := isBLatticeMorphism.Build d T d T idfun
idfun_is_bottom_morphism.

Fact comp_is_bottom_morphism : (f \o g) \bot = \bot.

Proof.

by rewrite /= !omorph0. Qed.

#[export]
HB.instance Definition _ := isBLatticeMorphism.Build d T d'' T'' (f \o g)
comp_is_bottom_morphism.

End IdCompFun.

End BLatticeMorphismTheory.
End BLatticeMorphismTheory.

Module Import TLatticeMorphismTheory.
Section TLatticeMorphismTheory.

Section Properties.

Variables (d : disp_t) (T : tLatticeType d).
Variables (d' : disp_t) (T' : tLatticeType d').
Variables (f : {tlmorphism T -> T'}).

Lemma omorph1 : f \top = \top.

Proof.

exact: omorph1_subproof. Qed.

End Properties.

Section IdCompFun.

Variables (d : disp_t) (T : tLatticeType d).
Variables (d' : disp_t) (T' : tLatticeType d').
Variables (d'' : disp_t) (T'' : tLatticeType d'').
Variables (f : {tlmorphism T' -> T''}) (g : {tlmorphism T -> T'}).

Fact idfun_is_top_morphism : (@idfun T) \top = \top

Proof.

by []. Qed.

#[export]
HB.instance Definition _ := isTLatticeMorphism.Build d T d T idfun
idfun_is_top_morphism.

Fact comp_is_top_morphism : (f \o g) \top = \top.

Proof.

by rewrite /= !omorph1. Qed.

Proof.

exact: opredI. Qed.

Lemma opredU (S : joinLatticeClosed T) : {in S &, forall u v, u `|` v \in S}.

Proof.

exact: opredU. Qed.

End LatticePred.

Section BLatticePred.

Variables (d : disp_t) (T : bLatticeType d).

Lemma opred0 (S : bLatticeClosed T) : \bot \in S.

Proof.

exact: opred0. Qed.

Lemma opred_joins (S : bJoinLatticeClosed T) I r (P : pred I) F :
(forall i, P i -> F i \in S) -> \join_(i <- r | P i) F i \in S.

Proof.

by move=> FS; elim/big_ind: _; [exact: opred0 | exact: opredU |]. Qed.

End BLatticePred.

Section TLatticePred.

Variables (d : disp_t) (T : tLatticeType d).

Lemma opred1 (S : tLatticeClosed T) : \top \in S.

Proof.

exact: opred1. Qed.

Lemma opred_meets (S : tMeetLatticeClosed T) I r (P : pred I) F :
(forall i, P i -> F i \in S) -> \meet_(i <- r | P i) F i \in S.

Proof.

by move=> FS; elim/big_ind: _; [exact: opred1 | exact: opredI |]. Qed.

Proof.

by move=> x y; apply: val_inj; rewrite !SubK meetC. Qed.

Let joinUC : commutative joinU.

Proof.

by move=> x y; apply: val_inj; rewrite !SubK joinC. Qed.

Let meetUA : associative meetU.

Proof.

by move=> x y z; apply: val_inj; rewrite !SubK meetA. Qed.

Let joinUA : associative joinU.

Proof.

by move=> x y z; apply: val_inj; rewrite !SubK joinA. Qed.

Lemma joinUKI y x : meetU x (joinU x y) = x.

Proof.

by apply: val_inj; rewrite !SubK joinKI. Qed.

Let meetUKU y x : joinU x (meetU x y) = x.

Proof.

by apply: val_inj; rewrite !SubK meetKU. Qed.

Let le_meetU x y : (x <= y) = (meetU x y == x).

Proof.

by rewrite -le_val -(inj_eq val_inj) SubK leEmeet. Qed.

HB.instance Definition _ := POrder_isLattice.Build d' U
meetUC joinUC meetUA joinUA joinUKI meetUKU le_meetU.

Fact valI : meet_morphism (val : U -> T).

Proof.

by move=> x y; rewrite !SubK. Qed.

Fact valU : join_morphism (val : U -> T).

Proof.

by move=> x y; rewrite !SubK. Qed.

Proof.

by rewrite -le_val /= SubK le0x. Qed.

HB.instance Definition _ := hasBottom.Build d' U le0x.

Fact val0 : (val : U -> T) \bot = \bot

Proof.

by rewrite SubK. Qed.

#[warning="-HB.no-new-instance"]
HB.instance Definition _ := isBSubLattice.Build d T S d' U val0.
HB.end.

HB.factory Record SubChoice_isBSubLattice
    d (T : bLatticeType d) S (d' : disp_t) U & SubChoice T S U := {
  opredI_subproof : meet_closed S;
  opredU_subproof : join_closed S;
  opred0_subproof : \bot \in S;
}.

HB.builders Context d T S d' U & SubChoice_isBSubLattice d T S d' U.
HB.instance Definition _ := SubChoice_isSubLattice.Build d T S d' U
  opredI_subproof opredU_subproof.
HB.instance Definition _ := SubPOrder_isBSubLattice.Build d T S d' U
  opred0_subproof.
HB.end.

HB.mixin Record isTSubLattice d (T : tLatticeType d) (S : pred T) d' U
    & SubType T S U & TLattice d' U := {
  val1_subproof : (val : U -> T) \top = \top;
}.

#[short(type="tMeetSubLattice")]
HB.structure Definition TMeetSubLattice d (T : tLatticeType d) S d' :=
  { U of @MeetSubLattice d T S d' U & TLattice d' U & isTSubLattice d T S d' U }.

#[short(type="tMeetSubBLattice")]
HB.structure Definition TMeetSubBLattice d (T : tLatticeType d) S d' :=
  { U of @TMeetSubLattice d T S d' U & TBLattice d' U }.

#[short(type="tSubLattice")]
HB.structure Definition TSubLattice d (T : tLatticeType d) S d' :=
  { U of @SubLattice d T S d' U & @TMeetSubLattice d T S d' U }.

#[short(type="tSubBLattice")]
HB.structure Definition TSubBLattice d (T : tLatticeType d) S d' :=
  { U of @TSubLattice d T S d' U & TBLattice d' U }.

#[export]
HB.instance Definition _ (d : disp_t) (T : tLatticeType d) (S : pred T)
    d' (U : TMeetSubLattice.type S d') :=
  isTLatticeMorphism.Build d' U d T val val1_subproof.

HB.factory Record SubPOrder_isTSubLattice d (T : tLatticeType d) S d' U
    & @SubPOrder d T S d' U & Lattice d' U := {
  opred1_subproof : \top \in S;
}.

HB.builders Context d T S d' U & SubPOrder_isTSubLattice d T S d' U.

Let inU v Sv : U := Sub v Sv.
Let oneU : U := inU opred1_subproof.

Fact lex1 x : x <= oneU

Proof.

by rewrite -le_val /= SubK lex1. Qed.

HB.instance Definition _ := hasTop.Build d' U lex1.

Fact val1 : (val : U -> T) \top = \top

Proof.

by rewrite SubK. Qed.

#[warning="-HB.no-new-instance"]
HB.instance Definition _ := isTSubLattice.Build d T S d' U val1.
HB.end.

HB.factory Record SubChoice_isTSubLattice
    d (T : tLatticeType d) S (d' : disp_t) U & SubChoice T S U := {
  opredI_subproof : meet_closed S;
  opredU_subproof : join_closed S;
  opred1_subproof : \top \in S;
}.

HB.builders Context d T S d' U & SubChoice_isTSubLattice d T S d' U.
HB.instance Definition _ := SubChoice_isSubLattice.Build d T S d' U
  opredI_subproof opredU_subproof.
HB.instance Definition _ := SubPOrder_isTSubLattice.Build d T S d' U
  opred1_subproof.
HB.end.

#[short(type="tbSubLattice")]
HB.structure Definition TBSubLattice d (T : tbLatticeType d) S d' :=
  { U of @BSubLattice d T S d' U & @TSubLattice d T S d' U}.

#[export]
HB.instance Definition _ (d : disp_t) (T : tbLatticeType d) (S : pred T) d'
    (U : TBSubLattice.type S d') := BLatticeMorphism.on (val : U -> T).

HB.factory Record SubPOrder_isTBSubLattice d (T : tbLatticeType d) S d' U
    & @SubPOrder d T S d' U & Lattice d' U := {
  opred0_subproof : \bot \in S;
  opred1_subproof : \top \in S;
}.

HB.builders Context d T S d' U & SubPOrder_isTBSubLattice d T S d' U.
HB.instance Definition _ := SubPOrder_isBSubLattice.Build d T S d' U
  opred0_subproof.
HB.instance Definition _ := SubPOrder_isTSubLattice.Build d T S d' U
  opred1_subproof.
HB.end.

HB.factory Record SubChoice_isTBSubLattice d (T : tbLatticeType d) S
    (d' : disp_t) U & SubChoice T S U := {
  opredI_subproof : meet_closed S;
  opredU_subproof : join_closed S;
  opred0_subproof : \bot \in S;
  opred1_subproof : \top \in S;
}.

HB.builders Context d T S d' U & SubChoice_isTBSubLattice d T S d' U.
HB.instance Definition _ := SubChoice_isSubLattice.Build d T S d' U
  opredI_subproof opredU_subproof.
HB.instance Definition _ := SubPOrder_isTBSubLattice.Build d T S d' U
  opred0_subproof opred1_subproof.
HB.end.

#[short(type="subOrder")]
HB.structure Definition SubOrder d (T : orderType d) S d' :=
  { U of @SubLattice d T S d' U & Total d' U }.

HB.factory Record SubLattice_isSubOrder d (T : orderType d) S d' U
    & @SubLattice d T S d' U := {}.

HB.builders Context d T S d' U & SubLattice_isSubOrder d T S d' U.
Lemma totalU : total (<=%O : rel U).

Proof.

by move=> x y; rewrite -!le_val le_total. Qed.

HB.instance Definition _ := Lattice_isTotal.Build d' U totalU.
HB.end.

HB.factory Record SubPOrder_isSubOrder d (T : orderType d) S d' U
& @SubPOrder d T S d' U := {}.

HB.builders Context d T S d' U & SubPOrder_isSubOrder d T S d' U.
Fact opredI : meet_closed S.

Proof.

by move=> x y Sx Sy; rewrite meetEtotal; case: leP. Qed.

Fact opredU : join_closed S.

Proof.

by move=> x y Sx Sy; rewrite joinEtotal; case: leP. Qed.

HB.instance Definition _ := SubPOrder_isSubLattice.Build d T S d' U opredI opredU.
HB.instance Definition _ := SubLattice_isSubOrder.Build d T S d' U.
HB.end.

HB.factory Record SubChoice_isSubOrder d (T : orderType d) S (d' : disp_t) U
    & @SubChoice T S U := {}.

HB.builders Context d T S d' U & SubChoice_isSubOrder d T S d' U.
HB.instance Definition _ := SubChoice_isSubPOrder.Build d T S d' U.
HB.instance Definition _ := SubPOrder_isSubOrder.Build d T S d' U.
HB.end.

Module SubOrderExports.

