Library mathcomp.ssreflect.order

(* (c) Copyright 2006-2019 Microsoft Corporation and Inria.                  
 Distributed under the terms of CeCILL-B.                                  *)

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.

This files defines types equipped with order relations.
Use one of the following modules implementing different theories: 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. Notations are accessible by opening the scope "order_scope" bound to the delimiting key "O".
We provide the following structures of ordered types porderType d == the type of partially ordered types latticeType d == the type of non-distributive lattices bLatticeType d == latticeType with a bottom element tbLatticeType d == latticeType with both a top and a bottom distrLatticeType d == the type of distributive lattices bDistrLatticeType d == distrLatticeType with a bottom element tbDistrLatticeType d == distrLatticeType with both a top and a bottom cbDistrLatticeType d == the type of sectionally complemented distributive lattices (lattices with bottom and a difference operation) ctbDistrLatticeType d == the type of complemented distributive lattices (lattices with top, bottom, difference, and complement) orderType d == the type of totally ordered types finPOrderType d == the type of partially ordered finite types finLatticeType d == the type of nonempty finite non-distributive lattices finDistrLatticeType d == the type of nonempty finite distributive lattices finCDistrLatticeType d == the type of nonempty finite complemented distributive lattices finOrderType d == the type of nonempty totally ordered finite types
Each generic partial order and lattice operations symbols also has a first argument which is the display, the second which is the minimal structure they operate on and then the operands. Here is the exhaustive list of all such symbols for partial orders and lattices together with their default display (as displayed by Check). We document their meaning in the paragraph after the next.
For porderType T @Order.le disp T == <=%O (in fun_scope) @Order.lt disp T == <%O (in fun_scope) @Order.comparable disp T == >=<%O (in fun_scope) @Order.ge disp T == >=%O (in fun_scope) @Order.gt disp T == >%O (in fun_scope) @Order.leif disp T == <?=%O (in fun_scope) @Order.lteif disp T == <?<=%O (in fun_scope) For latticeType T @Order.meet disp T x y == x `&` y (in order_scope) @Order.join disp T x y == x `|` y (in order_scope) For bLatticeType T @Order.bottom disp T == 0 (in order_scope) For tbLatticeType T @Order.top disp T == 1 (in order_scope) For cbDistrLatticeType T @Order.sub disp T x y == x `|` y (in order_scope) For ctbDistrLatticeType T @Order.compl disp T x == ~` x (in order_scope)
This first argument named either d, disp or display, of type unit, configures the printing of notations. Instantiating d with tt or an unknown key will lead to a default display for notations, i.e. we have: For x, y of type T, where T is canonically a porderType d: x <= y <-> x is less than or equal to y. x < y <-> x is less than y (:= (y != x) && (x <= y)). min x y <-> if x < y then x else y max x y <-> if x < y then y else x x >= y <-> x is greater than or equal to y (:= y <= x). x > y <-> x is greater than y (:= y < x). x <= y ?= iff C <-> x is less than y, or equal iff C is true. x < y ?<= if C <-> x is smaller than y, and strictly if C is false. x >=< y <-> x and y are comparable (:= (x <= y) || (y <= x)). x >< y <-> x and y are incomparable (:= ~~ x >=< y). f \min g <-> the function x |-> Order.min (f x) (g x); f \min g simplifies on application. f \max g <-> the function x |-> Order.max (f x) (g x); f \max g simplifies on application. For x, y of type T, where T is canonically a latticeType d: x `&` y == the meet of x and y. x `|` y == the join of x and y. In a type T, where T is canonically a bLatticeType d: 0 == the bottom element. \join<range> e == iterated join of a lattice with a bottom. In a type T, where T is canonically a tbLatticeType d: 1 == the top element. \meet<range> e == iterated meet of a lattice with a top. For x, y of type T, where T is canonically a cbDistrLatticeType d: x `\` y == the (sectional) complement of y in [0, x]. For x of type T, where T is canonically a ctbDistrLatticeType d: ~` x == the complement of x in [0, 1].
There are three distinct uses of the symbols <, <=, >, >=, _ <= _ ?= iff _, >=<, and >< in the default display: they can be 0-ary, unary (prefix), and binary (infix). 0. <%O, <=%O, >%O, >=%O, <?=%O, >=<%O, and ><%O stand respectively for lt, le, gt, ge, leif (_ <= _ ?= iff _), comparable, and incomparable. 1. (< x), (<= x), (> x), (>= x), (>=< x), and (>< x) stand respectively for (>%O x), (>=%O x), (<%O x), (<=%O x), (>=<%O x), and (><%O x). So (< x) is a predicate characterizing elements smaller than x. 2. (x < y), (x <= y), ... mean what they are expected to. These conventions are compatible with Haskell's, where ((< y) x) = (x < y) = ((<) x y), except that we write <%O instead of (<).
Alternative notation displays can be defined by : 1. declaring a new opaque definition of type unit. Using the idiom `Lemma my_display : unit. Proof. exact: tt. Qed.` 2. using this symbol to tag canonical porderType structures using `Canonical my_porderType := POrderType my_display my_type my_mixin`, 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.
One may use displays either for convenience or to disambiguate between different structures defined on "copies" of a type (as explained below.) We provide the following "copies" of types, the first one is a *documented example* natdvd := nat == a "copy" of 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) == a "copy" of 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' == a "copy" of 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] T' where prod_display adds an extra ^p to all notations T *lexi[d] T' := T * T' == a "copy" of 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] T' where lexi_display adds an extra ^l to all notations seqprod_with d T := seq T == a "copy" of 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 prod_display T n.-tupleprod T := n.-tuple[prod_display] T seqlexi_with d T := seq T == a "copy" of 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 lexi_display T n.-tuplelexi T := n.-tuple[lexi_display] T {subset[d] T} := {set T} == a "copy" of set which is canonically ordered by the subset order and displayed in display d {subset T} := {subset[subset_display] T}
Beware that canonical structure inference will not try to find the copy of the structures that fits the display one mentioned, but will rather determine which canonical structure and display to use depending on the copy of the type one provided. In this sense they are merely displays to inform the user of what the inference did, rather than additional input for the inference.
Existing displays are either dual_display d (where d is a display), dvd_display (both explained above), ring_display (from algebra/ssrnum to change the scope of the usual notations to ring_scope). We also provide lexi_display and prod_display for lexicographic and product order respectively. The default display is tt and users can define their own as explained above.
For porderType we provide the following operations [arg min(i < i0 | P) M] == a value i : T minimizing M : R, subject to the condition P (i may appear in P and M), and provided P holds for i0. [arg max(i > i0 | P) M] == a value i maximizing M subject to P and provided P holds for i0. [arg min(i < i0 in A) M] == an i \in A minimizing M if i0 \in A. [arg max(i > i0 in A) M] == an i \in A maximizing M if i0 \in A. [arg min(i < i0) M] == an i : T minimizing M, given i0 : T. [arg max(i > i0) M] == an i : T maximizing M, given i0 : T. with head symbols Order.arg_min and Order.arg_max The user may use extremumP or extremum_inP to eliminate them.
  • > patterns for contextual rewriting: leLHS := (X in (X <= _)%O)%pattern leRHS := (X in (_ <= X)%O)%pattern ltLHS := (X in (X < _)%O)%pattern ltRHS := (X in (_ < X)%O)%pattern
In order to build the above structures, one must provide the appropriate factory instance to the following structure constructors. The list of possible factories is indicated after each constructor. Each factory is documented in the next paragraph. NB: Since each mixim_of record of structure in this library is an internal interface that is not designed to be used by users directly, one should not build structure instances from their Mixin constructors.
POrderType disp T pord_mixin == builds a porderType from a canonical choiceType instance of T where pord_mixin can be of types lePOrderMixin, ltPOrderMixin, meetJoinMixin, leOrderMixin, or ltOrderMixin or computed using PcanPOrderMixin or CanPOrderMixin. disp is a display as explained above
LatticeType T lat_mixin == builds a latticeType from a porderType where lat_mixin can be of types latticeMixin, distrLatticePOrderMixin, totalPOrderMixin, meetJoinMixin, leOrderMixin, or ltOrderMixin or computed using IsoLatticeMixin.
BLatticeType T bot_mixin == builds a bLatticeType from a latticeType and bottom where bot_mixin is of type bottomMixin.
TBLatticeType T top_mixin == builds a tbLatticeType from a bLatticeType and top where top_mixin is of type topMixin.
DistrLatticeType T lat_mixin == builds a distrLatticeType from a porderType where lat_mixin can be of types distrLatticeMixin, distrLatticePOrderMixin, totalLatticeMixin, totalPOrderMixin, meetJoinMixin, leOrderMixin, or ltOrderMixin or computed using IsoLatticeMixin.
CBDistrLatticeType T sub_mixin == builds a cbDistrLatticeType from a bDistrLatticeType and a difference operation where sub_mixin is of type cbDistrLatticeMixin.
CTBDistrLatticeType T compl_mixin == builds a ctbDistrLatticeType from a tbDistrLatticeType and a complement operation where compl_mixin is of type ctbDistrLatticeMixin.
OrderType T ord_mixin == builds an orderType from a distrLatticeType where ord_mixin can be of types totalOrderMixin, totalPOrderMixin, totalLatticeMixin, leOrderMixin, or ltOrderMixin or computed using MonoTotalMixin.
Additionally:
  • [porderType of _ ] ... notations are available to recover structures on "copies" of the types, as in eqType, choiceType, ssralg...
  • [finPOrderType of _ ] ... notations to compute joins between finite types and ordered types
List of possible factories:
  • lePOrderMixin == on a choiceType, takes le, lt, reflexivity, antisymmetry and transitivity of le. (can build: porderType)
  • ltPOrderMixin == on a choiceType, takes le, lt, irreflexivity and transitivity of lt. (can build: porderType)
  • latticeMixin == on a porderType, takes meet, join, commutativity and associativity of meet and join, and some absorption laws. (can build: latticeType)
  • distrLatticeMixin == on a latticeType, takes distributivity of meet over join. (can build: distrLatticeType)
  • distrLatticePOrderMixin == on a porderType, takes meet, join, commutativity and associativity of meet and join, and the absorption and distributive laws. (can build: latticeType, distrLatticeType)
  • meetJoinMixin == on a choiceType, takes le, lt, meet, join, commutativity and associativity of meet and join, the absorption and distributive laws, and idempotence of meet. (can build: porderType, latticeType, distrLatticeType)
  • meetJoinLeMixin == on a porderType, takes meet, join, and a proof that those are respectvely the greatest lower bound and the least upper bound. (can build: latticeType)
  • leOrderMixin == on a choiceType, takes le, lt, meet, join, antisymmetry, transitivity and totality of le. (can build: porderType, latticeType, distrLatticeType, orderType)
  • ltOrderMixin == on a choiceType, takes le, lt, meet, join, irreflexivity, transitivity and totality of lt. (can build: porderType, latticeType, distrLatticeType, orderType)
  • totalPOrderMixin == on a porderType T, totality of the order of T := total (<=%O : rel T) (can build: latticeType, distrLatticeType, orderType)
  • totalLatticeMixin == on a latticeType T, totality of the order of T := total (<=%O : rel T) (can build distrLatticeType, orderType)
  • totalOrderMixin == on a distrLatticeType T, totality of the order of T := total (<=%O : rel T) (can build: orderType) NB: the above three mixins are kept separate from each other (even though they are convertible), in order to avoid ambiguous coercion paths.
  • bottomMixin, topMixin, cbDistrLatticeMixin, ctbDistrLatticeMixin == mixins with one extra operation (respectively bottom, top, difference, and complement)
Additionally:
  • [porderMixin of T by <: ] creates a porderMixin by subtyping.
  • [totalOrderMixin of T by <: ] creates the associated totalOrderMixin.
  • PCanPOrderMixin, CanPOrderMixin create porderMixin from cancellations
  • MonoTotalMixin creates a totalPOrderMixin from monotonicity
  • IsoLatticeMixin creates a distrLatticeMixin from an ordered structure isomorphism (i.e., cancel f f', cancel f' f, {mono f : x y / x <= y})
List of "big pack" notations:
  • DistrLatticeOfChoiceType builds a distrLatticeType from a choiceType and a meetJoinMixin.
  • DistrLatticeOfPOrderType builds a distrLatticeType from a porderType and a distrLatticePOrderMixin.
  • OrderOfChoiceType builds an orderType from a choiceType, and a leOrderMixin or a ltOrderMixin.
  • OrderOfPOrder builds an orderType from a porderType and a totalPOrderMixin.
  • OrderOfLattice builds an orderType from a latticeType and a totalLatticeMixin.
NB: These big pack notations should be used only to construct instances on the fly, e.g., in the middle of a proof, and should not be used to declare canonical instances. See field/algebraics_fundamentals.v for an example usage.
We provide the following canonical instances of ordered types
  • all possible structures on bool
  • porderType, latticeType, distrLatticeType, orderType and bLatticeType on nat for the leq order
  • porderType, latticeType, distrLatticeType, orderType and finPOrderType on 'I_n and bLatticeType, tbLatticeType, bDistrLatticeType, tbDistrLatticeType, finLatticeType, finDistrLatticeType and finOrderType on 'I_n.+1 (to guarantee it is nonempty).
  • porderType, latticeType, distrLatticeType, bLatticeType, tbLatticeType, on nat for the dvdn order, where meet and join are respectively gcdn and lcmn
  • porderType, latticeType, distrLatticeType, orderType, bLatticeType, tbLatticeType, cbDistrLatticeType, ctbDistrLatticeType on T *prod[disp] T' a "copy" of T * T' using product order (and T *p T' its specialization to prod_display)
  • porderType, latticeType, distrLatticeType, and orderType, on T *lexi[disp] T' another "copy" of T * T', with lexicographic ordering (and T *l T' its specialization to lexi_display)
  • porderType, latticeType, distrLatticeType, and orderType, on {t : T & T' x} with lexicographic ordering
  • porderType, latticeType, distrLatticeType, orderType, bLatticeType, tbLatticeType, cbDistrLatticeType, ctbDistrLatticeType on seqprod_with disp T a "copy" of seq T using product order (and seqprod T' its specialization to prod_display)
  • porderType, latticeType, distrLatticeType, and orderType, on seqlexi_with disp T another "copy" of seq T, with lexicographic ordering (and seqlexi T its specialization to lexi_display)
  • porderType, latticeType, distrLatticeType, orderType, bLatticeType, tbLatticeType, cbDistrLatticeType, ctbDistrLatticeType on n.-tupleprod[disp] a "copy" of n.-tuple T using product order (and n.-tupleprod T its specialization to prod_display)
  • porderType, latticeType, distrLatticeType, and orderType, on n.-tuplelexi[d] T another "copy" of n.-tuple T, with lexicographic ordering (and n.-tuplelexi T its specialization to lexi_display)
  • porderType, latticeType, distrLatticeType, orderType, bLatticeType, tbLatticeType, cbDistrLatticeType, ctbDistrLatticeType on {subset[disp] T} a "copy" of {set T} using subset order (and {subset T} its specialization to subset_display)
and all possible finite type instances
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.
On orderType, leP ltP 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.
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 monotonous 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)
This file is based on prior work by D. Dreyer, G. Gonthier, A. Nanevski, P-Y Strub, B. Ziliani

