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67 changes: 38 additions & 29 deletions classical/boolp.v
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Expand Up @@ -20,25 +20,28 @@ From mathcomp Require internal_Eqdep_dec.
(* Asymptotic Reasoning in Classical Analysis. JFR 2018 *)
(* *)
(* ## Axioms *)
(* ``` *)
(* functional_extensionality_dep == functional extensionality (on dependently *)
(* typed functions), i.e., functions that are pointwise *)
(* equal are equal *)
(* propositional_extensionality == propositional extensionality, i.e., iff *)
(* and equality are the same on Prop *)
(* constructive_indefinite_description == existential in Prop (ex) implies *)
(* existential in Type (sig) *)
(* cid := constructive_indefinite_description (shortcut) *)
(* ``` *)
(* `functional_extensionality_dep` *)
(* ~ functional extensionality (on dependently typed functions), *)
(* i.e., functions that are pointwise equal are equal *)
(* *)
(* `propositional_extensionality` *)
(* ~ propositional extensionality, i.e., iff and equality are the *)
(* same on `Prop` *)
(* *)
(* `constructive_indefinite_description` *)
(* ~ existential in `Prop` (`ex`) implies existential in `Type` *)
(* (`sig`) *)
(* *)
(* `cid` *)
(* ~ `constructive_indefinite_description` (shortcut) *)
(* *)
(* A number of properties are derived below from these axioms and are *)
(* often more pratical to use than directly using the axioms. For instance *)
(* propext, funext, the excluded middle (EM),... *)
(* *)
(* ## Boolean View of Prop *)
(* ``` *)
(* `[< P >] == boolean view of P : Prop, see all lemmas about asbool *)
(* ``` *)
(* `` `[< P >] `` *)
(* ~ boolean view of `P : Prop`, see all lemmas about `asbool` *)
(* *)
(* ## Mathematical Components Structures *)
(* *)
Expand All @@ -50,26 +53,32 @@ From mathcomp Require internal_Eqdep_dec.
(* and choiceType structures to let the user choose which definition of *)
(* equality to use and to avoid conflict with already declared instances. *)
(* *)
(* ``` *)
(* {classic T} == Endow T : Type with a canonical eqType/choiceType. *)
(* This is intended for local use. *)
(* E.g., T : Type |- A : {fset {classic T}} *)
(* Alternatively one may use elim/Pchoice: T => T in H *. *)
(* to substitute T with T : choiceType once and for all. *)
(* {eclassic T} == Endow T : eqType with a canonical choiceType. *)
(* On the model of {classic _}. *)
(* See also the lemmas Peq and eqPchoice. *)
(* ``` *)
(* `{classic T}` *)
(* ~ Endow `T : Type` with a canonical `eqType`/`choiceType`. *)
(* ~ This is intended for local use. *)
(* ~ E.g., `T : Type |- A : {fset {classic T}}` *)
(* ~ Alternatively one may use `elim/Pchoice: T => T in H *.` to *)
(* substitute `T` with `T : choiceType` once and for all. *)
(* *)
(* `{eclassic T}` *)
(* ~ Endow `T : eqType` with a canonical `choiceType`. *)
(* ~ On the model of `{classic _}`. *)
(* ~ See also the lemmas `Peq` and `eqPchoice`. *)
(* *)
(* Functions onto a porderType (resp. latticeType) are equipped with *)
(* a porderType (resp. latticeType), so that the generic notation (_ <= _)%O *)
(* (resp. `&`, `|`) from `order.v` can be used. *)
(* ``` *)
(* FunOrder.lef f g == pointwise large inequality between two functions *)
(* FunOrder.ltf f g == pointwise strict inequality between two functions *)
(* FunLattice.meetf f g == pointwise meet between two functions *)
(* FunLattice.joinf f g == pointwise join between two functions *)
(* ``` *)
(* `FunOrder.lef f g` *)
(* ~ pointwise large inequality between two functions *)
(* *)
(* `FunOrder.ltf f g` *)
(* ~ pointwise strict inequality between two functions *)
(* *)
(* `FunLattice.meetf f g` *)
(* ~ pointwise meet between two functions *)
(* *)
(* `FunLattice.joinf f g` *)
(* ~ pointwise join between two functions *)
(* *)
(******************************************************************************)