Notation "[ 'SubChoice_isSubPOrder' 'of' U 'by' <: ]" :=
  (SubChoice_isSubPOrder.Build _ _ _ _ U)
  (format "[ 'SubChoice_isSubPOrder' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubChoice_isSubPOrder' 'of' U 'by' <: 'with' disp ]" :=
  (SubChoice_isSubPOrder.Build _ _ _ disp U)
  (format "[ 'SubChoice_isSubPOrder' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: ]" :=
  (SubPOrder_isSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _))
  (format "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubPOrder_isSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _))
  (format "[ 'SubPOrder_isSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubChoice_isSubLattice' 'of' U 'by' <: ]" :=
  (SubChoice_isSubLattice.Build _ _ _ _ U (@opredI _ _ _) (@opredU _ _ _))
  (format "[ 'SubChoice_isSubLattice' 'of' U 'by' <: ]")
    : form_scope.
Notation "[ 'SubChoice_isSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubChoice_isSubLattice.Build _ _ _ disp U (@opredI _ _ _) (@opredU _ _ _))
  (format "[ 'SubChoice_isSubLattice' 'of' U 'by' <: 'with' disp ]")
    : form_scope.
Notation "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: ]" :=
  (SubPOrder_isBSubLattice.Build _ _ _ _ U (opred0 _))
  (format "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubPOrder_isBSubLattice.Build _ _ _ disp U (opred0 _))
  (format "[ 'SubPOrder_isBSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: ]" :=
  (SubChoice_isBSubLattice.Build _ _ _ _ U
     (@opredI _ _ _) (@opredU _ _ _) (opred0 _))
  (format "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubChoice_isBSubLattice.Build _ _ _ disp U
     (@opredI _ _ _) (@opredU _ _ _) (opred0 _))
  (format "[ 'SubChoice_isBSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: ]" :=
  (SubPOrder_isTSubLattice.Build _ _ _ _ U (opred1 _))
  (format "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubPOrder_isTSubLattice.Build _ _ _ disp U (opred1 _))
  (format "[ 'SubPOrder_isTSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: ]" :=
  (SubChoice_isTSubLattice.Build _ _ _ _ U
     (@opredI _ _ _) (@opredU _ _ _) (opred1 _))
  (format "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubChoice_isTSubLattice.Build _ _ _ disp U
     (@opredI _ _ _) (@opredU _ _ _) (opred1 _))
  (format "[ 'SubChoice_isTSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: ]" :=
  (SubPOrder_isTBSubLattice.Build _ _ _ _ U (opred0 _) (opred1 _))
  (format "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubPOrder_isTBSubLattice.Build _ _ _ disp U (opred0 _) (opred1 _))
  (format "[ 'SubPOrder_isTBSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: ]" :=
  (SubChoice_isTBSubLattice.Build _ _ _ _ U
     (@opredI _ _ _) (@opredU _ _ _) (opred0 _) (opred1 _))
  (format "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: 'with' disp ]" :=
  (SubChoice_isTBSubLattice.Build _ _ _ disp U
     (@opredI _ _ _) (@opredU _ _ _) (opred0 _) (opred1 _))
  (format "[ 'SubChoice_isTBSubLattice' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubLattice_isSubOrder' 'of' U 'by' <: ]" :=
  (SubLattice_isSubOrder.Build _ _ _ _ U)
  (format "[ 'SubLattice_isSubOrder' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubLattice_isSubOrder' 'of' U 'by' <: 'with' disp ]" :=
  (SubLattice_isSubOrder.Build _ _ _ disp U)
  (format "[ 'SubLattice_isSubOrder' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: ]" :=
  (SubPOrder_isSubOrder.Build _ _ _ _ U)
  (format "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: 'with' disp ]" :=
  (SubPOrder_isSubOrder.Build _ _ _ disp U)
  (format "[ 'SubPOrder_isSubOrder' 'of' U 'by' <: 'with' disp ]")
  : form_scope.
Notation "[ 'SubChoice_isSubOrder' 'of' U 'by' <: ]" :=
  (SubChoice_isSubOrder.Build _ _ _ _ U)
  (format "[ 'SubChoice_isSubOrder' 'of' U 'by' <: ]")
  : form_scope.
Notation "[ 'SubChoice_isSubOrder' 'of' U 'by' <: 'with' disp ]" :=
  (SubChoice_isSubOrder.Build _ _ _ disp U)
  (format "[ 'SubChoice_isSubOrder' 'of' U 'by' <: 'with' disp ]")
  : form_scope.

End SubOrderExports.
HB.export SubOrderExports.

Module DeprecatedSubOrder.

Section Total.
Context {disp : disp_t} {T : orderType disp} (P : {pred T}) (sT : subType P).

#[export]
HB.instance Definition _ :=
  SubPOrder_isSubOrder.Build disp T P disp (sub_type sT).

End Total.

Module Exports.
HB.reexport DeprecatedSubOrder.
Notation "[ 'POrder' 'of' T 'by' <: ]" :=
  (POrder.copy T%type (sub_type T%type))
  (format "[ 'POrder' 'of' T 'by' <: ]") : form_scope.
Notation "[ 'Order' 'of' T 'by' <: ]" :=
  (Total.copy T%type (sub_type T%type))
  (only parsing) : form_scope.
End Exports.
End DeprecatedSubOrder.
HB.export DeprecatedSubOrder.Exports.

INSTANCES

Instances on nat

This is an example of creation of multiple instances on the same type,
with distinct displays, using natural numbers.
We declare two distinct canonical orders:
- leq which is total, and where meet and join are minn and maxn, on nat
- dvdn which is partial, and where meet and join are gcdn and lcmn,
  on natdvd

The Module NatOrder defines leq as the canonical order on the type nat,
i.e., without creating an alias. We define and use nat_display and proceed
like a standard canonical structure declaration, except that we use this
display. We also use a single factory LeOrderMixin to instantiate three
different canonical declarations porderType, distrLatticeType, orderType.
We finish by providing theorems to convert the operations of ordered and
lattice types to their definition without structure abstraction.

Module NatOrder.
Section NatOrder.

Export NatOrder.

#[export]
HB.instance Definition _ :=
  Preorder_isPOrder.Build nat_display nat anti_leq.
#[export]
HB.instance Definition _ :=
  POrder_isTotal.Build nat_display nat leq_total.

End NatOrder.

Module Exports.
HB.reexport NatOrder.
End Exports.

End NatOrder.
HB.export NatOrder.Exports.

Module NatMonotonyTheory.
Section NatMonotonyTheory.

Export NatMonotonyTheory.

Context {disp : disp_t} {T : porderType disp}.
Variables (D : {pred nat}) (f : nat -> T).
Hypothesis Dconvex : {in D &, forall i j k, i < k < j -> k \in D}.

Lemma incn_inP : {in D, forall i, i.+1 \in D -> f i < f i.+1} ->
  {in D &, {mono f : i j / i <= j}}.

Proof.

by move=> f_inc; apply/le_mono_in/homo_ltn_lt_in. Qed.

Lemma decn_inP : {in D, forall i, i.+1 \in D -> f i > f i.+1} ->
{in D &, {mono f : i j /~ i <= j}}.

Proof.

by move=> f_dec; apply/le_nmono_in/nhomo_ltn_lt_in. Qed.

Lemma incnP : (forall i, f i < f i.+1) -> {mono f : i j / i <= j}.

Proof.

by move=> f_inc; apply/le_mono/homo_ltn_lt. Qed.

Lemma decnP : (forall i, f i > f i.+1) -> {mono f : i j /~ i <= j}.

Proof.

by move=> f_dec; apply/le_nmono/nhomo_ltn_lt. Qed.

End NatMonotonyTheory.
Arguments incn_inP {disp T} [D f].
Arguments decn_inP {disp T} [D f].
Arguments incnP {disp T} [f].
Arguments decnP {disp T} [f].
End NatMonotonyTheory.

The Module DvdSyntax introduces a new set of notations using the newly
created display dvd_display. We first define the display as an opaque
definition of type disp_t, and we use it as the first argument of the
operator which display we want to change from the default one (here le,
lt, dvd sdvd, meet, join, top and bottom, as well as big op notations on
gcd and lcm). This notations will now be used for any ordered type which
first parameter is set to dvd_display.

Module DvdSyntax.

Export DvdSyntax.

Notation gcd := (@meet dvd_display _).
Notation "@ 'gcd' T" := (@meet dvd_display T)
  (at level 10, T at level 8, only parsing) : function_scope.
Notation lcm := (@join dvd_display _).
Notation "@ 'lcm' T" := (@join dvd_display T)
  (at level 10, T at level 8, only parsing) : function_scope.

Notation "\gcd_ ( i <- r | P ) F" :=
  (\big[gcd/nat0]_(i <- r | P%B) F%O) : order_scope.
Notation "\gcd_ ( i <- r ) F" :=
  (\big[gcd/nat0]_(i <- r) F%O) : order_scope.
Notation "\gcd_ ( i | P ) F" :=
  (\big[gcd/nat0]_(i | P%B) F%O) : order_scope.
Notation "\gcd_ i F" :=
  (\big[gcd/nat0]_i F%O) : order_scope.
Notation "\gcd_ ( i : I | P ) F" :=
  (\big[gcd/nat0]_(i : I | P%B) F%O) (only parsing) :
  order_scope.
Notation "\gcd_ ( i : I ) F" :=
  (\big[gcd/nat0]_(i : I) F%O) (only parsing) : order_scope.
Notation "\gcd_ ( m <= i < n | P ) F" :=
  (\big[gcd/nat0]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\gcd_ ( m <= i < n ) F" :=
  (\big[gcd/nat0]_(m <= i < n) F%O) : order_scope.
Notation "\gcd_ ( i < n | P ) F" :=
  (\big[gcd/nat0]_(i < n | P%B) F%O) : order_scope.
Notation "\gcd_ ( i < n ) F" :=
  (\big[gcd/nat0]_(i < n) F%O) : order_scope.
Notation "\gcd_ ( i 'in' A | P ) F" :=
  (\big[gcd/nat0]_(i in A | P%B) F%O) : order_scope.
Notation "\gcd_ ( i 'in' A ) F" :=
  (\big[gcd/nat0]_(i in A) F%O) : order_scope.

Notation "\lcm_ ( i <- r | P ) F" :=
  (\big[lcm/nat1]_(i <- r | P%B) F%O) : order_scope.
Notation "\lcm_ ( i <- r ) F" :=
  (\big[lcm/nat1]_(i <- r) F%O) : order_scope.
Notation "\lcm_ ( i | P ) F" :=
  (\big[lcm/nat1]_(i | P%B) F%O) : order_scope.
Notation "\lcm_ i F" :=
  (\big[lcm/nat1]_i F%O) : order_scope.
Notation "\lcm_ ( i : I | P ) F" :=
  (\big[lcm/nat1]_(i : I | P%B) F%O) (only parsing) :
  order_scope.
Notation "\lcm_ ( i : I ) F" :=
  (\big[lcm/nat1]_(i : I) F%O) (only parsing) : order_scope.
Notation "\lcm_ ( m <= i < n | P ) F" :=
  (\big[lcm/nat1]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\lcm_ ( m <= i < n ) F" :=
  (\big[lcm/nat1]_(m <= i < n) F%O) : order_scope.
Notation "\lcm_ ( i < n | P ) F" :=
  (\big[lcm/nat1]_(i < n | P%B) F%O) : order_scope.
Notation "\lcm_ ( i < n ) F" :=
  (\big[lcm/nat1]_(i < n) F%O) : order_scope.
Notation "\lcm_ ( i 'in' A | P ) F" :=
  (\big[lcm/nat1]_(i in A | P%B) F%O) : order_scope.
Notation "\lcm_ ( i 'in' A ) F" :=
  (\big[lcm/nat1]_(i in A) F%O) : order_scope.

End DvdSyntax.

The Module NatDvd defines dvdn as the canonical order on NatDvd.t, which
is abbreviated using the notation natdvd at the end of the module.
We use the newly defined dvd_display, described above.
We first recover structures that are common to both nat and natdvd
(eqType, choiceType, countType) through the copy mechanism, then we use
a single factory MeetJoinMixin to instantiate both porderType and
distrLatticeType canonical structures, and end with top and bottom.
We finish by providing theorems to convert the operations of ordered and
lattice types to their definition without structure abstraction.

Module NatDvd.

Export NatDvd.

Section NatDvd.
Implicit Types (m n p : natdvd).

Lemma lcmnn n : lcmn n n = n.

Proof.

by case: n => // n; rewrite /lcmn gcdnn mulnK. Qed.

Lemma le_def m n : (m %| n) = (gcdn m n == m)%N.

Proof.

by apply/gcdn_idPl/eqP. Qed.

Lemma joinKI n m : gcdn m (lcmn m n) = m.

Proof.

by rewrite (gcdn_idPl _)// dvdn_lcml. Qed.

Lemma meetKU n m : lcmn m (gcdn m n) = m.

Proof.

by rewrite (lcmn_idPl _)// dvdn_gcdl. Qed.

Lemma meetUl : left_distributive gcdn lcmn.

Proof.

move=> [|m'] [|n'] [|p'] //=; rewrite ?lcmnn ?lcm0n ?lcmn0 ?gcd0n ?gcdn0//.
- by rewrite gcdnC meetKU.
- by rewrite lcmnC gcdnC meetKU.
apply: eqn_from_log; rewrite ?(gcdn_gt0, lcmn_gt0)//= => p.
by rewrite !(logn_gcd, logn_lcm) ?(gcdn_gt0, lcmn_gt0)// minn_maxl.
Qed.

Fact dvdn_anti : antisymmetric dvdn.