Set Implicit Arguments.

Declare Scope order_scope.

Delimit Scope order_scope with O.
Local Open Scope order_scope.

Reserved Notation "<= y" (at level 35).
Reserved Notation ">= y" (at level 35).
Reserved Notation "< y" (at level 35).
Reserved Notation "> y" (at level 35).
Reserved Notation "<= y :> T" (at level 35, y at next level).
Reserved Notation ">= y :> T" (at level 35, y at next level).
Reserved Notation "< y :> T" (at level 35, y at next level).
Reserved Notation "> y :> T" (at level 35, y at next level).
Reserved Notation "x >=< y" (at level 70, no associativity).
Reserved Notation ">=< y" (at level 35).
Reserved Notation ">=< y :> T" (at level 35, y at next level).
Reserved Notation "x >< y" (at level 70, no associativity).
Reserved Notation ">< x" (at level 35).
Reserved Notation ">< y :> T" (at level 35, y at next level).
Reserved Notation "f \min g" (at level 50, left associativity).
Reserved Notation "f \max g" (at level 50, left associativity).

Reserved Notation "x < y ?<= 'if' c" (at level 70, y, c at next level,
  format "x '[hv' < y '/' ?<= 'if' c ']'").
Reserved Notation "x < y ?<= 'if' c :> T" (at level 70, y, c at next level,
  format "x '[hv' < y '/' ?<= 'if' c :> T ']'").

Reserved notation for lattice operations.
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).

Notations for dual partial and total order
Reserved Notation "x <=^d y" (at level 70, y at next level).
Reserved Notation "x >=^d y" (at level 70, y at next level).
Reserved Notation "x <^d y" (at level 70, y at next level).
Reserved Notation "x >^d y" (at level 70, y at next level).
Reserved Notation "x <=^d y :> T" (at level 70, y at next level).
Reserved Notation "x >=^d y :> T" (at level 70, y at next level).
Reserved Notation "x <^d y :> T" (at level 70, y at next level).
Reserved Notation "x >^d y :> T" (at level 70, y at next level).
Reserved Notation "<=^d y" (at level 35).
Reserved Notation ">=^d y" (at level 35).
Reserved Notation "<^d y" (at level 35).
Reserved Notation ">^d y" (at level 35).
Reserved Notation "<=^d y :> T" (at level 35, y at next level).
Reserved Notation ">=^d y :> T" (at level 35, y at next level).
Reserved Notation "<^d y :> T" (at level 35, y at next level).
Reserved Notation ">^d y :> T" (at level 35, y at next level).
Reserved Notation "x >=<^d y" (at level 70, no associativity).
Reserved Notation ">=<^d y" (at level 35).
Reserved Notation ">=<^d y :> T" (at level 35, y at next level).
Reserved Notation "x ><^d y" (at level 70, no associativity).
Reserved Notation "><^d x" (at level 35).
Reserved Notation "><^d y :> T" (at level 35, y at next level).

Reserved Notation "x <=^d y <=^d z" (at level 70, y, z at next level).
Reserved Notation "x <^d y <=^d z" (at level 70, y, z at next level).
Reserved Notation "x <=^d y <^d z" (at level 70, y, z at next level).
Reserved Notation "x <^d y <^d z" (at level 70, y, z at next level).
Reserved Notation "x <=^d y ?= 'iff' c" (at level 70, y, c at next level,
  format "x '[hv' <=^d y '/' ?= 'iff' c ']'").
Reserved Notation "x <=^d y ?= 'iff' c :> T" (at level 70, y, c at next level,
  format "x '[hv' <=^d y '/' ?= 'iff' c :> T ']'").
Reserved Notation "x <^d y ?<= 'if' c" (at level 70, y, c at next level,
  format "x '[hv' <^d y '/' ?<= 'if' c ']'").
Reserved Notation "x <^d y ?<= 'if' c :> T" (at level 70, y, c at next level,
  format "x '[hv' <^d y '/' ?<= 'if' c :> T ']'").

Reserved notation for dual lattice operations.
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 "0^d" (at level 0).
Reserved Notation "1^d" (at level 0).