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55 changes: 36 additions & 19 deletions classical/cardinality.v
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Expand Up @@ -16,25 +16,42 @@ From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
(* only relations A #<= B and A #= B to compare the cardinals of two sets *)
(* (on two possibly different types). *)
(* *)
(* ``` *)
(* A #<= B == the cardinal of A is smaller or equal to the one of B *)
(* A #>= B := B #<= A *)
(* A #= B == the cardinal of A is equal to the cardinal of B *)
(* A #!= B := ~~ (A #= B) *)
(* finite_set A == the set A is finite *)
(* := exists n, A #= `I_n *)
(* <-> exists X : {fset T}, A = [set` X] *)
(* <-> ~ ([set: nat] #<= A) *)
(* infinite_set A := ~ finite_set A *)
(* cofinite_set A := finite_set (~` A) *)
(* countable A <-> A is countable *)
(* := A #<= [set: nat] *)
(* fset_set A == the finite set corresponding if A : set T is finite, *)
(* set0 otherwise (T : choiceType) *)
(* A.`1 := [fset x.1 | x in A] *)
(* A.`2 := [fset x.2 | x in A] *)
(* {fimfun aT >-> T} == type of functions with a finite image *)
(* ``` *)
(* `A #<= B` *)
(* ~ the cardinal of `A` is smaller or equal to the one of `B` *)
(* *)
(* `A #>= B` *)
(* ~ `B #<= A` *)
(* *)
(* `A #= B` *)
(* ~ the cardinal of `A` is equal to the cardinal of `B` *)
(* *)
(* `A #!= B` *)
(* ~ `~~ (A #= B)` *)
(* *)
(* `finite_set A` *)
(* ~ the set `A` is finite *)
(* ~ ``:= exists n, A #= `I_n`` *)
(* ~ ``<-> exists X : {fset T}, A = [set` X]`` *)
(* ~ `<-> ~ ([set: nat] #<= A)` *)
(* *)
(* `infinite_set A` *)
(* ~ `~ finite_set A` *)
(* *)
(* `cofinite_set A` *)
(* ~ ``finite_set (~` A)`` *)
(* *)
(* `countable A` *)
(* ~ `A` is countable *)
(* ~ `:= A #<= [set: nat]` *)
(* *)
(* ``A.`1`` *)
(* ~ `[fset x.1 | x in A]` *)
(* *)
(* ``A.`2`` *)
(* ~ `[fset x.2 | x in A]` *)
(* *)
(* `{fimfun aT >-> T}` *)
(* ~ type of functions with a finite image *)
(* *)
(******************************************************************************)

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21 changes: 14 additions & 7 deletions classical/classical_orders.v
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Expand Up @@ -8,13 +8,20 @@ From mathcomp Require Import functions set_interval.
(* *)
(* This file provides some additional order theory that requires stronger *)
(* axioms than mathcomp's own orders expect. *)
(* ``` *)
(* same_prefix n == two elements in a product space agree up n *)
(* first_diff x y == the first occurrence where x n != y n, or None *)
(* big_lexi_le == the (countably) infinite lexicographical ordering *)
(* big_lexi_order == an alias for attaching this order type *)
(* start_with n x y == x for the first n values, then switches to y *)
(* ``` *)
(* `same_prefix n` *)
(* ~ two elements in a product space agree up `n` *)
(* *)
(* `first_diff x y` *)
(* ~ the first occurrence where `x n != y n`, or `None` *)
(* *)
(* `big_lexi_le` *)
(* ~ the (countably) infinite lexicographical ordering *)
(* *)
(* `big_lexi_order` *)
(* ~ an alias for attaching this order type *)
(* *)
(* `start_with n x y` *)
(* ~ `x` for the first `n` values, then switches to `y` *)
(* *)
(******************************************************************************)

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