Proof.

by move=> a b => /andP[] /gcdn_idPl + /gcdn_idPr => ->. Qed.

#[export]
HB.instance Definition _ := Preorder_isPOrder.Build dvd_display t dvdn_anti.

#[export]
HB.instance Definition _ := @POrder_Meet_isDistrLattice.Build dvd_display t
gcdn lcmn gcdnC lcmnC gcdnA lcmnA joinKI meetKU le_def meetUl.

Import DvdSyntax.
Lemma sdvdE (m n : t) : (m %<| n) = (n != m) && (m %| n).

Proof.

exact/lt_def. Qed.

Lemma gcdE : gcd = gcdn :> (t -> t -> t)

Proof.

by []. Qed.

Lemma lcmE : lcm = lcmn :> (t -> t -> t)

Proof.

by []. Qed.

End NatDvd.
Module Exports.
HB.reexport NatDvd.
Definition sdvdEnat := sdvdE.
Definition gcdEnat := gcdE.
Definition lcmEnat := lcmE.
End Exports.
End NatDvd.
HB.export NatDvd.Exports.

Instances on ordinal

Module OrdinalOrder.

Export OrdinalOrder.

#[export]
HB.instance Definition _ (n : nat) :=
[SubChoice_isSubOrder of 'I_n by <: with ord_display].

End OrdinalOrder.
HB.export OrdinalOrder.

Instances on bool

Module BoolOrder.

Export BoolOrder.

Section BoolOrder.
Implicit Types (x y : bool).

Fact andbE x y : x && y = if (x < y)%N then x else y.

Proof.

by case: x y => [] []. Qed.

Fact orbE x y : x || y = if (x < y)%N then y else x.

Proof.

by case: x y => [] []. Qed.

Fact anti : antisymmetric (leq : rel bool).

Proof.

by move=> x y /anti_leq /(congr1 odd); rewrite !oddb. Qed.

#[export] HB.instance Definition _ := Preorder_isPOrder.Build bool_display bool
anti.

Fact orKb b a : a && (a || b) = a.

Proof.

by rewrite orbC orKb. Qed.

Fact andKb y x : x || x && y = x.

Proof.

by rewrite andbC andKb. Qed.

Fact leEmeet x y : (x <= y) = (x && y == x).

Proof.

by case: x; case: y. Qed.

#[export] HB.instance Definition _ := @POrder_Meet_isDistrLattice.Build
  bool_display bool andb orb andbC orbC andbA orbA orKb andKb leEmeet andb_orl.
#[export] HB.instance Definition _ := DistrLattice_isTotal.Build bool_display
  bool leq_total.
#[export] HB.instance Definition _ :=
  @TBDistrLattice_hasComplement.Build _ bool negb orbN andbN.

Lemma andEbool : meet = andb

Proof.

by []. Qed.

Lemma orEbool : meet = andb

Proof.

by []. Qed.

Lemma subEbool x y : x `\` y = x && ~~ y

Proof.

by []. Qed.

Lemma complEbool : compl = negb

Proof.

by []. Qed.

End BoolOrder.
Module Exports.
HB.reexport BoolOrder.
Definition leEbool := leEbool.
Definition ltEbool := ltEbool.
Definition andEbool := andEbool.
Definition orEbool := orEbool.
Definition subEbool := subEbool.
Definition complEbool := complEbool.
End Exports.
End BoolOrder.
HB.export BoolOrder.Exports.

Definition of prod_display

Module Import ProdSyntax.

Export ProdSyntax.

Local Notation "\bot" := (@bottom (prod_display _ _) _).
Local Notation "\top" := (@top (prod_display _ _) _).
Local Notation meet := (@meet (prod_display _ _) _).
Local Notation join := (@join (prod_display _ _) _).
Local Notation min := (@min (prod_display _ _) _).
Local Notation max := (@max (prod_display _ _) _).

Notation "x `&^p` y" := (meet x y) : order_scope.
Notation "x `|^p` y" := (join x y) : order_scope.

Notation "\join^p_ ( i <- r | P ) F" :=
  (\big[join / \bot]_(i <- r | P%B) F%O) : order_scope.
Notation "\join^p_ ( i <- r ) F" :=
  (\big[join / \bot]_(i <- r) F%O) : order_scope.
Notation "\join^p_ ( i | P ) F" :=
  (\big[join / \bot]_(i | P%B) F%O) : order_scope.
Notation "\join^p_ i F" :=
  (\big[join / \bot]_i F%O) : order_scope.
Notation "\join^p_ ( i : I | P ) F" :=
  (\big[join / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join^p_ ( i : I ) F" :=
  (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join^p_ ( m <= i < n | P ) F" :=
(\big[join / \bot]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\join^p_ ( m <= i < n ) F" :=
(\big[join / \bot]_(m <= i < n) F%O) : order_scope.
Notation "\join^p_ ( i < n | P ) F" :=
(\big[join / \bot]_(i < n | P%B) F%O) : order_scope.
Notation "\join^p_ ( i < n ) F" :=
(\big[join / \bot]_(i < n) F%O) : order_scope.
Notation "\join^p_ ( i 'in' A | P ) F" :=
(\big[join / \bot]_(i in A | P%B) F%O) : order_scope.
Notation "\join^p_ ( i 'in' A ) F" :=
(\big[join / \bot]_(i in A) F%O) : order_scope.

Notation "\meet^p_ ( i <- r | P ) F" :=
  (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet^p_ ( i <- r ) F" :=
  (\big[meet / \top]_(i <- r) F%O) : order_scope.
Notation "\meet^p_ ( i | P ) F" :=
  (\big[meet / \top]_(i | P%B) F%O) : order_scope.
Notation "\meet^p_ i F" :=
  (\big[meet / \top]_i F%O) : order_scope.
Notation "\meet^p_ ( i : I | P ) F" :=
  (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet^p_ ( i : I ) F" :=
  (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet^p_ ( m <= i < n | P ) F" :=
(\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\meet^p_ ( m <= i < n ) F" :=
(\big[meet / \top]_(m <= i < n) F%O) : order_scope.
Notation "\meet^p_ ( i < n | P ) F" :=
(\big[meet / \top]_(i < n | P%B) F%O) : order_scope.
Notation "\meet^p_ ( i < n ) F" :=
(\big[meet / \top]_(i < n) F%O) : order_scope.
Notation "\meet^p_ ( i 'in' A | P ) F" :=
(\big[meet / \top]_(i in A | P%B) F%O) : order_scope.
Notation "\meet^p_ ( i 'in' A ) F" :=
(\big[meet / \top]_(i in A) F%O) : order_scope.

Notation "\min^p_ i F" :=
  (\big[min/top]_i F) : order_scope.
Notation "\min^p_ ( i <- r | P ) F" :=
  (\big[min/top]_(i <- r | P%B) F%O) : order_scope.
Notation "\min^p_ ( i < r ) F" :=
  (\big[min/top]_(i <- r) F%O) : order_scope.
Notation "\min^p_ ( m <= i < n | P ) F" :=
  (\big[min/top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\min^p_ ( m <= i < n ) F" :=
  (\big[min/top]_(m <= i < n) F%O) : order_scope.
Notation "\min^p_ ( i | P ) F" :=
  (\big[min/top]_(i | P%B) F%O) : order_scope.
Notation "\min^p_ ( i : t | P ) F" :=
  (\big[min/top]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\min^p_ ( i : t ) F" :=
  (\big[min/top]_(i : t) F%O) (only parsing) : order_scope.
Notation "\min^p_ ( i < n | P ) F" :=
  (\big[min/top]_(i < n | P%B) F%O) : order_scope.
Notation "\min^p_ ( i < n ) F" :=
  (\big[min/top]_(i < n) F%O) : order_scope.
Notation "\min^p_ ( i 'in' A | P ) F" :=
  (\big[min/top]_(i in A | P%B) F%O) : order_scope.
Notation "\min^p_ ( i 'in' A ) F" :=
  (\big[min/top]_(i in A) F%O) : order_scope.

Notation "\max^p_ i F" :=
  (\big[max/bottom]_i F%O) : order_scope.
Notation "\max^p_ ( i <- r | P ) F" :=
  (\big[max/bottom]_(i <- r | P%B) F%O) : order_scope.
Notation "\max^p_ ( i < r ) F" :=
  (\big[max/bottom]_(i <- r) F%O) : order_scope.
Notation "\max^p_ ( m <= i < n | P ) F" :=
  (\big[max/bottom]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\max^p_ ( m <= i < n ) F" :=
  (\big[max/bottom]_(m <= i < n) F%O) : order_scope.
Notation "\max^p_ ( i | P ) F" :=
  (\big[max/bottom]_(i | P%B) F%O) : order_scope.
Notation "\max^p_ ( i : t | P ) F" :=
  (\big[max/bottom]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\max^p_ ( i : t ) F" :=
  (\big[max/bottom]_(i : t) F%O) (only parsing) : order_scope.
Notation "\max^p_ ( i < n | P ) F" :=
  (\big[max/bottom]_(i < n | P%B) F%O) : order_scope.
Notation "\max^p_ ( i < n ) F" :=
  (\big[max/bottom]_(i < n) F%O) : order_scope.
Notation "\max^p_ ( i 'in' A | P ) F" :=
  (\big[max/bottom]_(i in A | P%B) F%O) : order_scope.
Notation "\max^p_ ( i 'in' A ) F" :=
  (\big[max/bottom]_(i in A) F%O) : order_scope.

End ProdSyntax.

Module Import SeqProdSyntax.

Export SeqProdSyntax.

Local Notation "\bot" := (@bottom (seqprod_display _) _).
Local Notation "\top" := (@top (seqprod_display _) _).
Local Notation meet := (@meet (seqprod_display _) _).
Local Notation join := (@join (seqprod_display _) _).
Local Notation min := (@min (seqprod_display _) _).
Local Notation max := (@max (seqprod_display _) _).

Notation "x `&^sp` y" := (meet x y) : order_scope.
Notation "x `|^sp` y" := (join x y) : order_scope.

Notation "\join^sp_ ( i <- r | P ) F" :=
  (\big[join / \bot]_(i <- r | P%B) F%O) : order_scope.
Notation "\join^sp_ ( i <- r ) F" :=
  (\big[join / \bot]_(i <- r) F%O) : order_scope.
Notation "\join^sp_ ( i | P ) F" :=
  (\big[join / \bot]_(i | P%B) F%O) : order_scope.
Notation "\join^sp_ i F" :=
  (\big[join / \bot]_i F%O) : order_scope.
Notation "\join^sp_ ( i : I | P ) F" :=
  (\big[join / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join^sp_ ( i : I ) F" :=
  (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join^sp_ ( m <= i < n | P ) F" :=
(\big[join / \bot]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\join^sp_ ( m <= i < n ) F" :=
(\big[join / \bot]_(m <= i < n) F%O) : order_scope.
Notation "\join^sp_ ( i < n | P ) F" :=
(\big[join / \bot]_(i < n | P%B) F%O) : order_scope.
Notation "\join^sp_ ( i < n ) F" :=
(\big[join / \bot]_(i < n) F%O) : order_scope.
Notation "\join^sp_ ( i 'in' A | P ) F" :=
(\big[join / \bot]_(i in A | P%B) F%O) : order_scope.
Notation "\join^sp_ ( i 'in' A ) F" :=
(\big[join / \bot]_(i in A) F%O) : order_scope.

Notation "\meet^sp_ ( i <- r | P ) F" :=
  (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet^sp_ ( i <- r ) F" :=
  (\big[meet / \top]_(i <- r) F%O) : order_scope.
Notation "\meet^sp_ ( i | P ) F" :=
  (\big[meet / \top]_(i | P%B) F%O) : order_scope.
Notation "\meet^sp_ i F" :=
  (\big[meet / \top]_i F%O) : order_scope.
Notation "\meet^sp_ ( i : I | P ) F" :=
  (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet^sp_ ( i : I ) F" :=
  (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet^sp_ ( m <= i < n | P ) F" :=
(\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\meet^sp_ ( m <= i < n ) F" :=
(\big[meet / \top]_(m <= i < n) F%O) : order_scope.
Notation "\meet^sp_ ( i < n | P ) F" :=
(\big[meet / \top]_(i < n | P%B) F%O) : order_scope.
Notation "\meet^sp_ ( i < n ) F" :=
(\big[meet / \top]_(i < n) F%O) : order_scope.
Notation "\meet^sp_ ( i 'in' A | P ) F" :=
(\big[meet / \top]_(i in A | P%B) F%O) : order_scope.
Notation "\meet^sp_ ( i 'in' A ) F" :=
(\big[meet / \top]_(i in A) F%O) : order_scope.