Reserved notations for product ordering of prod or seq
Reserved Notation "x <=^p y" (at level 70, y at next level).
Reserved Notation "x >=^p y" (at level 70, y at next level).
Reserved Notation "x <^p y" (at level 70, y at next level).
Reserved Notation "x >^p y" (at level 70, y at next level).
Reserved Notation "x <=^p y :> T" (at level 70, y at next level).
Reserved Notation "x >=^p y :> T" (at level 70, y at next level).
Reserved Notation "x <^p y :> T" (at level 70, y at next level).
Reserved Notation "x >^p y :> T" (at level 70, y at next level).
Reserved Notation "<=^p y" (at level 35).
Reserved Notation ">=^p y" (at level 35).
Reserved Notation "<^p y" (at level 35).
Reserved Notation ">^p y" (at level 35).
Reserved Notation "<=^p y :> T" (at level 35, y at next level).
Reserved Notation ">=^p y :> T" (at level 35, y at next level).
Reserved Notation "<^p y :> T" (at level 35, y at next level).
Reserved Notation ">^p y :> T" (at level 35, y at next level).
Reserved Notation "x >=<^p y" (at level 70, no associativity).
Reserved Notation ">=<^p x" (at level 35).
Reserved Notation ">=<^p y :> T" (at level 35, y at next level).
Reserved Notation "x ><^p y" (at level 70, no associativity).
Reserved Notation "><^p x" (at level 35).
Reserved Notation "><^p y :> T" (at level 35, y at next level).

Reserved Notation "x <=^p y <=^p z" (at level 70, y, z at next level).
Reserved Notation "x <^p y <=^p z" (at level 70, y, z at next level).
Reserved Notation "x <=^p y <^p z" (at level 70, y, z at next level).
Reserved Notation "x <^p y <^p z" (at level 70, y, z at next level).
Reserved Notation "x <=^p y ?= 'iff' c" (at level 70, y, c at next level,
  format "x '[hv' <=^p y '/' ?= 'iff' c ']'").
Reserved Notation "x <=^p y ?= 'iff' c :> T" (at level 70, y, c at next level,
  format "x '[hv' <=^p y '/' ?= 'iff' c :> T ']'").

Reserved notation for dual lattice operations.
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 notations for lexicographic ordering of prod or seq
Reserved Notation "x <=^l y" (at level 70, y at next level).
Reserved Notation "x >=^l y" (at level 70, y at next level).
Reserved Notation "x <^l y" (at level 70, y at next level).
Reserved Notation "x >^l y" (at level 70, y at next level).
Reserved Notation "x <=^l y :> T" (at level 70, y at next level).
Reserved Notation "x >=^l y :> T" (at level 70, y at next level).
Reserved Notation "x <^l y :> T" (at level 70, y at next level).
Reserved Notation "x >^l y :> T" (at level 70, y at next level).
Reserved Notation "<=^l y" (at level 35).
Reserved Notation ">=^l y" (at level 35).
Reserved Notation "<^l y" (at level 35).
Reserved Notation ">^l y" (at level 35).
Reserved Notation "<=^l y :> T" (at level 35, y at next level).
Reserved Notation ">=^l y :> T" (at level 35, y at next level).
Reserved Notation "<^l y :> T" (at level 35, y at next level).
Reserved Notation ">^l y :> T" (at level 35, y at next level).
Reserved Notation "x >=<^l y" (at level 70, no associativity).
Reserved Notation ">=<^l x" (at level 35).
Reserved Notation ">=<^l y :> T" (at level 35, y at next level).
Reserved Notation "x ><^l y" (at level 70, no associativity).
Reserved Notation "><^l x" (at level 35).
Reserved Notation "><^l y :> T" (at level 35, y at next level).

Reserved Notation "x <=^l y <=^l z" (at level 70, y, z at next level).
Reserved Notation "x <^l y <=^l z" (at level 70, y, z at next level).
Reserved Notation "x <=^l y <^l z" (at level 70, y, z at next level).
Reserved Notation "x <^l y <^l z" (at level 70, y, z at next level).
Reserved Notation "x <=^l y ?= 'iff' c" (at level 70, y, c at next level,
  format "x '[hv' <=^l y '/' ?= 'iff' c ']'").
Reserved Notation "x <=^l y ?= 'iff' c :> T" (at level 70, y, c at next level,
  format "x '[hv' <=^l y '/' ?= 'iff' c :> T ']'").

Reserved notations for divisibility
Reserved Notation "x %<| y" (at level 70, no associativity).

Reserved Notation "\gcd_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \gcd_ i '/ ' F ']'").
Reserved Notation "\gcd_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \gcd_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \gcd_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\gcd_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \gcd_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \gcd_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \gcd_ ( i | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\gcd_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\gcd_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \gcd_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \gcd_ ( i < n ) F ']'").
Reserved Notation "\gcd_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \gcd_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\gcd_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \gcd_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\lcm_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \lcm_ i '/ ' F ']'").
Reserved Notation "\lcm_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \lcm_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \lcm_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\lcm_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \lcm_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \lcm_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \lcm_ ( i | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\lcm_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\lcm_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \lcm_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \lcm_ ( i < n ) F ']'").
Reserved Notation "\lcm_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \lcm_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\lcm_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \lcm_ ( i 'in' A ) '/ ' F ']'").

Reserved notation for dual lattice operations.
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 "\meet_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \meet_ i '/ ' F ']'").
Reserved Notation "\meet_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \meet_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \meet_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \meet_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \meet_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \meet_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\meet_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\meet_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \meet_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \meet_ ( i < n ) F ']'").
Reserved Notation "\meet_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \meet_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \meet_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \join_ i '/ ' F ']'").
Reserved Notation "\join_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \join_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \join_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \join_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \join_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \join_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\join_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\join_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \join_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \join_ ( i < n ) F ']'").
Reserved Notation "\join_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \join_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \join_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\min_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \min_ i '/ ' F ']'").
Reserved Notation "\min_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \min_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\min_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \min_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\min_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \min_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\min_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \min_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\min_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \min_ ( i | P ) '/ ' F ']'").
Reserved Notation "\min_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\min_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\min_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \min_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\min_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \min_ ( i < n ) F ']'").
Reserved Notation "\min_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \min_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\min_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \min_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\max_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \max_ i '/ ' F ']'").
Reserved Notation "\max_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \max_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \max_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\max_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \max_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\max_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \max_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\max_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \max_ ( i | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\max_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\max_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \max_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \max_ ( i < n ) F ']'").
Reserved Notation "\max_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \max_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\max_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \max_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet^d_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \meet^d_ i '/ ' F ']'").
Reserved Notation "\meet^d_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \meet^d_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \meet^d_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \meet^d_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \meet^d_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \meet^d_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\meet^d_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\meet^d_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \meet^d_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \meet^d_ ( i < n ) F ']'").
Reserved Notation "\meet^d_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \meet^d_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet^d_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \meet^d_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join^d_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \join^d_ i '/ ' F ']'").
Reserved Notation "\join^d_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \join^d_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \join^d_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join^d_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \join^d_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \join^d_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \join^d_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\join^d_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\join^d_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \join^d_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \join^d_ ( i < n ) F ']'").
Reserved Notation "\join^d_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \join^d_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join^d_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \join^d_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\min^d_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \min^d_ i '/ ' F ']'").
Reserved Notation "\min^d_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \min^d_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\min^d_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \min^d_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\min^d_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \min^d_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\min^d_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \min^d_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\min^d_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \min^d_ ( i | P ) '/ ' F ']'").
Reserved Notation "\min^d_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\min^d_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\min^d_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \min^d_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\min^d_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \min^d_ ( i < n ) F ']'").
Reserved Notation "\min^d_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \min^d_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\min^d_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \min^d_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\max^d_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \max^d_ i '/ ' F ']'").
Reserved Notation "\max^d_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \max^d_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\max^d_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \max^d_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\max^d_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \max^d_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\max^d_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \max^d_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\max^d_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \max^d_ ( i | P ) '/ ' F ']'").
Reserved Notation "\max^d_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\max^d_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\max^d_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \max^d_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\max^d_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \max^d_ ( i < n ) F ']'").
Reserved Notation "\max^d_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \max^d_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\max^d_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \max^d_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\meet^p_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \meet^p_ i '/ ' F ']'").
Reserved Notation "\meet^p_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \meet^p_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \meet^p_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \meet^p_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \meet^p_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \meet^p_ ( i | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\meet^p_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\meet^p_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \meet^p_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \meet^p_ ( i < n ) F ']'").
Reserved Notation "\meet^p_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \meet^p_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\meet^p_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \meet^p_ ( i 'in' A ) '/ ' F ']'").

Reserved Notation "\join^p_ i F"
  (at level 41, F at level 41, i at level 0,
           format "'[' \join^p_ i '/ ' F ']'").
Reserved Notation "\join^p_ ( i <- r | P ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \join^p_ ( i <- r | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i <- r ) F"
  (at level 41, F at level 41, i, r at level 50,
           format "'[' \join^p_ ( i <- r ) '/ ' F ']'").
Reserved Notation "\join^p_ ( m <= i < n | P ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \join^p_ ( m <= i < n | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( m <= i < n ) F"
  (at level 41, F at level 41, i, m, n at level 50,
           format "'[' \join^p_ ( m <= i < n ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i | P ) F"
  (at level 41, F at level 41, i at level 50,
           format "'[' \join^p_ ( i | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i : t | P ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\join^p_ ( i : t ) F"
  (at level 41, F at level 41, i at level 50).
Reserved Notation "\join^p_ ( i < n | P ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \join^p_ ( i < n | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i < n ) F"
  (at level 41, F at level 41, i, n at level 50,
           format "'[' \join^p_ ( i < n ) F ']'").
Reserved Notation "\join^p_ ( i 'in' A | P ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \join^p_ ( i 'in' A | P ) '/ ' F ']'").
Reserved Notation "\join^p_ ( i 'in' A ) F"
  (at level 41, F at level 41, i, A at level 50,
           format "'[' \join^p_ ( i 'in' A ) '/ ' F ']'").

Module Order.

STRUCTURES

Module POrder.
Section ClassDef.

Record mixin_of (T0 : Type) (b : Equality.class_of T0)
                (T := Equality.Pack b) := Mixin {
  le : rel T;
  lt : rel T;
  _ : x y, lt x y = (y != x) && (le x y);
  _ : reflexive le;
  _ : antisymmetric le;
  _ : transitive le;
}.

