(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
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(* This file contains the definitions of:                                     *)
(*   choiceType == interface for types with a choice operator.                *)
(*    countType == interface for countable types (implies choiceType).        *)
(* subCountType == interface for types that are both subType and countType.   *)
(*  xchoose exP == a standard x such that P x, given exP : exists x : T, P x  *)
(*                 when T is a choiceType. The choice depends only on the     *)
(*                 extent of P (in particular, it is independent of exP).     *)
(*   choose P x0 == if P x0, a standard x such that P x.                      *)
(*      pickle x == a nat encoding the value x : T, where T is a countType.   *)
(*    unpickle n == a partial inverse to pickle: unpickle (pickle x) = Some x *)
(*  pickle_inv n == a sharp partial inverse to pickle pickle_inv n = Some x   *)
(*                  if and only if pickle x = n.                              *)
(*            choiceMixin T == type of choice mixins; the exact contents is   *)
(*                        documented below in the Choice submodule.           *)
(*           ChoiceType T m == the packed choiceType class for T and mixin m. *)
(* [choiceType of T for cT] == clone for T of the choiceType cT.              *)
(*        [choiceType of T] == clone for T of the choiceType inferred for T.  *)
(*            CountType T m == the packed countType class for T and mixin m.  *)
(*  [countType of T for cT] == clone for T of the countType cT.               *)
(*        [count Type of T] == clone for T of the countType inferred for T.   *)
(* [choiceMixin of T by <:] == Choice mixin for T when T has a subType p      *)
(*                        structure with p : pred cT and cT has a Choice      *)
(*                        structure; the corresponding structure is Canonical.*)
(*  [countMixin of T by <:] == Count mixin for a subType T of a countType.    *)
(*  PcanChoiceMixin fK == Choice mixin for T, given f : T -> cT where cT has  *)
(*                        a Choice structure, a left inverse partial function *)
(*                        g and fK : pcancel f g.                             *)
(*   CanChoiceMixin fK == Choice mixin for T, given f : T -> cT, g and        *)
(*                        fK : cancel f g.                                    *)
(*   PcanCountMixin fK == Count mixin for T, given f : T -> cT where cT has   *)
(*                        a Countable structure, a left inverse partial       *)
(*                        function g and fK : pcancel f g.                    *)
(*    CanCountMixin fK == Count mixin for T, given f : T -> cT, g and         *)
(*                        fK : cancel f g.                                    *)
(*      GenTree.tree T == generic n-ary tree type with nat-labeled nodes and  *)
(*                        T-labeled leaves, for example GenTree.Leaf (x : T), *)
(*                        GenTree.Node 5 [:: t; t']. GenTree.tree is equipped *)
(*                        with canonical eqType, choiceType, and countType    *)
(*                        instances, and so simple datatypes can be similarly *)
(*                        equipped by encoding into GenTree.tree and using    *)
(*                        the mixins above.                                   *)
(*        CodeSeq.code == bijection from seq nat to nat.                      *)
(*      CodeSeq.decode == bijection inverse to CodeSeq.code.                  *)
(* In addition to the lemmas relevant to these definitions, this file also    *)
(* contains definitions of a Canonical choiceType and countType instances for *)
(* all basic datatypes (e.g., nat, bool, subTypes, pairs, sums, etc.).        *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(* Technical definitions about coding and decoding of nat sequences, which    *)
(* are used below to define various Canonical instances of the choice and     *)
(* countable interfaces.                                                      *)
Module CodeSeq.
(* Goedel-style one-to-one encoding of seq nat into nat.                      *)
(* The code for [:: n1; ...; nk] has binary representation                    *)
(*          1 0 ... 0 1 ... 1 0 ... 0 1 0 ... 0                               *)
(*            <----->         <----->   <----->                               *)
(*             nk 0s           n2 0s     n1 0s                                *)
Definition code := foldr (fun n m => 2 ^ n * m.*2.+1) 0.
Fixpoint decode_rec (v q r : nat) {struct q} :=
match q, r with
| 0, _         => [:: v]
| q'.+1, 0     => v :: [rec 0, q', q']
| q'.+1, 1     => [rec v.+1, q', q']
| q'.+1, r'.+2 => [rec v, q', r']
end where "[ 'rec' v , q , r ]" := (decode_rec v q r).
Definition decode n := if n is 0 then [::] else [rec 0, n.-1, n.-1].

Lemma ltn_code s : all (fun j => j < code s) s.