Notation "\min^sp_ i F" :=
  (\big[min/top]_i F) : order_scope.
Notation "\min^sp_ ( i <- r | P ) F" :=
  (\big[min/top]_(i <- r | P%B) F%O) : order_scope.
Notation "\min^sp_ ( i < r ) F" :=
  (\big[min/top]_(i <- r) F%O) : order_scope.
Notation "\min^sp_ ( m <= i < n | P ) F" :=
  (\big[min/top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\min^sp_ ( m <= i < n ) F" :=
  (\big[min/top]_(m <= i < n) F%O) : order_scope.
Notation "\min^sp_ ( i | P ) F" :=
  (\big[min/top]_(i | P%B) F%O) : order_scope.
Notation "\min^sp_ ( i : t | P ) F" :=
  (\big[min/top]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\min^sp_ ( i : t ) F" :=
  (\big[min/top]_(i : t) F%O) (only parsing) : order_scope.
Notation "\min^sp_ ( i < n | P ) F" :=
  (\big[min/top]_(i < n | P%B) F%O) : order_scope.
Notation "\min^sp_ ( i < n ) F" :=
  (\big[min/top]_(i < n) F%O) : order_scope.
Notation "\min^sp_ ( i 'in' A | P ) F" :=
  (\big[min/top]_(i in A | P%B) F%O) : order_scope.
Notation "\min^sp_ ( i 'in' A ) F" :=
  (\big[min/top]_(i in A) F%O) : order_scope.

Notation "\max^sp_ i F" :=
  (\big[max/bottom]_i F%O) : order_scope.
Notation "\max^sp_ ( i <- r | P ) F" :=
  (\big[max/bottom]_(i <- r | P%B) F%O) : order_scope.
Notation "\max^sp_ ( i < r ) F" :=
  (\big[max/bottom]_(i <- r) F%O) : order_scope.
Notation "\max^sp_ ( m <= i < n | P ) F" :=
  (\big[max/bottom]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\max^sp_ ( m <= i < n ) F" :=
  (\big[max/bottom]_(m <= i < n) F%O) : order_scope.
Notation "\max^sp_ ( i | P ) F" :=
  (\big[max/bottom]_(i | P%B) F%O) : order_scope.
Notation "\max^sp_ ( i : t | P ) F" :=
  (\big[max/bottom]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\max^sp_ ( i : t ) F" :=
  (\big[max/bottom]_(i : t) F%O) (only parsing) : order_scope.
Notation "\max^sp_ ( i < n | P ) F" :=
  (\big[max/bottom]_(i < n | P%B) F%O) : order_scope.
Notation "\max^sp_ ( i < n ) F" :=
  (\big[max/bottom]_(i < n) F%O) : order_scope.
Notation "\max^sp_ ( i 'in' A | P ) F" :=
  (\big[max/bottom]_(i in A | P%B) F%O) : order_scope.
Notation "\max^sp_ ( i 'in' A ) F" :=
  (\big[max/bottom]_(i in A) F%O) : order_scope.

End SeqProdSyntax.

Module Import LexiSyntax.

Export LexiSyntax.

Notation meetlexi := (@meet (lexi_display _ _) _).
Notation joinlexi := (@join (lexi_display _ _) _).

Notation "x `&^l` y" := (meetlexi x y) : order_scope.
Notation "x `|^l` y" := (joinlexi x y) : order_scope.

Local Notation "\bot" := (@bottom (lexi_display _ _) _).
Local Notation "\top" := (@top (lexi_display _ _) _).
Local Notation meet := (@meet (lexi_display _ _) _).
Local Notation join := (@join (lexi_display _ _) _).
Local Notation min := (@min (lexi_display _ _) _).
Local Notation max := (@max (lexi_display _ _) _).

Notation "\join^l_ ( i <- r | P ) F" :=
  (\big[join / \bot]_(i <- r | P%B) F%O) : order_scope.
Notation "\join^l_ ( i <- r ) F" :=
  (\big[join / \bot]_(i <- r) F%O) : order_scope.
Notation "\join^l_ ( i | P ) F" :=
  (\big[join / \bot]_(i | P%B) F%O) : order_scope.
Notation "\join^l_ i F" :=
  (\big[join / \bot]_i F%O) : order_scope.
Notation "\join^l_ ( i : I | P ) F" :=
  (\big[join / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join^l_ ( i : I ) F" :=
  (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join^l_ ( m <= i < n | P ) F" :=
(\big[join / \bot]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\join^l_ ( m <= i < n ) F" :=
(\big[join / \bot]_(m <= i < n) F%O) : order_scope.
Notation "\join^l_ ( i < n | P ) F" :=
(\big[join / \bot]_(i < n | P%B) F%O) : order_scope.
Notation "\join^l_ ( i < n ) F" :=
(\big[join / \bot]_(i < n) F%O) : order_scope.
Notation "\join^l_ ( i 'in' A | P ) F" :=
(\big[join / \bot]_(i in A | P%B) F%O) : order_scope.
Notation "\join^l_ ( i 'in' A ) F" :=
(\big[join / \bot]_(i in A) F%O) : order_scope.

Notation "\meet^l_ ( i <- r | P ) F" :=
  (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet^l_ ( i <- r ) F" :=
  (\big[meet / \top]_(i <- r) F%O) : order_scope.
Notation "\meet^l_ ( i | P ) F" :=
  (\big[meet / \top]_(i | P%B) F%O) : order_scope.
Notation "\meet^l_ i F" :=
  (\big[meet / \top]_i F%O) : order_scope.
Notation "\meet^l_ ( i : I | P ) F" :=
  (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet^l_ ( i : I ) F" :=
  (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet^l_ ( m <= i < n | P ) F" :=
(\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\meet^l_ ( m <= i < n ) F" :=
(\big[meet / \top]_(m <= i < n) F%O) : order_scope.
Notation "\meet^l_ ( i < n | P ) F" :=
(\big[meet / \top]_(i < n | P%B) F%O) : order_scope.
Notation "\meet^l_ ( i < n ) F" :=
(\big[meet / \top]_(i < n) F%O) : order_scope.
Notation "\meet^l_ ( i 'in' A | P ) F" :=
(\big[meet / \top]_(i in A | P%B) F%O) : order_scope.
Notation "\meet^l_ ( i 'in' A ) F" :=
(\big[meet / \top]_(i in A) F%O) : order_scope.

Notation "\min^l_ i F" :=
  (\big[min/top]_i F) : order_scope.
Notation "\min^l_ ( i <- r | P ) F" :=
  (\big[min/top]_(i <- r | P%B) F%O) : order_scope.
Notation "\min^l_ ( i < r ) F" :=
  (\big[min/top]_(i <- r) F%O) : order_scope.
Notation "\min^l_ ( m <= i < n | P ) F" :=
  (\big[min/top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\min^l_ ( m <= i < n ) F" :=
  (\big[min/top]_(m <= i < n) F%O) : order_scope.
Notation "\min^l_ ( i | P ) F" :=
  (\big[min/top]_(i | P%B) F%O) : order_scope.
Notation "\min^l_ ( i : t | P ) F" :=
  (\big[min/top]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\min^l_ ( i : t ) F" :=
  (\big[min/top]_(i : t) F%O) (only parsing) : order_scope.
Notation "\min^l_ ( i < n | P ) F" :=
  (\big[min/top]_(i < n | P%B) F%O) : order_scope.
Notation "\min^l_ ( i < n ) F" :=
  (\big[min/top]_(i < n) F%O) : order_scope.
Notation "\min^l_ ( i 'in' A | P ) F" :=
  (\big[min/top]_(i in A | P%B) F%O) : order_scope.
Notation "\min^l_ ( i 'in' A ) F" :=
  (\big[min/top]_(i in A) F%O) : order_scope.

Notation "\max^l_ i F" :=
  (\big[max/bottom]_i F%O) : order_scope.
Notation "\max^l_ ( i <- r | P ) F" :=
  (\big[max/bottom]_(i <- r | P%B) F%O) : order_scope.
Notation "\max^l_ ( i < r ) F" :=
  (\big[max/bottom]_(i <- r) F%O) : order_scope.
Notation "\max^l_ ( m <= i < n | P ) F" :=
  (\big[max/bottom]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\max^l_ ( m <= i < n ) F" :=
  (\big[max/bottom]_(m <= i < n) F%O) : order_scope.
Notation "\max^l_ ( i | P ) F" :=
  (\big[max/bottom]_(i | P%B) F%O) : order_scope.
Notation "\max^l_ ( i : t | P ) F" :=
  (\big[max/bottom]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\max^l_ ( i : t ) F" :=
  (\big[max/bottom]_(i : t) F%O) (only parsing) : order_scope.
Notation "\max^l_ ( i < n | P ) F" :=
  (\big[max/bottom]_(i < n | P%B) F%O) : order_scope.
Notation "\max^l_ ( i < n ) F" :=
  (\big[max/bottom]_(i < n) F%O) : order_scope.
Notation "\max^l_ ( i 'in' A | P ) F" :=
  (\big[max/bottom]_(i in A | P%B) F%O) : order_scope.
Notation "\max^l_ ( i 'in' A ) F" :=
  (\big[max/bottom]_(i in A) F%O) : order_scope.

End LexiSyntax.

Module Import SeqLexiSyntax.

Export SeqLexiSyntax.

Notation meetlexi := (@meet (seqlexi_display _) _).
Notation joinlexi := (@join (seqlexi_display _) _).

Notation "x `&^l` y" := (meetlexi x y) : order_scope.
Notation "x `|^l` y" := (joinlexi x y) : order_scope.

Local Notation "\bot" := (@bottom (lexi_display _ _) _).
Local Notation "\top" := (@top (lexi_display _ _) _).
Local Notation meet := (@meet (lexi_display _ _) _).
Local Notation join := (@join (lexi_display _ _) _).
Local Notation min := (@min (lexi_display _ _) _).
Local Notation max := (@max (lexi_display _ _) _).

Notation "\join^l_ ( i <- r | P ) F" :=
  (\big[join / \bot]_(i <- r | P%B) F%O) : order_scope.
Notation "\join^l_ ( i <- r ) F" :=
  (\big[join / \bot]_(i <- r) F%O) : order_scope.
Notation "\join^l_ ( i | P ) F" :=
  (\big[join / \bot]_(i | P%B) F%O) : order_scope.
Notation "\join^l_ i F" :=
  (\big[join / \bot]_i F%O) : order_scope.
Notation "\join^l_ ( i : I | P ) F" :=
  (\big[join / \bot]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join^l_ ( i : I ) F" :=
  (\big[join / \bot]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join^l_ ( m <= i < n | P ) F" :=
(\big[join / \bot]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\join^l_ ( m <= i < n ) F" :=
(\big[join / \bot]_(m <= i < n) F%O) : order_scope.
Notation "\join^l_ ( i < n | P ) F" :=
(\big[join / \bot]_(i < n | P%B) F%O) : order_scope.
Notation "\join^l_ ( i < n ) F" :=
(\big[join / \bot]_(i < n) F%O) : order_scope.
Notation "\join^l_ ( i 'in' A | P ) F" :=
(\big[join / \bot]_(i in A | P%B) F%O) : order_scope.
Notation "\join^l_ ( i 'in' A ) F" :=
(\big[join / \bot]_(i in A) F%O) : order_scope.

Notation "\meet^l_ ( i <- r | P ) F" :=
  (\big[meet / \top]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet^l_ ( i <- r ) F" :=
  (\big[meet / \top]_(i <- r) F%O) : order_scope.
Notation "\meet^l_ ( i | P ) F" :=
  (\big[meet / \top]_(i | P%B) F%O) : order_scope.
Notation "\meet^l_ i F" :=
  (\big[meet / \top]_i F%O) : order_scope.
Notation "\meet^l_ ( i : I | P ) F" :=
  (\big[meet / \top]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet^l_ ( i : I ) F" :=
  (\big[meet / \top]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet^l_ ( m <= i < n | P ) F" :=
(\big[meet / \top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\meet^l_ ( m <= i < n ) F" :=
(\big[meet / \top]_(m <= i < n) F%O) : order_scope.
Notation "\meet^l_ ( i < n | P ) F" :=
(\big[meet / \top]_(i < n | P%B) F%O) : order_scope.
Notation "\meet^l_ ( i < n ) F" :=
(\big[meet / \top]_(i < n) F%O) : order_scope.
Notation "\meet^l_ ( i 'in' A | P ) F" :=
(\big[meet / \top]_(i in A | P%B) F%O) : order_scope.
Notation "\meet^l_ ( i 'in' A ) F" :=
(\big[meet / \top]_(i in A) F%O) : order_scope.