Record class_of (T : Type) := Class {
  base : Choice.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (Choice.class bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> Choice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Canonical eqType.
Canonical choiceType.
Notation porderType := type.
Notation POrderType disp T m := (@pack T disp _ _ id m).
Notation "[ 'porderType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'porderType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'porderType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0, format "[ 'porderType' 'of' T 'for' cT 'with' disp ]") :
  form_scope.
Notation "[ 'porderType' 'of' T ]" := [porderType of T for _]
  (at level 0, format "[ 'porderType' 'of' T ]") : form_scope.
Notation "[ 'porderType' 'of' T 'with' disp ]" :=
  [porderType of T for _ with disp]
  (at level 0, format "[ 'porderType' 'of' T 'with' disp ]") : form_scope.
End Exports.

End POrder.
Import POrder.Exports.

Section POrderDef.

Variable (disp : unit) (T : porderType disp).

Definition le : rel T := POrder.le (POrder.class T).

Definition lt : rel T := POrder.lt (POrder.class T).

Definition comparable : rel T := fun (x y : T) ⇒ (x y) || (y x).

Definition ge : simpl_rel T := [rel x y | y x].
Definition gt : simpl_rel T := [rel x y | y < x].
Definition leif (x y : T) C : Prop := ((x y) × ((x == y) = C))%type.

Definition le_of_leif x y C (le_xy : @leif x y C) := le_xy.1 : le x y.

Definition lteif x y C := if C then x y else x < y.

Variant le_xor_gt (x y : T) :
  T T T T bool bool Set :=
  | LeNotGt of x y : le_xor_gt x y x x y y true false
  | GtNotLe of y < x : le_xor_gt x y y y x x false true.

Variant lt_xor_ge (x y : T) :
  T T T T bool bool Set :=
  | LtNotGe of x < y : lt_xor_ge x y x x y y false true
  | GeNotLt of y x : lt_xor_ge x y y y x x true false.

Definition min x y := if x < y then x else y.
Definition max x y := if x < y then y else x.

Variant compare (x y : T) :
   T T T T
   bool bool bool bool bool bool Set :=
  | CompareLt of x < y : compare x y
    x x y y false false false true false true
  | CompareGt of y < x : compare x y
    y y x x false false true false true false
  | CompareEq of x = y : compare x y
    x x x x true true true true false false.

Variant incompare (x y : T) :
   T T T T
  bool bool bool bool bool bool bool bool Set :=
  | InCompareLt of x < y : incompare x y
    x x y y false false false true false true true true
  | InCompareGt of y < x : incompare x y
    y y x x false false true false true false true true
  | InCompare of x >< y : incompare x y
    x y y x false false false false false false false false
  | InCompareEq of x = y : incompare x y
    x x x x true true true true false false true true.

Definition arg_min {I : finType} := @extremum T I le.
Definition arg_max {I : finType} := @extremum T I ge.

Lifted min/max operations.
Section LiftedPOrder.
Variable T' : Type.
Implicit Type f : T' T.
Definition min_fun f g x := min (f x) (g x).
Definition max_fun f g x := min (f x) (g x).
End LiftedPOrder.

End POrderDef.

Arguments ge {_ _}.
Arguments gt {_ _}.
Arguments min {_ _}.
Arguments max {_ _}.
Arguments comparable {_ _}.
Arguments min_fun {_ _ _} f g _ /.
Arguments max_fun {_ _ _} f g _ /.

Module Import POSyntax.

Notation "<=%O" := le : fun_scope.
Notation ">=%O" := ge : fun_scope.
Notation "<%O" := lt : fun_scope.
Notation ">%O" := gt : fun_scope.
Notation "<?=%O" := leif : fun_scope.
Notation "<?<=%O" := lteif : fun_scope.
Notation ">=<%O" := comparable : fun_scope.
Notation "><%O" := (fun x y~~ (comparable x y)) : fun_scope.

Notation "<= y" := (ge y) : order_scope.
Notation "<= y :> T" := ( (y : T)) (only parsing) : order_scope.
Notation ">= y" := (le y) : order_scope.
Notation ">= y :> T" := ( (y : T)) (only parsing) : order_scope.

Notation "< y" := (gt y) : order_scope.
Notation "< y :> T" := (< (y : T)) (only parsing) : order_scope.
Notation "> y" := (lt y) : order_scope.
Notation "> y :> T" := (> (y : T)) (only parsing) : order_scope.

Notation "x <= y" := (le x y) : order_scope.
Notation "x <= y :> T" := ((x : T) (y : T)) (only parsing) : order_scope.
Notation "x >= y" := (y x) (only parsing) : order_scope.
Notation "x >= y :> T" := ((x : T) (y : T)) (only parsing) : order_scope.

Notation "x < y" := (lt x y) : order_scope.
Notation "x < y :> T" := ((x : T) < (y : T)) (only parsing) : order_scope.
Notation "x > y" := (y < x) (only parsing) : order_scope.
Notation "x > y :> T" := ((x : T) > (y : T)) (only parsing) : order_scope.

Notation "x <= y <= z" := ((x y) && (y z)) : order_scope.
Notation "x < y <= z" := ((x < y) && (y z)) : order_scope.
Notation "x <= y < z" := ((x y) && (y < z)) : order_scope.
Notation "x < y < z" := ((x < y) && (y < z)) : order_scope.

Notation "x <= y ?= 'iff' C" := (leif x y C) : order_scope.
Notation "x <= y ?= 'iff' C :> T" := ((x : T) (y : T) ?= iff C)
  (only parsing) : order_scope.

Notation "x < y ?<= 'if' C" := (lteif x y C) : order_scope.
Notation "x < y ?<= 'if' C :> T" := ((x : T) < (y : T) ?<= if C)
  (only parsing) : order_scope.

Notation ">=< y" := [pred x | comparable x y] : order_scope.
Notation ">=< y :> T" := (>=< (y : T)) (only parsing) : order_scope.
Notation "x >=< y" := (comparable x y) : order_scope.

Notation ">< y" := [pred x | ~~ comparable x y] : order_scope.
Notation ">< y :> T" := (>< (y : T)) (only parsing) : order_scope.
Notation "x >< y" := (~~ (comparable x y)) : order_scope.

Notation "[ 'arg' 'min_' ( i < i0 | P ) F ]" :=
    (arg_min i0 (fun iP%B) (fun iF))
  (at level 0, i, i0 at level 10,
   format "[ 'arg' 'min_' ( i < i0 | P ) F ]") : order_scope.

Notation "[ 'arg' 'min_' ( i < i0 'in' A ) F ]" :=
    [arg min_(i < i0 | i \in A) F]
  (at level 0, i, i0 at level 10,
   format "[ 'arg' 'min_' ( i < i0 'in' A ) F ]") : order_scope.

Notation "[ 'arg' 'min_' ( i < i0 ) F ]" := [arg min_(i < i0 | true) F]
  (at level 0, i, i0 at level 10,
   format "[ 'arg' 'min_' ( i < i0 ) F ]") : order_scope.

Notation "[ 'arg' 'max_' ( i > i0 | P ) F ]" :=
     (arg_max i0 (fun iP%B) (fun iF))
  (at level 0, i, i0 at level 10,
   format "[ 'arg' 'max_' ( i > i0 | P ) F ]") : order_scope.

Notation "[ 'arg' 'max_' ( i > i0 'in' A ) F ]" :=
    [arg max_(i > i0 | i \in A) F]
  (at level 0, i, i0 at level 10,
   format "[ 'arg' 'max_' ( i > i0 'in' A ) F ]") : order_scope.

Notation "[ 'arg' 'max_' ( i > i0 ) F ]" := [arg max_(i > i0 | true) F]
  (at level 0, i, i0 at level 10,
   format "[ 'arg' 'max_' ( i > i0 ) F ]") : order_scope.

Notation "f \min g" := (min_fun f g) : order_scope.
Notation "f \max g" := (max_fun f g) : order_scope.

Notation leLHS := (X in (X _)%O)%pattern.
Notation leRHS := (X in (_ X)%O)%pattern.
Notation ltLHS := (X in (X < _)%O)%pattern.
Notation ltRHS := (X in (_ < X)%O)%pattern.

End POSyntax.

Module POCoercions.
Coercion le_of_leif : leif >-> is_true.
End POCoercions.

Module Lattice.
Section ClassDef.

Record mixin_of (T0 : Type) (b : POrder.class_of T0)
                (T := POrder.Pack tt b) := Mixin {
  meet : T T T;
  join : T T T;
  _ : commutative meet;
  _ : commutative join;
  _ : associative meet;
  _ : associative join;
  _ : y x, meet x (join x y) = x;
  _ : y x, join x (meet x y) = x;
  _ : x y, (x y) = (meet x y == x);
}.

Record class_of (T : Type) := Class {
  base : POrder.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@POrder.class disp bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> POrder.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Notation latticeType := type.
Notation LatticeType T m := (@pack T _ _ _ id m).
Notation "[ 'latticeType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'latticeType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'latticeType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0, format "[ 'latticeType' 'of' T 'for' cT 'with' disp ]") :
  form_scope.
Notation "[ 'latticeType' 'of' T ]" := [latticeType of T for _]
  (at level 0, format "[ 'latticeType' 'of' T ]") : form_scope.
Notation "[ 'latticeType' 'of' T 'with' disp ]" :=
  [latticeType of T for _ with disp]
  (at level 0, format "[ 'latticeType' 'of' T 'with' disp ]") : form_scope.
End Exports.
End Lattice.
Export Lattice.Exports.

Section LatticeDef.
Context {disp : unit} {T : latticeType disp}.
Definition meet : T T T := Lattice.meet (Lattice.class T).
Definition join : T T T := Lattice.join (Lattice.class T).

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 Import LatticeSyntax.

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

End LatticeSyntax.

Module BLattice.
Section ClassDef.

Record mixin_of (T : Type) (b : POrder.class_of T)
                (T := POrder.Pack tt b) := Mixin {
  bottom : T;
  _ : x, bottom x;
}.