Notation "\min^l_ i F" :=
  (\big[min/top]_i F) : order_scope.
Notation "\min^l_ ( i <- r | P ) F" :=
  (\big[min/top]_(i <- r | P%B) F%O) : order_scope.
Notation "\min^l_ ( i < r ) F" :=
  (\big[min/top]_(i <- r) F%O) : order_scope.
Notation "\min^l_ ( m <= i < n | P ) F" :=
  (\big[min/top]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\min^l_ ( m <= i < n ) F" :=
  (\big[min/top]_(m <= i < n) F%O) : order_scope.
Notation "\min^l_ ( i | P ) F" :=
  (\big[min/top]_(i | P%B) F%O) : order_scope.
Notation "\min^l_ ( i : t | P ) F" :=
  (\big[min/top]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\min^l_ ( i : t ) F" :=
  (\big[min/top]_(i : t) F%O) (only parsing) : order_scope.
Notation "\min^l_ ( i < n | P ) F" :=
  (\big[min/top]_(i < n | P%B) F%O) : order_scope.
Notation "\min^l_ ( i < n ) F" :=
  (\big[min/top]_(i < n) F%O) : order_scope.
Notation "\min^l_ ( i 'in' A | P ) F" :=
  (\big[min/top]_(i in A | P%B) F%O) : order_scope.
Notation "\min^l_ ( i 'in' A ) F" :=
  (\big[min/top]_(i in A) F%O) : order_scope.

Notation "\max^l_ i F" :=
  (\big[max/bottom]_i F%O) : order_scope.
Notation "\max^l_ ( i <- r | P ) F" :=
  (\big[max/bottom]_(i <- r | P%B) F%O) : order_scope.
Notation "\max^l_ ( i < r ) F" :=
  (\big[max/bottom]_(i <- r) F%O) : order_scope.
Notation "\max^l_ ( m <= i < n | P ) F" :=
  (\big[max/bottom]_(m <= i < n | P%B) F%O) : order_scope.
Notation "\max^l_ ( m <= i < n ) F" :=
  (\big[max/bottom]_(m <= i < n) F%O) : order_scope.
Notation "\max^l_ ( i | P ) F" :=
  (\big[max/bottom]_(i | P%B) F%O) : order_scope.
Notation "\max^l_ ( i : t | P ) F" :=
  (\big[max/bottom]_(i : t | P%B) F%O) (only parsing) : order_scope.
Notation "\max^l_ ( i : t ) F" :=
  (\big[max/bottom]_(i : t) F%O) (only parsing) : order_scope.
Notation "\max^l_ ( i < n | P ) F" :=
  (\big[max/bottom]_(i < n | P%B) F%O) : order_scope.
Notation "\max^l_ ( i < n ) F" :=
  (\big[max/bottom]_(i < n) F%O) : order_scope.
Notation "\max^l_ ( i 'in' A | P ) F" :=
  (\big[max/bottom]_(i in A | P%B) F%O) : order_scope.
Notation "\max^l_ ( i 'in' A ) F" :=
  (\big[max/bottom]_(i in A) F%O) : order_scope.

End SeqLexiSyntax.

We declare an alias of the cartesian product
which has canonical product order.

Module ProdOrder.

Export ProdOrder.

Local Open Scope type_scope.

Section POrder.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : porderType disp1) (T2 : porderType disp2).

Fact anti : antisymmetric (@le disp1 disp2 disp2 T1 T2).

Proof.

case=> [? ?] [? ?].
by rewrite andbAC andbA andbAC -andbA => /= /andP [] /le_anti -> /le_anti ->.
Qed.

End POrder.

Section POrder.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : porderType disp1) (T2 : porderType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Let T1' : Type := T1.
HB.instance Definition _ := POrder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := POrder.on T2'.

#[export]
HB.instance Definition _ :=
Preorder_isDuallyPOrder.Build disp3 (T1 * T2)
(@anti _ _ T1' T2') (@anti _ _ T1^d T2^d).

Lemma ltEprod x y : (x < y) = [&& x != y, x.1 <= y.1 & x.2 <= y.2].

Proof.

by rewrite lt_neqAle. Qed.

Lemma lt_pair (x1 y1 : T1) (x2 y2 : T2) : ((x1, x2) < (y1, y2) :> T1 * T2) =
[&& (x1 != y1) || (x2 != y2), x1 <= y1 & x2 <= y2].

Proof.

by rewrite ltEprod negb_and. Qed.

End POrder.

Section MeetSemilattice.
Context (disp1 disp2 : disp_t).
Context (T1 : meetSemilatticeType disp1) (T2 : meetSemilatticeType disp2).
Local Notation "T1 * T2" := (type (Disp tt tt) T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Let meet x y := (x.1 `&` y.1, x.2 `&` y.2).

Fact lexI x y z : (x <= meet y z) = (x <= y) && (x <= z).

Proof.

by rewrite leEprod !lexI andbACA. Qed.

End MeetSemilattice.

Section MeetSemilattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : meetSemilatticeType disp1) (T2 : meetSemilatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Let T1' : Type := T1.
HB.instance Definition _ := MeetSemilattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := MeetSemilattice.on T2'.

Definition meet x y := (x.1 `&` y.1, x.2 `&` y.2).

#[export]
HB.instance Definition _ :=
@POrder_isMeetSemilattice.Build disp3 (T1 * T2) meet (@lexI _ _ T1' T2').

Lemma meetEprod x y : x `&` y = (x.1 `&` y.1, x.2 `&` y.2)

Proof.

by []. Qed.

End MeetSemilattice.

Section JoinSemilattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : joinSemilatticeType disp1) (T2 : joinSemilatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Definition join x y := (x.1 `|` y.1, x.2 `|` y.2).

#[export]
HB.instance Definition _ :=
@POrder_isJoinSemilattice.Build disp3 (T1 * T2) join
(fun x y z => @lexI _ _ T1^d T2^d z x y).

Lemma joinEprod x y : x `|` y = (x.1 `|` y.1, x.2 `|` y.2)

Proof.

by []. Qed.

Proof.

by move=> ? ? ?; rewrite meetEprod !meetUl. Qed.

End DistrLattice.

Section DistrLattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : distrLatticeType disp1) (T2 : distrLatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.

Let T1' : Type := T1.
HB.instance Definition _ := DistrLattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := DistrLattice.on T2'.

#[export]
HB.instance Definition _ := Lattice_isDistributive.Build disp3 (T1 * T2)
  (@meetUl _ _ T1' T2') (@meetUl _ _ T1^d T2^d).

End DistrLattice.

#[export]
HB.instance Definition _ (disp1 disp2 disp3 : disp_t)
  (T1 : bDistrLatticeType disp1) (T2 : bDistrLatticeType disp2) :=
  POrder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (disp1 disp2 disp3 : disp_t)
  (T1 : tDistrLatticeType disp1) (T2 : tDistrLatticeType disp2) :=
  POrder.on (type disp3 T1 T2).
#[export]
HB.instance Definition _ (disp1 disp2 disp3 : disp_t)
  (T1 : tbDistrLatticeType disp1) (T2 : tbDistrLatticeType disp2) :=
  POrder.on (type disp3 T1 T2).

Section CDistrLattice.
Context (disp1 disp2 : disp_t).
Context (T1 : cDistrLatticeType disp1) (T2 : cDistrLatticeType disp2).
Local Notation "T1 * T2" := (type (Disp tt tt) T1 T2) : type_scope.
Implicit Types (x y z : T1 * T2).

Let rcompl x y z := (rcompl x.1 y.1 z.1, rcompl x.2 y.2 z.2).

Fact rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof.

by rewrite !(meetEprod, joinEprod) !rcomplPmeet. Qed.

End CDistrLattice.

Section CDistrLattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : cDistrLatticeType disp1) (T2 : cDistrLatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y z : T1 * T2).

Let T1' : Type := T1.
HB.instance Definition _ := CDistrLattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := CDistrLattice.on T2'.

Definition rcompl x y z := (rcompl x.1 y.1 z.1, rcompl x.2 y.2 z.2).

#[export]
HB.instance Definition _ :=
  @DistrLattice_hasRelativeComplement.Build disp3 (T1 * T2)
    rcompl (@rcomplPmeet _ _ T1' T2')
    (fun x y => @rcomplPmeet _ _ T1^d T2^d y x).

Lemma rcomplEprod x y z :
  rcompl x y z = (Order.rcompl x.1 y.1 z.1, Order.rcompl x.2 y.2 z.2).

Proof.

by []. Qed.

End CDistrLattice.

Section CBDistrLattice.
Context (disp1 disp2 : disp_t).
Context (T1 : cbDistrLatticeType disp1) (T2 : cbDistrLatticeType disp2).
Local Notation "T1 * T2" := (type (Disp tt tt) T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Let diff x y := (diff x.1 y.1, diff x.2 y.2).

Fact diffErcompl x y : diff x y = rcompl \bot x y.

Proof.

by rewrite /diff !diffErcompl. Qed.

End CBDistrLattice.

Section CBDistrLattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : cbDistrLatticeType disp1) (T2 : cbDistrLatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Let T1' : Type := T1.
HB.instance Definition _ := CBDistrLattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := CBDistrLattice.on T2'.

Definition diff x y := (diff x.1 y.1, diff x.2 y.2).

#[export]
HB.instance Definition _ :=
@CDistrLattice_hasSectionalComplement.Build disp3 (T1 * T2)
diff (@diffErcompl _ _ T1' T2').

Lemma diffEprod x y : x `\` y = (x.1 `\` y.1, x.2 `\` y.2)

Proof.

by []. Qed.

End CBDistrLattice.

Section CTDistrLattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : ctDistrLatticeType disp1) (T2 : ctDistrLatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Definition codiff x y := (codiff x.1 y.1, codiff x.2 y.2).

#[export]
HB.instance Definition _ :=
  @CDistrLattice_hasDualSectionalComplement.Build disp3 (T1 * T2)
    codiff (@diffErcompl _ _ T1^d T2^d).

Lemma codiffEprod x y :
  codiff x y = (Order.codiff x.1 y.1, Order.codiff x.2 y.2).

Proof.

by []. Qed.

End CTDistrLattice.

Section CTBDistrLattice.
Context (disp1 disp2 : disp_t).
Context (T1 : ctbDistrLatticeType disp1) (T2 : ctbDistrLatticeType disp2).
Local Notation "T1 * T2" := (type (Disp tt tt) T1 T2) : type_scope.
Implicit Types (x : T1 * T2).

Let compl x := (~` x.1, ~` x.2).

Fact complEdiff x : compl x = (\top : T1 * T2) `\` x.

Proof.

by rewrite /compl !complEdiff. Qed.

End CTBDistrLattice.

Section CTBDistrLattice.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : ctbDistrLatticeType disp1) (T2 : ctbDistrLatticeType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x : T1 * T2).

Let T1' : Type := T1.
HB.instance Definition _ := CTBDistrLattice.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := CTBDistrLattice.on T2'.

Definition compl x := (~` x.1, ~` x.2).

#[export]
HB.instance Definition _ := @CDistrLattice_hasComplement.Build _ (T1 * T2) compl
(@complEdiff _ _ T1' T2') (@complEdiff _ _ T1^d T2^d).

Lemma complEprod x : ~` x = (~` x.1, ~` x.2)

Proof.

by []. Qed.

We declare lexicographic ordering on dependent pairs.

Module SigmaOrder.
Section SigmaOrder.

Context {disp1 disp2 : disp_t}.

Section POrder.
Context (T : porderType disp1) (T' : T -> porderType disp2).
Implicit Types (x y : {t : T & T' t}).

Definition le x y := (tag x <= tag y) &&
((tag x >= tag y) ==> (tagged x <= tagged_as x y)).
Definition lt x y := (tag x <= tag y) &&
((tag x >= tag y) ==> (tagged x < tagged_as x y)).

Fact refl : reflexive le.

Proof.

by move=> [x x']; rewrite /le tagged_asE/= !lexx. Qed.

Fact anti : antisymmetric le.

Proof.

rewrite /le => -[x x'] [y y']/=; case: comparableP => //= eq_xy.
by case: _ / eq_xy in y' *; rewrite !tagged_asE => /le_anti ->.
Qed.

Fact trans : transitive le.

Proof.

move=> [y y'] [x x'] [z z'] /andP[/= lexy lexy'] /andP[/= leyz leyz'].
rewrite /= /le (le_trans lexy) //=; apply/implyP => lezx.
elim: _ / (@le_anti _ _ x y) in y' z' lexy' leyz' *; last first.
by rewrite lexy (le_trans leyz).
elim: _ / (@le_anti _ _ x z) in z' leyz' *; last by rewrite (le_trans lexy).
by rewrite lexx !tagged_asE/= in lexy' leyz' *; rewrite (le_trans lexy').
Qed.

Fact lt_le_def x y : lt x y = le x y && ~~ le y x.

Proof.

rewrite /lt /le; case: x y => [x x'] [y y']//=.
case: (comparableP x y) => //= xy.
by subst y; rewrite !tagged_asE lt_le_def.
Qed.

#[export]
HB.instance Definition _ :=
  isPreorder.Build disp2 {t : T & T' t} lt_le_def refl trans.
#[export]
HB.instance Definition _ :=
  Preorder_isPOrder.Build disp2 {t : T & T' t} anti.

Lemma leEsig x y : (x <= y) =
  (tag x <= tag y) && ((tag x >= tag y) ==> (tagged x <= tagged_as x y)).

Proof.

by []. Qed.

Lemma ltEsig x y : (x < y) =
(tag x <= tag y) && ((tag x >= tag y) ==> (tagged x < tagged_as x y)).

Proof.

by []. Qed.

Lemma le_Taggedl x (u : T' (tag x)) : (Tagged T' u <= x) = (u <= tagged x).

Proof.

by case: x => [t v]/= in u *; rewrite leEsig/= lexx/= tagged_asE. Qed.

Lemma le_Taggedr x (u : T' (tag x)) : (x <= Tagged T' u) = (tagged x <= u).

Proof.

by case: x => [t v]/= in u *; rewrite leEsig/= lexx/= tagged_asE. Qed.

Lemma lt_Taggedl x (u : T' (tag x)) : (Tagged T' u < x) = (u < tagged x).

Proof.

by case: x => [t v]/= in u *; rewrite ltEsig/= lexx/= tagged_asE. Qed.

Lemma lt_Taggedr x (u : T' (tag x)) : (x < Tagged T' u) = (tagged x < u).

Proof.

by case: x => [t v]/= in u *; rewrite ltEsig/= lexx/= tagged_asE. Qed.

End POrder.

Section BPOrder.
Context (T : bPOrderType disp1) (T' : T -> bPOrderType disp2).

Fact le0x (x : {t : T & T' t}) : Tagged T' (\bot : T' \bot) <= x.

Proof.

by rewrite leEsig /= !le0x implybT. Qed.

#[export]
HB.instance Definition _ := hasBottom.Build _ {t : T & T' t} le0x.

Lemma botEsig : \bot = Tagged T' (\bot : T' \bot)

Proof.

by []. Qed.

End BPOrder.

Section TPOrder.
Context (T : tPOrderType disp1) (T' : T -> tPOrderType disp2).

Fact lex1 (x : {t : T & T' t}) : x <= Tagged T' (\top : T' \top).

Proof.

rewrite leEsig /=; case: comparableP (lex1 (tag x)) => //=.
by case: x => //= x px x0; rewrite x0 in px *; rewrite tagged_asE lex1.
Qed.

#[export]
HB.instance Definition _ := hasTop.Build _ {t : T & T' t} lex1.

Lemma topEsig : \top = Tagged T' (\top : T' \top)

Proof.

by []. Qed.

End TPOrder.

Section Total.
Context (T : orderType disp1) (T' : T -> orderType disp2).
Implicit Types (x y : {t : T & T' t}).

Fact total : total (<=%O : rel {t : T & T' t}).

Proof.

move=> x y; rewrite !leEsig; case: (ltgtP (tag x) (tag y)) => //=.
case: x y => [x x'] [y y']/= eqxy; elim: _ /eqxy in y' *.
by rewrite !tagged_asE le_total.
Qed.

We declare an alias of the cartesian product,
which has canonical lexicographic order.

Module ProdLexiOrder.

Export ProdLexiOrder.

Local Open Scope type_scope.

Section POrder.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : porderType disp1) (T2 : porderType disp2).

Fact anti : antisymmetric (@le disp1 disp2 disp2 T1 T2).

Proof.

by rewrite /le => -[x x'] [y y'] /=; case: comparableP => //= -> /le_anti->.
Qed.

End POrder.

Section POrder.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : porderType disp1) (T2 : porderType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.
Implicit Types (x y : T1 * T2).

Let T1' : Type := T1.
HB.instance Definition _ := POrder.on T1'.
Let T2' : Type := T2.
HB.instance Definition _ := POrder.on T2'.

#[export]
HB.instance Definition _ :=
Preorder_isDuallyPOrder.Build disp3 (T1 * T2)
(@anti _ _ T1' T2') (@anti _ _ T1^d T2^d).

End POrder.

Section Total.
Context (disp1 disp2 disp3 : disp_t).
Context (T1 : orderType disp1) (T2 : orderType disp2).
Local Notation "T1 * T2" := (type disp3 T1 T2) : type_scope.

Fact total : total (<=%O : rel (T1 * T2)).

Proof.

by move=> x y; rewrite !leEprodlexi; case: ltgtP => //= _; exact: le_total.
Qed.

We declare an alias of the sequences,
which has canonical product order.

Module SeqProdOrder.

Export SeqProdOrder.

Section SeqProdOrder.

Context {disp disp' : disp_t}.

Local Notation seq := (type disp').

Section POrder.
Variable T : porderType disp.
Implicit Types s : seq T.

Fact anti : antisymmetric (@le disp disp' T).

Proof.

by elim=> [|x s ihs] [|y s'] //=; rewrite andbACA => /andP[/le_anti-> /ihs->].
Qed.

#[export]
HB.instance Definition _ := Preorder_isPOrder.Build disp' (seq T) anti.

End POrder.

Section MeetSemilattice.
Context (T : meetSemilatticeType disp).
Implicit Types (s : seq T).

Fixpoint meet s1 s2 :=
  match s1, s2 with
    | x1 :: s1', x2 :: s2' => (x1 `&` x2) :: meet s1' s2'
    | _, _ => [::]
  end.

Fact lexI s1 s2 s3 : (s1 <= meet s2 s3) = (s1 <= s2) && (s1 <= s3).

Proof.

elim: s1 s2 s3 => [|x s1 IHs1] [|y s2] [|z s3] //=; first by rewrite andbF.
by rewrite leEseq lexI IHs1 andbACA.
Qed.

#[export]
HB.instance Definition _ := @POrder_isMeetSemilattice.Build _ (seq T) meet lexI.

Lemma meetEseq s1 s2 : s1 `&` s2 = [seq x.1 `&` x.2 | x <- zip s1 s2].

Proof.

by elim: s1 s2 => [|x s1 ihs1] [|y s2]//=; rewrite -ihs1. Qed.

Lemma meet_cons x1 s1 x2 s2 :
(x1 :: s1 : seq T) `&` (x2 :: s2) = (x1 `&` x2) :: s1 `&` s2.

Proof.

by []. Qed.

End MeetSemilattice.

Section JoinSemilattice.
Context (T : joinSemilatticeType disp).
Implicit Types (s : seq T).

Fixpoint join s1 s2 :=
  match s1, s2 with
    | [::], _ => s2 | _, [::] => s1
    | x1 :: s1', x2 :: s2' => (x1 `|` x2) :: join s1' s2'
  end.

Fact leUx s1 s2 s3 : (join s1 s2 <= s3) = (s1 <= s3) && (s2 <= s3).

Proof.

elim : s1 s2 s3 => [|x s1 IHs1] [|y s2] [|z s3] //=; first by rewrite andbT.
by rewrite leEseq leUx IHs1 andbACA.
Qed.

Proof.

by case: s1. Qed.

Lemma join_cons x1 s1 x2 s2 :
(x1 :: s1 : seq T) `|` (x2 :: s2) = (x1 `|` x2) :: s1 `|` s2.

Proof.

by []. Qed.

End JoinSemilattice.

#[export]
HB.instance Definition _ (T : latticeType disp) := POrder.on (seq T).

Section DistrLattice.
Context (T : distrLatticeType disp).

Fact meetUl : left_distributive (@meet T) (@join T).

Proof.

by elim=> [|? ? ih] [|? ?] [|? ?] //=; rewrite meetUl ih. Qed.

#[export]
HB.instance Definition _ := Lattice_Meet_isDistrLattice.Build _ (seq T) meetUl.

End DistrLattice.

End SeqProdOrder.

Module Exports.

HB.reexport SeqProdOrder.

Definition meetEseq := @meetEseq.
Definition meet_cons := @meet_cons.
Definition joinEseq := @joinEseq.

End Exports.
End SeqProdOrder.
HB.export SeqProdOrder.Exports.

Module DefaultSeqProdOrder.

Export DefaultSeqProdOrder.

Section DefaultSeqProdOrder.
Context {disp : disp_t}.

HB.instance Definition _ (T : porderType disp) :=
  POrder.copy (seq T) (seqprod T).
HB.instance Definition _ (T : meetSemilatticeType disp) :=
  BMeetSemilattice.copy (seq T) (seqprod T).
HB.instance Definition _ (T : joinSemilatticeType disp) :=
  BJoinSemilattice.copy (seq T) (seqprod T).
HB.instance Definition _ (T : latticeType disp) :=
  BLattice.copy (seq T) (seqprod T).
HB.instance Definition _ (T : distrLatticeType disp) :=
  BDistrLattice.copy (seq T) (seqprod T).

End DefaultSeqProdOrder.
End DefaultSeqProdOrder.

We declare an alias of the sequences,
which has canonical lexicographic order.

Module SeqLexiOrder.

Export SeqLexiOrder.

Section SeqLexiOrder.

Context {disp disp' : disp_t}.

Local Notation seq := (type disp').

Section POrder.
Variable T : porderType disp.
Implicit Types s : seq T.

Fact anti: antisymmetric (@le disp disp' T).

Proof.

move=> x y /andP []; elim: x y => [|x sx ih] [|y sy] //=.
by case: comparableP => //= -> lesxsy /(ih _ lesxsy) ->.
Qed.

#[export]
HB.instance Definition _ := Preorder_isPOrder.Build disp' (seq T) anti.

Lemma neqhead_lexiE (x y : T) s1 s2 : x != y ->
(x :: s1 <= y :: s2 :> seq _) = (x < y).

Proof.

by rewrite lexi_cons; case: comparableP. Qed.

Lemma neqhead_ltxiE (x y : T) s1 s2 : x != y ->
(x :: s1 < y :: s2 :> seq _) = (x < y).

Proof.

by rewrite ltxi_cons; case: (comparableP x y). Qed.

End POrder.

Section Total.
Context (T : orderType disp).

Fact total : total (<=%O : rel (seq T)).

Proof.

by elim=> [|x1 s1 ihs1] [|x2 s2]//=; rewrite !lexi_cons; case: ltgtP => /=.
Qed.

#[export]
HB.instance Definition _ := POrder_isTotal.Build _ (seq T) total.

End Total.

End SeqLexiOrder.

Module Exports.

HB.reexport SeqLexiOrder.

Definition neqhead_lexiE := @neqhead_lexiE.
Definition neqhead_ltxiE := @neqhead_ltxiE.

End Exports.
End SeqLexiOrder.
HB.export SeqLexiOrder.Exports.

Module DefaultSeqLexiOrder.

Export DefaultSeqLexiOrder.

Section DefaultSeqLexiOrder.
Context {disp : disp_t}.

HB.instance Definition _ (T : porderType disp) :=
POrder.copy (seq T) (seqlexi T).
HB.instance Definition _ (T : orderType disp) :=
BTotal.copy (seq T) (seqlexi T).

End DefaultSeqLexiOrder.
End DefaultSeqLexiOrder.