Record class_of (T : Type) := Class {
  base : Lattice.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@Lattice.class disp bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> Lattice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Notation bLatticeType := type.
Notation BLatticeType T m := (@pack T _ _ _ id m).
Notation "[ 'bLatticeType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'bLatticeType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'bLatticeType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0,
   format "[ 'bLatticeType' 'of' T 'for' cT 'with' disp ]") :
  form_scope.
Notation "[ 'bLatticeType' 'of' T ]" := [bLatticeType of T for _]
  (at level 0, format "[ 'bLatticeType' 'of' T ]") : form_scope.
Notation "[ 'bLatticeType' 'of' T 'with' disp ]" :=
  [bLatticeType of T for _ with disp]
  (at level 0, format "[ 'bLatticeType' 'of' T 'with' disp ]") :
  form_scope.
End Exports.

End BLattice.
Export BLattice.Exports.

Definition bottom {disp : unit} {T : bLatticeType disp} : T :=
  BLattice.bottom (BLattice.class T).

Module Import BLatticeSyntax.
Notation "0" := bottom : order_scope.

Notation "\join_ ( i <- r | P ) F" :=
  (\big[@join _ _/0%O]_(i <- r | P%B) F%O) : order_scope.
Notation "\join_ ( i <- r ) F" :=
  (\big[@join _ _/0%O]_(i <- r) F%O) : order_scope.
Notation "\join_ ( i | P ) F" :=
  (\big[@join _ _/0%O]_(i | P%B) F%O) : order_scope.
Notation "\join_ i F" :=
  (\big[@join _ _/0%O]_i F%O) : order_scope.
Notation "\join_ ( i : I | P ) F" :=
  (\big[@join _ _/0%O]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join_ ( i : I ) F" :=
  (\big[@join _ _/0%O]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join_ ( m <= i < n | P ) F" :=
  (\big[@join _ _/0%O]_(m i < n | P%B) F%O) : order_scope.
Notation "\join_ ( m <= i < n ) F" :=
  (\big[@join _ _/0%O]_(m i < n) F%O) : order_scope.
Notation "\join_ ( i < n | P ) F" :=
  (\big[@join _ _/0%O]_(i < n | P%B) F%O) : order_scope.
Notation "\join_ ( i < n ) F" :=
  (\big[@join _ _/0%O]_(i < n) F%O) : order_scope.
Notation "\join_ ( i 'in' A | P ) F" :=
  (\big[@join _ _/0%O]_(i in A | P%B) F%O) : order_scope.
Notation "\join_ ( i 'in' A ) F" :=
  (\big[@join _ _/0%O]_(i in A) F%O) : order_scope.

End BLatticeSyntax.

Module TBLattice.
Section ClassDef.

Record mixin_of (T0 : Type) (b : POrder.class_of T0)
                (T := POrder.Pack tt b) := Mixin {
  top : T;
  _ : x, x top;
}.

Record class_of (T : Type) := Class {
  base : BLattice.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@BLattice.class disp bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> BLattice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Canonical bLatticeType.
Notation tbLatticeType := type.
Notation TBLatticeType T m := (@pack T _ _ _ id m).
Notation "[ 'tbLatticeType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'tbLatticeType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'tbLatticeType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0,
   format "[ 'tbLatticeType' 'of' T 'for' cT 'with' disp ]") : form_scope.
Notation "[ 'tbLatticeType' 'of' T ]" := [tbLatticeType of T for _]
  (at level 0, format "[ 'tbLatticeType' 'of' T ]") : form_scope.
Notation "[ 'tbLatticeType' 'of' T 'with' disp ]" :=
  [tbLatticeType of T for _ with disp]
  (at level 0, format "[ 'tbLatticeType' 'of' T 'with' disp ]") : form_scope.
End Exports.

End TBLattice.
Export TBLattice.Exports.

Definition top disp {T : tbLatticeType disp} : T :=
  TBLattice.top (TBLattice.class T).

Module Import TBLatticeSyntax.

Notation "1" := top : order_scope.

Notation "\meet_ ( i <- r | P ) F" :=
  (\big[meet/1]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet_ ( i <- r ) F" :=
  (\big[meet/1]_(i <- r) F%O) : order_scope.
Notation "\meet_ ( i | P ) F" :=
  (\big[meet/1]_(i | P%B) F%O) : order_scope.
Notation "\meet_ i F" :=
  (\big[meet/1]_i F%O) : order_scope.
Notation "\meet_ ( i : I | P ) F" :=
  (\big[meet/1]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet_ ( i : I ) F" :=
  (\big[meet/1]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet_ ( m <= i < n | P ) F" :=
 (\big[meet/1]_(m i < n | P%B) F%O) : order_scope.
Notation "\meet_ ( m <= i < n ) F" :=
 (\big[meet/1]_(m i < n) F%O) : order_scope.
Notation "\meet_ ( i < n | P ) F" :=
 (\big[meet/1]_(i < n | P%B) F%O) : order_scope.
Notation "\meet_ ( i < n ) F" :=
 (\big[meet/1]_(i < n) F%O) : order_scope.
Notation "\meet_ ( i 'in' A | P ) F" :=
 (\big[meet/1]_(i in A | P%B) F%O) : order_scope.
Notation "\meet_ ( i 'in' A ) F" :=
 (\big[meet/1]_(i in A) F%O) : order_scope.

End TBLatticeSyntax.

Module DistrLattice.
Section ClassDef.

Record mixin_of (T0 : Type) (b : Lattice.class_of T0)
                (T := Lattice.Pack tt b) := Mixin {
  _ : @left_distributive T T meet join;
}.

Record class_of (T : Type) := Class {
  base : Lattice.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@Lattice.class disp bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> Lattice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Notation distrLatticeType := type.
Notation DistrLatticeType T m := (@pack T _ _ _ id m).
Notation "[ 'distrLatticeType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'distrLatticeType' 'of' T 'for' cT ]") :
  form_scope.
Notation "[ 'distrLatticeType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0,
   format "[ 'distrLatticeType' 'of' T 'for' cT 'with' disp ]") :
  form_scope.
Notation "[ 'distrLatticeType' 'of' T ]" := [distrLatticeType of T for _]
  (at level 0, format "[ 'distrLatticeType' 'of' T ]") : form_scope.
Notation "[ 'distrLatticeType' 'of' T 'with' disp ]" :=
  [distrLatticeType of T for _ with disp]
  (at level 0, format "[ 'distrLatticeType' 'of' T 'with' disp ]") :
  form_scope.
End Exports.

End DistrLattice.
Export DistrLattice.Exports.

Module BDistrLattice.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : DistrLattice.class_of T;
  mixin : BLattice.mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT b & phant_id (@DistrLattice.class disp bT) b
  fun mT m & phant_id (@BLattice.class disp mT) (BLattice.Class m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition nb_distrLatticeType := @DistrLattice.Pack disp bLatticeType class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> DistrLattice.class_of.
Coercion base2 : class_of >-> BLattice.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical distrLatticeType.
Canonical nb_distrLatticeType.
Notation bDistrLatticeType := type.
Notation "[ 'bDistrLatticeType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'bDistrLatticeType' 'of' T ]") : form_scope.
End Exports.

End BDistrLattice.
Export BDistrLattice.Exports.

Module TBDistrLattice.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : BDistrLattice.class_of T;
  mixin : TBLattice.mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT (b : BDistrLattice.class_of T)
      & phant_id (@BDistrLattice.class disp bT) b
  fun mT m & phant_id (@TBLattice.class disp mT) (@TBLattice.Class _ b m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition tbLatticeType := @TBLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition bDistrLatticeType := @BDistrLattice.Pack disp cT class.
Definition ntb_distrLatticeType := @DistrLattice.Pack disp tbLatticeType class.
Definition ntb_bDistrLatticeType :=
  @BDistrLattice.Pack disp tbLatticeType class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> BDistrLattice.class_of.
Coercion base2 : class_of >-> TBLattice.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion tbLatticeType : type >-> TBLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Coercion bDistrLatticeType : type >-> BDistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical tbLatticeType.
Canonical distrLatticeType.
Canonical bDistrLatticeType.
Canonical ntb_distrLatticeType.
Canonical ntb_bDistrLatticeType.
Notation tbDistrLatticeType := type.
Notation "[ 'tbDistrLatticeType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'tbDistrLatticeType' 'of' T ]") : form_scope.
End Exports.

End TBDistrLattice.
Export TBDistrLattice.Exports.

Module CBDistrLattice.
Section ClassDef.

Record mixin_of (T0 : Type) (b : BDistrLattice.class_of T0)
                (T := BDistrLattice.Pack tt b) := Mixin {
  sub : T T T;
  _ : x y, y `&` sub x y = bottom;
  _ : x y, (x `&` y) `|` sub x y = x
}.

Record class_of (T : Type) := Class {
  base : BDistrLattice.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@BDistrLattice.class disp bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition bDistrLatticeType := @BDistrLattice.Pack disp cT class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> BDistrLattice.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Coercion bDistrLatticeType : type >-> BDistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical distrLatticeType.
Canonical bDistrLatticeType.
Notation cbDistrLatticeType := type.
Notation CBDistrLatticeType T m := (@pack T _ _ _ id m).
Notation "[ 'cbDistrLatticeType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'cbDistrLatticeType' 'of' T 'for' cT ]") :
  form_scope.
Notation "[ 'cbDistrLatticeType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0,
   format "[ 'cbDistrLatticeType' 'of' T 'for' cT 'with' disp ]") :
  form_scope.
Notation "[ 'cbDistrLatticeType' 'of' T ]" := [cbDistrLatticeType of T for _]
  (at level 0, format "[ 'cbDistrLatticeType' 'of' T ]") : form_scope.
Notation "[ 'cbDistrLatticeType' 'of' T 'with' disp ]" :=
  [cbDistrLatticeType of T for _ with disp]
  (at level 0, format "[ 'cbDistrLatticeType' 'of' T 'with' disp ]") :
  form_scope.
End Exports.

End CBDistrLattice.
Export CBDistrLattice.Exports.

Definition sub {disp : unit} {T : cbDistrLatticeType disp} : T T T :=
  CBDistrLattice.sub (CBDistrLattice.class T).

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

Module CTBDistrLattice.
Section ClassDef.

Record mixin_of (T0 : Type) (b : TBDistrLattice.class_of T0)
                (T := TBDistrLattice.Pack tt b) (sub : T T T) := Mixin {
  compl : T T;
  _ : x, compl x = sub top x
}.

Record class_of (T : Type) := Class {
  base : TBDistrLattice.class_of T;
  mixin1 : CBDistrLattice.mixin_of base;
  mixin2 : @mixin_of _ base (CBDistrLattice.sub mixin1);
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack disp T c.
Definition clone_with disp' c of phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@TBDistrLattice.class disp bT) b
  fun mT m0 & phant_id (@CBDistrLattice.class disp mT) (CBDistrLattice.Class m0) ⇒
  fun m1Pack disp (@Class T b m0 m1).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition tbLatticeType := @TBLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition bDistrLatticeType := @BDistrLattice.Pack disp cT class.
Definition tbDistrLatticeType := @TBDistrLattice.Pack disp cT class.
Definition cbDistrLatticeType := @CBDistrLattice.Pack disp cT class.
Definition cb_tbLatticeType := @TBLattice.Pack disp cbDistrLatticeType class.
Definition cb_tbDistrLatticeType :=
  @TBDistrLattice.Pack disp cbDistrLatticeType class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> TBDistrLattice.class_of.
Coercion base2 : class_of >-> CBDistrLattice.class_of.
Coercion mixin1 : class_of >-> CBDistrLattice.mixin_of.
Coercion mixin2 : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion tbLatticeType : type >-> TBLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Coercion bDistrLatticeType : type >-> BDistrLattice.type.
Coercion tbDistrLatticeType : type >-> TBDistrLattice.type.
Coercion cbDistrLatticeType : type >-> CBDistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical tbLatticeType.
Canonical distrLatticeType.
Canonical bDistrLatticeType.
Canonical tbDistrLatticeType.
Canonical cbDistrLatticeType.
Canonical cb_tbLatticeType.
Canonical cb_tbDistrLatticeType.
Notation ctbDistrLatticeType := type.
Notation CTBDistrLatticeType T m := (@pack T _ _ _ id _ _ id m).
Notation "[ 'ctbDistrLatticeType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'ctbDistrLatticeType' 'of' T 'for' cT ]") :
  form_scope.
Notation "[ 'ctbDistrLatticeType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0,
   format "[ 'ctbDistrLatticeType' 'of' T 'for' cT 'with' disp ]")
  : form_scope.
Notation "[ 'ctbDistrLatticeType' 'of' T ]" := [ctbDistrLatticeType of T for _]
  (at level 0, format "[ 'ctbDistrLatticeType' 'of' T ]") : form_scope.
Notation "[ 'ctbDistrLatticeType' 'of' T 'with' disp ]" :=
  [ctbDistrLatticeType of T for _ with disp]
  (at level 0, format "[ 'ctbDistrLatticeType' 'of' T 'with' disp ]") :
  form_scope.
Notation "[ 'default_ctbDistrLatticeType' 'of' T ]" :=
  (@pack T _ _ _ id _ _ id (Mixin (fun erefl)))
  (at level 0, format "[ 'default_ctbDistrLatticeType' 'of' T ]") :
  form_scope.
End Exports.

End CTBDistrLattice.
Export CTBDistrLattice.Exports.

Definition compl {disp : unit} {T : ctbDistrLatticeType disp} : T T :=
  CTBDistrLattice.compl (CTBDistrLattice.class T).

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

Module Total.
Section ClassDef.

Definition mixin_of T0 (b : POrder.class_of T0) (T := POrder.Pack tt b) :=
  total (<=%O : rel T).

Record class_of (T : Type) := Class {
  base : DistrLattice.class_of T;
  mixin : mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c & phant_id class c := @Pack disp T c.
Definition clone_with disp' c & phant_id class c := @Pack disp' T c.

Definition pack :=
  fun bT b & phant_id (@DistrLattice.class disp bT) b
  fun mPack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.

End ClassDef.

Module Exports.
Coercion base : class_of >-> DistrLattice.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion porderType : type >-> POrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical porderType.
Canonical latticeType.
Canonical distrLatticeType.
Notation orderType := type.
Notation OrderType T m := (@pack T _ _ _ id m).
Notation "[ 'orderType' 'of' T 'for' cT ]" := (@clone T _ cT _ id)
  (at level 0, format "[ 'orderType' 'of' T 'for' cT ]") : form_scope.
Notation "[ 'orderType' 'of' T 'for' cT 'with' disp ]" :=
  (@clone_with T _ cT disp _ id)
  (at level 0, format "[ 'orderType' 'of' T 'for' cT 'with' disp ]") :
  form_scope.
Notation "[ 'orderType' 'of' T ]" := [orderType of T for _]
  (at level 0, format "[ 'orderType' 'of' T ]") : form_scope.
Notation "[ 'orderType' 'of' T 'with' disp ]" :=
  [orderType of T for _ with disp]
  (at level 0, format "[ 'orderType' 'of' T 'with' disp ]") : form_scope.
End Exports.

End Total.
Import Total.Exports.

FINITE

Module FinPOrder.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : POrder.class_of T;
  mixin : Finite.mixin_of (Equality.Pack base)
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT b & phant_id (@POrder.class disp bT) b
  fun mT m & phant_id (@Finite.class mT) (@Finite.Class _ _ m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition countType := @Countable.Pack cT class.
Definition finType := @Finite.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition count_porderType := @POrder.Pack disp countType class.
Definition fin_porderType := @POrder.Pack disp finType class.
End ClassDef.

Module Exports.
Coercion base : class_of >-> POrder.class_of.
Coercion base2 : class_of >-> Finite.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion countType : type >-> Countable.type.
Coercion finType : type >-> Finite.type.
Coercion porderType : type >-> POrder.type.
Canonical eqType.
Canonical choiceType.
Canonical countType.
Canonical finType.
Canonical porderType.
Canonical count_porderType.
Canonical fin_porderType.
Notation finPOrderType := type.
Notation "[ 'finPOrderType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'finPOrderType' 'of' T ]") : form_scope.
End Exports.

End FinPOrder.
Import FinPOrder.Exports.

Module FinLattice.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : TBLattice.class_of T;
  mixin : Finite.mixin_of (Equality.Pack base);
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT b & phant_id (@TBLattice.class disp bT) b
  fun mT m & phant_id (@Finite.class mT) (@Finite.Class _ _ m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition countType := @Countable.Pack cT class.
Definition finType := @Finite.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition finPOrderType := @FinPOrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition tbLatticeType := @TBLattice.Pack disp cT class.
Definition count_latticeType := @Lattice.Pack disp countType class.
Definition count_bLatticeType := @BLattice.Pack disp countType class.
Definition count_tbLatticeType := @TBLattice.Pack disp countType class.
Definition fin_latticeType := @Lattice.Pack disp finType class.
Definition fin_bLatticeType := @BLattice.Pack disp finType class.
Definition fin_tbLatticeType := @TBLattice.Pack disp finType class.
Definition finPOrder_latticeType := @Lattice.Pack disp finPOrderType class.
Definition finPOrder_bLatticeType := @BLattice.Pack disp finPOrderType class.
Definition finPOrder_tbLatticeType := @TBLattice.Pack disp finPOrderType class.

End ClassDef.

Module Exports.
Coercion base : class_of >-> TBLattice.class_of.
Coercion base2 : class_of >-> FinPOrder.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion countType : type >-> Countable.type.
Coercion finType : type >-> Finite.type.
Coercion porderType : type >-> POrder.type.
Coercion finPOrderType : type >-> FinPOrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion tbLatticeType : type >-> TBLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical countType.
Canonical finType.
Canonical porderType.
Canonical finPOrderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical tbLatticeType.
Canonical count_latticeType.
Canonical count_bLatticeType.
Canonical count_tbLatticeType.
Canonical fin_latticeType.
Canonical fin_bLatticeType.
Canonical fin_tbLatticeType.
Canonical finPOrder_latticeType.
Canonical finPOrder_bLatticeType.
Canonical finPOrder_tbLatticeType.
Notation finLatticeType := type.
Notation "[ 'finLatticeType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'finLatticeType' 'of' T ]") : form_scope.
End Exports.

End FinLattice.
Export FinLattice.Exports.

Module FinDistrLattice.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : TBDistrLattice.class_of T;
  mixin : Finite.mixin_of (Equality.Pack base);
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT b & phant_id (@TBDistrLattice.class disp bT) b
  fun mT m & phant_id (@Finite.class mT) (@Finite.Class _ _ m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition countType := @Countable.Pack cT class.
Definition finType := @Finite.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition finPOrderType := @FinPOrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition tbLatticeType := @TBLattice.Pack disp cT class.
Definition finLatticeType := @FinLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition bDistrLatticeType := @BDistrLattice.Pack disp cT class.
Definition tbDistrLatticeType := @TBDistrLattice.Pack disp cT class.
Definition count_distrLatticeType := @DistrLattice.Pack disp countType class.
Definition count_bDistrLatticeType := @BDistrLattice.Pack disp countType class.
Definition count_tbDistrLatticeType :=
  @TBDistrLattice.Pack disp countType class.
Definition fin_distrLatticeType := @DistrLattice.Pack disp finType class.
Definition fin_bDistrLatticeType := @BDistrLattice.Pack disp finType class.
Definition fin_tbDistrLatticeType := @TBDistrLattice.Pack disp finType class.
Definition finPOrder_distrLatticeType :=
  @DistrLattice.Pack disp finPOrderType class.
Definition finPOrder_bDistrLatticeType :=
  @BDistrLattice.Pack disp finPOrderType class.
Definition finPOrder_tbDistrLatticeType :=
  @TBDistrLattice.Pack disp finPOrderType class.
Definition finLattice_distrLatticeType :=
  @DistrLattice.Pack disp finLatticeType class.
Definition finLattice_bDistrLatticeType :=
  @BDistrLattice.Pack disp finLatticeType class.
Definition finLattice_tbDistrLatticeType :=
  @TBDistrLattice.Pack disp finLatticeType class.