We declare an alias of the tuples,
which has canonical product order.

Module TupleProdOrder.

Export TupleProdOrder.

Import DefaultSeqProdOrder.

Section TupleProdOrder.

Context {disp disp' : disp_t}.
Local Notation "n .-tuple" := (type disp' n) : type_scope.

Section POrder.
Implicit Types (T : porderType disp).

#[export, warning="-HB.no-new-instance"]
HB.instance Definition _ n T := SubChoice.on (n.-tuple T).
#[export]
HB.instance Definition _ n T :=
  [SubChoice_isSubPOrder of n.-tuple T by <: with disp'].

End POrder.

#[export]
HB.instance Definition _ (n : nat) (T : bPOrderType disp) :=
  POrder.on (n.-tuple T).

#[export]
HB.instance Definition _ (n : nat) (T : tPOrderType disp) :=
  POrder.on (n.-tuple T).
#[export]
HB.instance Definition _ (n : nat) (T : tbPOrderType disp) :=
  POrder.on (n.-tuple T).

Section MeetSemilattice.
Context (n : nat) (T : meetSemilatticeType disp).
Implicit Types (t : n.-tuple T).

Definition meet t1 t2 : n.-tuple T := [tuple tnth t1 i `&` tnth t2 i | i < n].

Fact lexI t1 t2 t3 : (t1 <= meet t2 t3) = (t1 <= t2) && (t1 <= t3).

Proof.

rewrite !leEtprod; apply/forallP/andP => [H|[Ht12 Ht13] i]; last first.
by rewrite tnth_mktuple lexI (forallP Ht12) (forallP Ht13).
by split; apply/forallP => i; move: (H i); rewrite tnth_mktuple lexI => /andP[].
Qed.

#[export]
HB.instance Definition _ :=
@POrder_isMeetSemilattice.Build _ (n.-tuple T) meet lexI.

Lemma tnth_meet t1 t2 i : tnth (t1 `&` t2) i = tnth t1 i `&` tnth t2 i.

Proof.

exact: tnth_mktuple. Qed.

Lemma meetEtprod t1 t2 : t1 `&` t2 = [tuple tnth t1 i `&` tnth t2 i | i < n].

Proof.

by []. Qed.

End MeetSemilattice.

Section JoinSemilattice.
Context (n : nat) (T : joinSemilatticeType disp).
Implicit Types (t : n.-tuple T).

Definition join t1 t2 : n.-tuple T := [tuple tnth t1 i `|` tnth t2 i | i < n].

Fact leUx t1 t2 t3 : (join t1 t2 <= t3) = (t1 <= t3) && (t2 <= t3).

Proof.

rewrite !leEtprod; apply/forallP/andP => [H|[Ht13 Ht23] i]; last first.
by rewrite tnth_mktuple leUx (forallP Ht13) (forallP Ht23).
by split; apply/forallP => i; move: (H i); rewrite tnth_mktuple leUx => /andP[].
Qed.

#[export]
HB.instance Definition _ :=
@POrder_isJoinSemilattice.Build _ (n.-tuple T) join leUx.

Lemma tnth_join t1 t2 i : tnth (t1 `|` t2) i = tnth t1 i `|` tnth t2 i.

Proof.

exact: tnth_mktuple. Qed.

Lemma joinEtprod t1 t2 : t1 `|` t2 = [tuple tnth t1 i `|` tnth t2 i | i < n].

Proof.

by []. Qed.

Proof.

by move=> t1 t2 t3; apply: eq_from_tnth => i; rewrite !tnth_mktuple meetUl.
Qed.

#[export]
HB.instance Definition _ :=
  Lattice_Meet_isDistrLattice.Build _ (n.-tuple T) meetUl.

End DistrLattice.

#[export]
HB.instance Definition _ (n : nat) (T : bDistrLatticeType disp) :=
  DistrLattice.on (n.-tuple T).
#[export]
HB.instance Definition _ (n : nat) (T : tDistrLatticeType disp) :=
  DistrLattice.on (n.-tuple T).
#[export]
HB.instance Definition _ (n : nat) (T : tbDistrLatticeType disp) :=
  DistrLattice.on (n.-tuple T).

Section CDistrLattice.
Context (n : nat) (T : cDistrLatticeType disp).
Implicit Types (t : n.-tuple T).

Definition rcompl t1 t2 t3 :=
  [tuple rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i) | i < n].

Fact rcomplPmeet x y z : ((x `&` y) `|` z) `&` rcompl x y z = x `&` y.

Proof.

by apply: eq_from_tnth => i; rewrite !tnth_mktuple rcomplPmeet. Qed.

Fact rcomplPjoin x y z : ((y `|` x) `&` z) `|` rcompl x y z = y `|` x.

Proof.

by apply: eq_from_tnth => i; rewrite !tnth_mktuple rcomplPjoin. Qed.

#[export]
HB.instance Definition _ :=
  @DistrLattice_hasRelativeComplement.Build _ (n.-tuple T)
    rcompl rcomplPmeet rcomplPjoin.

Lemma tnth_rcompl t1 t2 t3 i :
  tnth (Order.rcompl t1 t2 t3) i =
    Order.rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i).

Proof.

exact: tnth_mktuple. Qed.

Lemma rcomplEtprod t1 t2 t3 :
Order.rcompl t1 t2 t3 =
[tuple Order.rcompl (tnth t1 i) (tnth t2 i) (tnth t3 i) | i < n].

Proof.

by []. Qed.

End CDistrLattice.

Section CBDistrLattice.
Context (n : nat) (T : cbDistrLatticeType disp).
Implicit Types (t : n.-tuple T).

Definition diff t1 t2 : n.-tuple T := [tuple tnth t1 i `\` tnth t2 i | i < n].

Fact diffErcompl t1 t2 : diff t1 t2 = rcompl \bot t1 t2.

Proof.

by apply: eq_from_tnth => i; rewrite !tnth_mktuple diffErcompl. Qed.

#[export] HB.instance Definition _ :=
@CDistrLattice_hasSectionalComplement.Build _ (n.-tuple T) diff diffErcompl.

Lemma tnth_diff t1 t2 i : tnth (diff t1 t2) i = tnth t1 i `\` tnth t2 i.

Proof.

exact: tnth_mktuple. Qed.

Lemma diffEtprod t1 t2 : t1 `\` t2 = [tuple tnth t1 i `\` tnth t2 i | i < n].

Proof.

by []. Qed.

End CBDistrLattice.

Section CTDistrLattice.
Context (n : nat) (T : ctDistrLatticeType disp).
Implicit Types (t : n.-tuple T).

Definition codiff t1 t2 : n.-tuple T :=
[tuple Order.codiff (tnth t1 i) (tnth t2 i) | i < n].

Fact codiffErcompl t1 t2 : codiff t1 t2 = rcompl t1 \top t2.

Proof.

by apply: eq_from_tnth => i; rewrite !tnth_mktuple codiffErcompl. Qed.

#[export] HB.instance Definition _ :=
  @CDistrLattice_hasDualSectionalComplement.Build _ (n.-tuple T)
    codiff codiffErcompl.

Lemma tnth_codiff t1 t2 i :
  tnth (Order.codiff t1 t2) i = Order.codiff (tnth t1 i) (tnth t2 i).

Proof.

exact: tnth_mktuple. Qed.

Lemma codiffEtprod t1 t2 :
Order.codiff t1 t2 = [tuple Order.codiff (tnth t1 i) (tnth t2 i) | i < n].

Proof.

by []. Qed.

End CTDistrLattice.

Section CTBDistrLattice.
Context (n : nat) (T : ctbDistrLatticeType disp).
Implicit Types (t : n.-tuple T).

Definition compl t : n.-tuple T := map_tuple compl t.

Fact complEdiff t : compl t = (\top : n.-tuple T) `\` t.

Proof.

by apply: eq_from_tnth => i; rewrite tnth_map !tnth_mktuple complEdiff.
Qed.

Fact complEcodiff t : compl t = codiff (\bot : n.-tuple T) t.

Proof.

by apply: eq_from_tnth => i; rewrite tnth_map !tnth_mktuple complEcodiff.
Qed.

#[export] HB.instance Definition _ :=
@CDistrLattice_hasComplement.Build _ (n.-tuple T)
compl complEdiff complEcodiff.

Lemma tnth_compl t i : tnth (~` t) i = ~` tnth t i.

Proof.

by rewrite tnth_map. Qed.

Lemma complEtprod t : ~` t = map_tuple Order.compl t.

Proof.

by []. Qed.

We declare an alias of the tuples,
which has canonical lexicographic order.

Module TupleLexiOrder.

Export TupleLexiOrder.

Section TupleLexiOrder.
Import DefaultSeqLexiOrder.

Context {disp disp' : disp_t}.
Local Notation "n .-tuple" := (type disp' n) : type_scope.

Section POrder.
Implicit Types (n : nat) (T : porderType disp).

#[export] HB.instance Definition _ n T :=
  [SubChoice_isSubPOrder of n.-tuple T by <: with disp'].

Lemma lexi_tupleP n T (t1 t2 : n.-tuple T) :
   reflect (exists k : 'I_n.+1, forall i : 'I_n, (i <= k)%N ->
               tnth t1 i <= tnth t2 i ?= iff (i != k :> nat)) (t1 <= t2).

Proof.

elim: n => [|n IHn] in t1 t2 *.
  by rewrite tuple0 [t2]tuple0/= lexx; constructor; exists ord0 => -[].
case: (tupleP t1) (tupleP t2) => [x1 {}t1] [x2 {}t2].
rewrite [_ <= _]lexi_cons; apply: (iffP idP) => [|[k leif_xt12]].
  case: comparableP => //= [ltx12 _|-> /IHn[k kP]].
    exists ord0 => i; rewrite leqn0 => /eqP/(@ord_inj n.+1 i ord0)->.
    by apply/leifP; rewrite !tnth0.
  exists (lift ord0 k) => i; case: (unliftP ord0 i) => [j ->|-> _].
    by rewrite !ltnS => /kP; rewrite !tnthS.
  by apply/leifP; rewrite !tnth0 eqxx.
have /= := leif_xt12 ord0 isT; rewrite !tnth0 => leif_x12.
rewrite leif_x12/=; move: leif_x12 leif_xt12 => /leifP.
case: (unliftP ord0 k) => {k} [k-> /eqP<-{x2}|-> /lt_geF->//] leif_xt12.
rewrite lexx implyTb; apply/IHn; exists k => i le_ik.
by have := leif_xt12 (lift ord0 i) le_ik; rewrite !tnthS.
Qed.

Lemma ltxi_tupleP n T (t1 t2 : n.-tuple T) :
reflect (exists k : 'I_n, forall i : 'I_n, (i <= k)%N ->
tnth t1 i <= tnth t2 i ?= iff (i != k :> nat)) (t1 < t2).

Proof.

elim: n => [|n IHn] in t1 t2 *.
  by rewrite tuple0 [t2]tuple0/= ltxx; constructor => - [] [].
case: (tupleP t1) (tupleP t2) => [x1 {}t1] [x2 {}t2].
rewrite [_ < _]ltxi_cons; apply: (iffP idP) => [|[k leif_xt12]].
  case: (comparableP x1 x2) => //= [ltx12 _|-> /IHn[k kP]].
    exists ord0 => i; rewrite leqn0 => /eqP/(@ord_inj n.+1 i ord0)->.
    by apply/leifP; rewrite !tnth0.
  exists (lift ord0 k) => i; case: (unliftP ord0 i) => {i} [i ->|-> _].
    by rewrite !ltnS => /kP; rewrite !tnthS.
  by apply/leifP; rewrite !tnth0 eqxx.
have /= := leif_xt12 ord0 isT; rewrite !tnth0 => leif_x12.
rewrite leif_x12/=; move: leif_x12 leif_xt12 => /leifP.
case: (unliftP ord0 k) => {k} [k-> /eqP<-{x2}|-> /lt_geF->//] leif_xt12.
rewrite lexx implyTb; apply/IHn; exists k => i le_ik.
by have := leif_xt12 (lift ord0 i) le_ik; rewrite !tnthS.
Qed.

Lemma ltxi_tuplePlt n T (t1 t2 : n.-tuple T) : reflect
  (exists2 k : 'I_n, forall i : 'I_n, (i < k)%N -> tnth t1 i = tnth t2 i
                                                 & tnth t1 k < tnth t2 k)
  (t1 < t2).