End ClassDef.

Module Exports.
Coercion base : class_of >-> TBDistrLattice.class_of.
Coercion base2 : class_of >-> FinLattice.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion countType : type >-> Countable.type.
Coercion finType : type >-> Finite.type.
Coercion porderType : type >-> POrder.type.
Coercion finPOrderType : type >-> FinPOrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion tbLatticeType : type >-> TBLattice.type.
Coercion finLatticeType : type >-> FinLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Coercion bDistrLatticeType : type >-> BDistrLattice.type.
Coercion tbDistrLatticeType : type >-> TBDistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical countType.
Canonical finType.
Canonical porderType.
Canonical finPOrderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical tbLatticeType.
Canonical finLatticeType.
Canonical distrLatticeType.
Canonical bDistrLatticeType.
Canonical tbDistrLatticeType.
Canonical count_distrLatticeType.
Canonical count_bDistrLatticeType.
Canonical count_tbDistrLatticeType.
Canonical fin_distrLatticeType.
Canonical fin_bDistrLatticeType.
Canonical fin_tbDistrLatticeType.
Canonical finPOrder_distrLatticeType.
Canonical finPOrder_bDistrLatticeType.
Canonical finPOrder_tbDistrLatticeType.
Canonical finLattice_distrLatticeType.
Canonical finLattice_bDistrLatticeType.
Canonical finLattice_tbDistrLatticeType.
Notation finDistrLatticeType := type.
Notation "[ 'finDistrLatticeType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'finDistrLatticeType' 'of' T ]") : form_scope.
End Exports.

End FinDistrLattice.
Export FinDistrLattice.Exports.

Module FinCDistrLattice.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : CTBDistrLattice.class_of T;
  mixin : Finite.mixin_of (Equality.Pack base);
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT b & phant_id (@CTBDistrLattice.class disp bT) b
  fun mT m & phant_id (@Finite.class mT) (@Finite.Class _ _ m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition countType := @Countable.Pack cT class.
Definition finType := @Finite.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition finPOrderType := @FinPOrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition tbLatticeType := @TBLattice.Pack disp cT class.
Definition finLatticeType := @FinLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition bDistrLatticeType := @BDistrLattice.Pack disp cT class.
Definition tbDistrLatticeType := @TBDistrLattice.Pack disp cT class.
Definition finDistrLatticeType := @FinDistrLattice.Pack disp cT class.
Definition cbDistrLatticeType := @CBDistrLattice.Pack disp cT class.
Definition ctbDistrLatticeType := @CTBDistrLattice.Pack disp cT class.
Definition count_cbDistrLatticeType :=
  @CBDistrLattice.Pack disp countType class.
Definition count_ctbDistrLatticeType :=
  @CTBDistrLattice.Pack disp countType class.
Definition fin_cbDistrLatticeType := @CBDistrLattice.Pack disp finType class.
Definition fin_ctbDistrLatticeType := @CTBDistrLattice.Pack disp finType class.
Definition finPOrder_cbDistrLatticeType :=
  @CBDistrLattice.Pack disp finPOrderType class.
Definition finPOrder_ctbDistrLatticeType :=
  @CTBDistrLattice.Pack disp finPOrderType class.
Definition finLattice_cbDistrLatticeType :=
  @CBDistrLattice.Pack disp finLatticeType class.
Definition finLattice_ctbDistrLatticeType :=
  @CTBDistrLattice.Pack disp finLatticeType class.
Definition finDistrLattice_cbDistrLatticeType :=
  @CBDistrLattice.Pack disp finDistrLatticeType class.
Definition finDistrLattice_ctbDistrLatticeType :=
  @CTBDistrLattice.Pack disp finDistrLatticeType class.

End ClassDef.

Module Exports.
Coercion base : class_of >-> CTBDistrLattice.class_of.
Coercion base2 : class_of >-> FinDistrLattice.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion countType : type >-> Countable.type.
Coercion finType : type >-> Finite.type.
Coercion porderType : type >-> POrder.type.
Coercion finPOrderType : type >-> FinPOrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion tbLatticeType : type >-> TBLattice.type.
Coercion finLatticeType : type >-> FinLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Coercion bDistrLatticeType : type >-> BDistrLattice.type.
Coercion tbDistrLatticeType : type >-> TBDistrLattice.type.
Coercion finDistrLatticeType : type >-> FinDistrLattice.type.
Coercion cbDistrLatticeType : type >-> CBDistrLattice.type.
Coercion ctbDistrLatticeType : type >-> CTBDistrLattice.type.
Canonical eqType.
Canonical choiceType.
Canonical countType.
Canonical finType.
Canonical porderType.
Canonical finPOrderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical tbLatticeType.
Canonical finLatticeType.
Canonical distrLatticeType.
Canonical bDistrLatticeType.
Canonical tbDistrLatticeType.
Canonical finDistrLatticeType.
Canonical cbDistrLatticeType.
Canonical ctbDistrLatticeType.
Canonical count_cbDistrLatticeType.
Canonical count_ctbDistrLatticeType.
Canonical fin_cbDistrLatticeType.
Canonical fin_ctbDistrLatticeType.
Canonical finPOrder_cbDistrLatticeType.
Canonical finPOrder_ctbDistrLatticeType.
Canonical finLattice_cbDistrLatticeType.
Canonical finLattice_ctbDistrLatticeType.
Canonical finDistrLattice_cbDistrLatticeType.
Canonical finDistrLattice_ctbDistrLatticeType.
Notation finCDistrLatticeType := type.
Notation "[ 'finCDistrLatticeType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'finCDistrLatticeType' 'of' T ]") : form_scope.
End Exports.

End FinCDistrLattice.
Export FinCDistrLattice.Exports.

Module FinTotal.
Section ClassDef.

Record class_of (T : Type) := Class {
  base : FinDistrLattice.class_of T;
  mixin : Total.mixin_of base;
}.


Structure type (disp : unit) := Pack { sort; _ : class_of sort }.


Variables (T : Type) (disp : unit) (cT : type disp).

Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.

Definition pack :=
  fun bT b & phant_id (@FinDistrLattice.class disp bT) b
  fun mT m & phant_id (@Total.class disp mT) (Total.Class m) ⇒
  Pack disp (@Class T b m).

Definition eqType := @Equality.Pack cT class.
Definition choiceType := @Choice.Pack cT class.
Definition countType := @Countable.Pack cT class.
Definition finType := @Finite.Pack cT class.
Definition porderType := @POrder.Pack disp cT class.
Definition finPOrderType := @FinPOrder.Pack disp cT class.
Definition latticeType := @Lattice.Pack disp cT class.
Definition bLatticeType := @BLattice.Pack disp cT class.
Definition tbLatticeType := @TBLattice.Pack disp cT class.
Definition finLatticeType := @FinLattice.Pack disp cT class.
Definition distrLatticeType := @DistrLattice.Pack disp cT class.
Definition bDistrLatticeType := @BDistrLattice.Pack disp cT class.
Definition tbDistrLatticeType := @TBDistrLattice.Pack disp cT class.
Definition finDistrLatticeType := @FinDistrLattice.Pack disp cT class.
Definition orderType := @Total.Pack disp cT class.
Definition order_countType := @Countable.Pack orderType class.
Definition order_finType := @Finite.Pack orderType class.
Definition order_finPOrderType := @FinPOrder.Pack disp orderType class.
Definition order_bLatticeType := @BLattice.Pack disp orderType class.
Definition order_tbLatticeType := @TBLattice.Pack disp orderType class.
Definition order_finLatticeType := @FinLattice.Pack disp orderType class.
Definition order_bDistrLatticeType := @BDistrLattice.Pack disp orderType class.
Definition order_tbDistrLatticeType :=
  @TBDistrLattice.Pack disp orderType class.
Definition order_finDistrLatticeType :=
  @FinDistrLattice.Pack disp orderType class.

End ClassDef.

Module Exports.
Coercion base : class_of >-> FinDistrLattice.class_of.
Coercion base2 : class_of >-> Total.class_of.
Coercion sort : type >-> Sortclass.
Coercion eqType : type >-> Equality.type.
Coercion choiceType : type >-> Choice.type.
Coercion countType : type >-> Countable.type.
Coercion finType : type >-> Finite.type.
Coercion porderType : type >-> POrder.type.
Coercion finPOrderType : type >-> FinPOrder.type.
Coercion latticeType : type >-> Lattice.type.
Coercion bLatticeType : type >-> BLattice.type.
Coercion tbLatticeType : type >-> TBLattice.type.
Coercion finLatticeType : type >-> FinLattice.type.
Coercion distrLatticeType : type >-> DistrLattice.type.
Coercion bDistrLatticeType : type >-> BDistrLattice.type.
Coercion tbDistrLatticeType : type >-> TBDistrLattice.type.
Coercion finDistrLatticeType : type >-> FinDistrLattice.type.
Coercion orderType : type >-> Total.type.
Canonical eqType.
Canonical choiceType.
Canonical countType.
Canonical finType.
Canonical porderType.
Canonical finPOrderType.
Canonical latticeType.
Canonical bLatticeType.
Canonical tbLatticeType.
Canonical finLatticeType.
Canonical distrLatticeType.
Canonical bDistrLatticeType.
Canonical tbDistrLatticeType.
Canonical finDistrLatticeType.
Canonical orderType.
Canonical order_countType.
Canonical order_finType.
Canonical order_finPOrderType.
Canonical order_bLatticeType.
Canonical order_tbLatticeType.
Canonical order_finLatticeType.
Canonical order_bDistrLatticeType.
Canonical order_tbDistrLatticeType.
Canonical order_finDistrLatticeType.
Notation finOrderType := type.
Notation "[ 'finOrderType' 'of' T ]" := (@pack T _ _ _ id _ _ id)
  (at level 0, format "[ 'finOrderType' 'of' T ]") : form_scope.
End Exports.