Proof.

apply: (iffP (ltxi_tupleP _ _)) => [[k kP]|[k kP ltk12]].
exists k => [i i_lt|]; last by rewrite (lt_leif (kP _ _)) ?eqxx ?leqnn.
by have /eqTleif->// := kP i (ltnW i_lt); rewrite ltn_eqF.
by exists k => i; case: ltngtP => //= [/kP-> _|/ord_inj-> _]; apply/leifP.
Qed.

We declare an alias of the sets,
which is canonically ordered by inclusion

Module SetSubsetOrder.

Export SetSubsetOrder.

Section SetSubsetOrder.

Context {disp : disp_t} {T : finType}.
Local Notation "{ 'subset' T }" := (type disp T).
Implicit Type (A B C : type disp T).

Lemma setKUC B A : A :&: (A :|: B) = A.

Proof.

by rewrite setUC setKU. Qed.

Lemma setKIC B A : A :|: (A :&: B) = A.

Proof.

by rewrite setIC setKI. Qed.

Fact le_anti : antisymmetric (fun A B => A \subset B).

Proof.

by move=> A B ABA; apply/eqP; rewrite eqEsubset. Qed.

#[export]
HB.instance Definition _ := Preorder_isPOrder.Build disp {subset T} le_anti.
#[export]
HB.instance Definition _ := POrder_Meet_isDistrLattice.Build disp {subset T}
(@setIC _) (@setUC _) (@setIA _) (@setUA _) setKUC setKIC le_def (@setIUl _).

Lemma setIDv A B : B :&: (A :\: B) = set0.

Proof.

apply/eqP; rewrite -subset0; apply/subsetP => x.
by rewrite !inE => /and3P[->].
Qed.

#[export]
HB.instance Definition _ :=
@BDistrLattice_hasSectionalComplement.Build disp {subset T}
(@setD _) setIDv (@setID _).

Lemma setTDsym A : ~: A = setT :\: A.

Proof.

by rewrite setTD. Qed.

#[export]
HB.instance Definition _ :=
CBDistrLattice_hasComplement.Build disp {subset T} setTDsym.

Lemma meetEsubset A B : A `&` B = A :&: B.

Proof.

by []. Qed.

Lemma joinEsubset A B : A `|` B = A :|: B.

Proof.

by []. Qed.

Lemma botEsubset : \bot = set0 :> {subset T}.

Proof.

by []. Qed.

Lemma topEsubset : \top = setT :> {subset T}.

Proof.

by []. Qed.

Lemma subEsubset A B : A `\` B = A :\: B.

Proof.

by []. Qed.

Lemma complEsubset A : ~` A = ~: A.

Proof.

by []. Qed.

End SetSubsetOrder.

Module Exports.

HB.reexport.

Definition meetEsubset := @meetEsubset.
Definition joinEsubset := @joinEsubset.
Definition botEsubset := @botEsubset.
Definition topEsubset := @topEsubset.
Definition subEsubset := @subEsubset.
Definition complEsubset := @complEsubset.

End Exports.
End SetSubsetOrder.
Export SetSubsetOrder.Exports.

Module DefaultSetSubsetOrder.

Export DefaultSetSubsetOrder.

HB.instance Definition _ (T : finType) :=
  CTBDistrLattice.copy {set T} {subset T}.

End DefaultSetSubsetOrder.

Lemma mono_unique d (T T' : finPOrderType d) (f g : T -> T') :
    total (<=%O : rel T) -> (#|T'| <= #|T|)%N ->
    {mono f : x y / x <= y} -> {mono g : x y / x <= y} ->
  f =1 g.

Proof.

move=> le_total leT'T lef leg x0; move: {+}x0.
suff: finfun f = finfun g by move=> /ffunP + x => /(_ x); rewrite !ffunE.
apply: (can_inj fgraphK); apply/val_inj => /=; rewrite !codomE.
under eq_map do rewrite ffunE; under [RHS]eq_map do rewrite ffunE.
have [finj ginj] := (inc_inj lef, inc_inj leg).
have [f' fK f'K] := inj_card_bij finj leT'T.
have [g' gK g'K] := inj_card_bij ginj leT'T.
apply/eqP; have : [seq f i | i <- enum T] = [seq g i | i <- enum T].
  apply: (@sorted_eq _ <=%O le_trans le_anti); rewrite ?mono_sorted_enum//.
  apply: uniq_perm; rewrite ?map_inj_uniq ?sort_uniq ?fintype.enum_uniq//.
  move=> x; apply/mapP/mapP => -[y _ ->].
    by exists (g' (f y)); rewrite ?mem_enum.
  by exists (f' (g y)); rewrite ?mem_enum.
move=> /eqP; rewrite !eq_map_all all_map [in X in _ -> X]all_map.
by have /permPl/perm_all-> := perm_sort <=%O (fintype.enum T).
Qed.

Module Import EnumVal.

Export EnumVal.

Section EnumVal.
Import OrdinalOrder.Exports.
Variables (d : disp_t) (T : finPOrderType d).
Implicit Types (x : T) (A : {pred T}).

Section total.
Hypothesis (leT_total : total (<=%O : rel T)).

Lemma le_enum_val A : {mono @enum_val _ _ A : i j / i <= j}.

Proof.

apply: le_mono => i j le_ij.
rewrite /enum_val (set_nth_default (enum_default j)) -?cardE//.
apply: (sorted_ltn_nth lt_trans); rewrite -?topredE/= -?cardE//.
by rewrite lt_sorted_uniq_le enum_uniq/= sort_sorted.
Qed.

Lemma le_enum_rank_in x0 A (Ax0 : x0 \in A) :
{in A &, {mono enum_rank_in Ax0 : x y / x <= y}}.

Proof.

apply: can_mono_in (@in2W _ _ predT predT _ (@le_enum_val A)) => //.
exact/onW_can_in/enum_rankK_in.
Qed.

Lemma le_enum_rank : {mono @enum_rank d T : i j / i <= j}.

Proof.

exact: (can_mono (@enum_rankK _ _) (@le_enum_val predT)). Qed.

Proof.

rewrite card_tagged sumnE/= big_map big_enum.
by apply: eq_bigr => i _; rewrite card_ord.
Qed.

Definition sig : ordsum -> T := enum_val \o cast_ord (esym card).
Definition rank : T -> ordsum := cast_ord card \o enum_rank.

Lemma sigK : cancel sig rank.

Proof.

by move=> s; rewrite /sig/rank/= enum_valK cast_ord_comp cast_ord_id.
Qed.

Lemma sig_inj : injective sig

Proof.

exact: can_inj sigK. Qed.

Lemma rankK : cancel rank sig.

Proof.

by move=> p; rewrite /sig/rank/= cast_ord_comp cast_ord_id enum_rankK.
Qed.

Lemma rank_inj : injective rank

Proof.

exact: can_inj rankK. Qed.

Definition sig1 s : 'I_n := tag (sig s).
Definition sig2 s : 'I_(p_ (sig1 s)) := tagged (sig s).
Definition Rank i (j : 'I_(p_ i)) := rank (Tagged _ j).

Lemma sigE12 s : sig s = @Tagged _ (sig1 s) _ (sig2 s).

Proof.

by rewrite /sig1 /sig2; case: sig. Qed.

Lemma rankE p : rank p = @Rank (tag p) (tagged p)

Proof.

by case: p. Qed.

Lemma sig2K s : Rank (sig2 s) = s

Proof.

by rewrite -rankE sigK. Qed.

Lemma Rank1K i0 (k : 'I_(p_ i0)) : sig1 (Rank k) = i0.

Proof.

by rewrite /sig1 /Rank/= rankK/=. Qed.

Lemma Rank2K i0 (k : 'I_(p_ i0)) :
sig2 (Rank k) = cast_ord (congr1 p_ (esym (Rank1K k))) k.

Proof.

by apply: val_inj; rewrite /sig2/sig1/Rank/= rankK. Qed.

#[local] Hint Resolve sigK rankK : core.

Lemma rank_bij : bijective rank

Proof.

by exists sig. Qed.

Lemma sig_bij : bijective sig

Proof.

by exists rank. Qed.

Lemma rank_bij_on : {on [pred _ | true], bijective rank}.

Proof.

exact/onW_bij/rank_bij. Qed.

Lemma sig_bij_on : {on [pred _ | true], bijective sig}.

Proof.

exact/onW_bij/sig_bij. Qed.

Lemma le_sig : {mono sig : i j / i <= j}.

Proof.

by move=> i j; rewrite /sig/= le_enum_val//; apply: le_total. Qed.

Lemma le_sig1 : {homo sig1 : i j / i <= j}.

Proof.

by move=> i j; rewrite /sig1/= -le_sig leEsig/=; case: leP. Qed.

Lemma le_rank : {mono rank : p q / p <= q}.

Proof.

exact: can_mono le_sig. Qed.

Lemma le_Rank i : {mono @Rank i : j k / j <= k}.

Proof.

by move=> j k; rewrite /Rank le_rank/= leEsig/= tagged_asE lexx. Qed.

Lemma lt_sig : {mono sig : i j / i < j}.

Proof.

by move=> i j; rewrite !ltNge le_sig. Qed.

Lemma lt_rank : {mono rank : p q / p < q}.

Proof.

by move=> p q; rewrite !ltNge le_rank. Qed.

Lemma lt_Rank i : {mono @Rank i : j k / j < k}.

Proof.

by move=> j k; rewrite !ltNge le_Rank. Qed.

Lemma eq_Rank i i' (j : 'I_(p_ i)) (j': 'I_(p_ i')) :
(Rank j == Rank j' :> nat) = (i == i') && (j == j' :> nat).

Proof.

rewrite val_eqE /Rank -(can_eq sigK) !rankK.
case: (i =P i') => ii' /=; last by case: eqVneq => // -[].
by case: _ / ii' in j' *; rewrite eq_Tagged.
Qed.

Lemma rankEsum p : rank p = \sum_(i < n | (i < tag p)%N) p_ i + tagged p :> nat.

Proof.

pose sum p := \sum_(i < n | (i < tag p)%N) p_ i + tagged p.
rewrite -/(sum _); have sumlt : forall p, (sum p < \sum_i p_ i)%N.
  rewrite /sum => -[/= i j].
  rewrite [ltnRHS](bigID [pred i' : 'I__ | (i' < i)%N])/= ltn_add2l.
  by rewrite (bigD1 i) ?ltnn//= ltn_addr.
suff: rank =1 (fun p => Ordinal (sumlt p)) by move=> /(_ p)/(congr1 val).
apply: (Order.mono_unique _ _ le_rank) => //=.
- exact: le_total.
- by rewrite card card_ord.
apply: le_mono => /= -[i j] -[i' j']; rewrite ltEsig/= !ltEord/= /sum leEord/=.
case: (ltngtP i i') => //= [ltii' _|/val_inj ii']; last first.
  by rewrite -ii' in j' *; rewrite tagged_asE => ltjj'; rewrite ltn_add2l.
rewrite ltn_addr// (@leq_trans (\sum_(i0 < n | (i0 < i)%N) p_ i0 + p_ i))%N//.
  by rewrite ltn_add2l.
rewrite [leqRHS](bigID [pred i' : 'I__ | (i' < i)%N])/=.
rewrite leq_add//; last first.
  by rewrite (bigD1 i) ?ltnn ?ltii'//= leq_addr.
rewrite [leqRHS](eq_bigl [pred k : 'I_n | (k < i)%N])// => k/=.
by case: (ltnP k i); rewrite ?andbF// => /ltn_trans->.
Qed.

Lemma RankEsum i j : @Rank i j = \sum_(k < n | (k < i)%N) p_ k + j :> nat.

Proof.

by rewrite /Rank rankEsum/=. Qed.

Lemma rect s : s = \sum_(i < n | (i < sig1 s)%N) p_ i + sig2 s :> nat.

Proof.

by rewrite -[s]sigK rankEsum /= sigK. Qed.

Lemma eqRank (i0 j : nat) (li0 : (i0 < n)%N) (lj : (j < p_ (Ordinal li0))%N) :
(\sum_(i < n | (i < i0)%N) p_ i) + j = Rank (Ordinal lj) :> nat.

Proof.

by rewrite RankEsum. Qed.

End tagnat.
End tagnat.
Arguments tagnat.Rank {n p_}.

Files

Module mathcomp.order.order