End FinTotal.
Export FinTotal.Exports.

DUAL

Definition dual T : Type := T.
Definition dual_display : unit unit.

Notation dual_le := (@le (dual_display _) _).
Notation dual_lt := (@lt (dual_display _) _).
Notation dual_comparable := (@comparable (dual_display _) _).
Notation dual_ge := (@ge (dual_display _) _).
Notation dual_gt := (@gt (dual_display _) _).
Notation dual_leif := (@leif (dual_display _) _).
Notation dual_lteif := (@lteif (dual_display _) _).
Notation dual_max := (@max (dual_display _) _).
Notation dual_min := (@min (dual_display _) _).
Notation dual_meet := (@meet (dual_display _) _).
Notation dual_join := (@join (dual_display _) _).
Notation dual_bottom := (@bottom (dual_display _) _).
Notation dual_top := (@top (dual_display _) _).

Module Import DualSyntax.

Notation "T ^d" := (dual T) (at level 2, format "T ^d") : type_scope.
Notation "<=^d%O" := dual_le : fun_scope.
Notation ">=^d%O" := dual_ge : fun_scope.
Notation "<^d%O" := dual_lt : fun_scope.
Notation ">^d%O" := dual_gt : fun_scope.
Notation "<?=^d%O" := dual_leif : fun_scope.
Notation "<?<=^d%O" := dual_lteif : fun_scope.
Notation ">=<^d%O" := dual_comparable : fun_scope.
Notation "><^d%O" := (fun x y~~ dual_comparable x y) : fun_scope.

Notation "<=^d y" := (>=^d%O y) : order_scope.
Notation "<=^d y :> T" := (<=^d (y : T)) (only parsing) : order_scope.
Notation ">=^d y" := (<=^d%O y) : order_scope.
Notation ">=^d y :> T" := (>=^d (y : T)) (only parsing) : order_scope.

Notation "<^d y" := (>^d%O y) : order_scope.
Notation "<^d y :> T" := (<^d (y : T)) (only parsing) : order_scope.
Notation ">^d y" := (<^d%O y) : order_scope.
Notation ">^d y :> T" := (>^d (y : T)) (only parsing) : order_scope.

Notation "x <=^d y" := (<=^d%O x y) : order_scope.
Notation "x <=^d y :> T" := ((x : T) <=^d (y : T)) (only parsing) : order_scope.
Notation "x >=^d y" := (y <=^d x) (only parsing) : order_scope.
Notation "x >=^d y :> T" := ((x : T) >=^d (y : T)) (only parsing) : order_scope.

Notation "x <^d y" := (<^d%O x y) : order_scope.
Notation "x <^d y :> T" := ((x : T) <^d (y : T)) (only parsing) : order_scope.
Notation "x >^d y" := (y <^d x) (only parsing) : order_scope.
Notation "x >^d y :> T" := ((x : T) >^d (y : T)) (only parsing) : order_scope.

Notation "x <=^d y <=^d z" := ((x <=^d y) && (y <=^d z)) : order_scope.
Notation "x <^d y <=^d z" := ((x <^d y) && (y <=^d z)) : order_scope.
Notation "x <=^d y <^d z" := ((x <=^d y) && (y <^d z)) : order_scope.
Notation "x <^d y <^d z" := ((x <^d y) && (y <^d z)) : order_scope.

Notation "x <=^d y ?= 'iff' C" := (<?=^d%O x y C) : order_scope.
Notation "x <=^d y ?= 'iff' C :> T" := ((x : T) <=^d (y : T) ?= iff C)
  (only parsing) : order_scope.

Notation "x <^d y ?<= 'if' C" := (<?<=^d%O x y C) : order_scope.
Notation "x <^d y ?<= 'if' C :> T" := ((x : T) <^d (y : T) ?<= if C)
  (only parsing) : order_scope.

Notation ">=<^d x" := (>=<^d%O x) : order_scope.
Notation ">=<^d y :> T" := (>=<^d (y : T)) (only parsing) : order_scope.
Notation "x >=<^d y" := (>=<^d%O x y) : order_scope.

Notation "><^d y" := [pred x | ~~ dual_comparable x y] : order_scope.
Notation "><^d y :> T" := (><^d (y : T)) (only parsing) : order_scope.
Notation "x ><^d y" := (~~ (><^d%O x y)) : order_scope.

Notation "x `&^d` y" := (dual_meet x y) : order_scope.
Notation "x `|^d` y" := (dual_join x y) : order_scope.

Notation "0^d" := dual_bottom : order_scope.
Notation "1^d" := dual_top : order_scope.

The following Local Notations are here to define the \join^d and \meet^d notations later. Do not remove them.

Notation "\join^d_ ( i <- r | P ) F" :=
  (\big[join/0]_(i <- r | P%B) F%O) : order_scope.
Notation "\join^d_ ( i <- r ) F" :=
  (\big[join/0]_(i <- r) F%O) : order_scope.
Notation "\join^d_ ( i | P ) F" :=
  (\big[join/0]_(i | P%B) F%O) : order_scope.
Notation "\join^d_ i F" :=
  (\big[join/0]_i F%O) : order_scope.
Notation "\join^d_ ( i : I | P ) F" :=
  (\big[join/0]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\join^d_ ( i : I ) F" :=
  (\big[join/0]_(i : I) F%O) (only parsing) : order_scope.
Notation "\join^d_ ( m <= i < n | P ) F" :=
 (\big[join/0]_(m i < n | P%B) F%O) : order_scope.
Notation "\join^d_ ( m <= i < n ) F" :=
 (\big[join/0]_(m i < n) F%O) : order_scope.
Notation "\join^d_ ( i < n | P ) F" :=
 (\big[join/0]_(i < n | P%B) F%O) : order_scope.
Notation "\join^d_ ( i < n ) F" :=
 (\big[join/0]_(i < n) F%O) : order_scope.
Notation "\join^d_ ( i 'in' A | P ) F" :=
 (\big[join/0]_(i in A | P%B) F%O) : order_scope.
Notation "\join^d_ ( i 'in' A ) F" :=
 (\big[join/0]_(i in A) F%O) : order_scope.

Notation "\meet^d_ ( i <- r | P ) F" :=
  (\big[meet/1]_(i <- r | P%B) F%O) : order_scope.
Notation "\meet^d_ ( i <- r ) F" :=
  (\big[meet/1]_(i <- r) F%O) : order_scope.
Notation "\meet^d_ ( i | P ) F" :=
  (\big[meet/1]_(i | P%B) F%O) : order_scope.
Notation "\meet^d_ i F" :=
  (\big[meet/1]_i F%O) : order_scope.
Notation "\meet^d_ ( i : I | P ) F" :=
  (\big[meet/1]_(i : I | P%B) F%O) (only parsing) : order_scope.
Notation "\meet^d_ ( i : I ) F" :=
  (\big[meet/1]_(i : I) F%O) (only parsing) : order_scope.
Notation "\meet^d_ ( m <= i < n | P ) F" :=
 (\big[meet/1]_(m i < n | P%B) F%O) : order_scope.
Notation "\meet^d_ ( m <= i < n ) F" :=
 (\big[meet/1]_(m i < n) F%O) : order_scope.
Notation "\meet^d_ ( i < n | P ) F" :=
 (\big[meet/1]_(i < n | P%B) F%O) : order_scope.
Notation "\meet^d_ ( i < n ) F" :=
 (\big[meet/1]_(i < n) F%O) : order_scope.
Notation "\meet^d_ ( i 'in' A | P ) F" :=
 (\big[meet/1]_(i in A | P%B) F%O) : order_scope.
Notation "\meet^d_ ( i 'in' A ) F" :=
 (\big[meet/1]_(i in A) F%O) : order_scope.

End DualSyntax.

THEORY

Module Import POrderTheory.
Section POrderTheory.

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

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

Lemma geE x y : ge x y = (y x).
Lemma gtE x y : gt x y = (y < x).

Lemma lexx (x : T) : x x.
Hint Resolve lexx : core.

Definition le_refl : reflexive le := lexx.
Definition ge_refl : reflexive ge := lexx.
Hint Resolve le_refl : core.

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

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

Lemma le_trans: transitive (<=%O : rel T).

Lemma ge_trans: transitive (>=%O : rel T).

Lemma le_le_trans x y z t : z x y t x y z t.

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

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

Lemma ltxx x: x < x = false.

Definition lt_irreflexive : irreflexive lt := ltxx.
Hint Resolve lt_irreflexive : core.

Definition ltexx := (lexx, ltxx).

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

Lemma lt_eqF x y: x < y x == y = false.

Lemma gt_eqF x y : y < x x == y = false.

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

Lemma ltW x y: x < y x y.

Lemma lt_le_trans y x z: x < y y z x < z.

Lemma lt_trans: transitive (<%O : rel T).

Lemma le_lt_trans y x z: x y y < z x < z.

Lemma lt_nsym x y : x < y y < x False.

Lemma lt_asym x y : x < y < x = false.

Lemma le_gtF x y: x y y < x = false.

Lemma lt_geF x y : (x < y) y x = false.

Definition lt_gtF x y hxy := le_gtF (@ltW x y hxy).

Lemma lt_leAnge x y : (x < y) = (x y) && ~~ (y x).

Lemma lt_le_asym x y : x < y x = false.

Lemma le_lt_asym x y : x y < x = false.

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

Lemma le_path_min x s : path <=%O x s all ( x) s.

Lemma lt_path_min x s : path <%O x s all (> x) s.

Lemma le_path_sortedE x s : path <=%O x s = all ( x) s && sorted <=%O s.

Lemma lt_path_sortedE x s : path <%O x s = all (> x) s && sorted <%O s.

Lemma le_sorted_pairwise s : sorted <=%O s = pairwise <=%O s.

Lemma lt_sorted_pairwise s : sorted <%O s = pairwise <%O s.

Lemma le_path_pairwise x s : path <=%O x s = pairwise <=%O (