✨ Param_bool

_CoqProject

@@ -15,6 +15,7 @@ theories/Database.v
 theories/Param.v
 theories/Trocq.v
 theories/Vernac.v
+theories/Param_bool.v
 theories/Param_nat.v
 theories/Param_paths.v
 theories/Param_sigma.v

theories/Param_bool.v

@@ -0,0 +1,92 @@
+(*****************************************************************************)
+(*                            *                    Trocq                     *)
+(*  _______                   *           Copyright (C) 2023 MERCE           *)
+(* |__   __|                  *    (Mitsubishi Electric R&D Centre Europe)   *)
+(*    | |_ __ ___   ___ __ _  *       Cyril Cohen <[email protected]>     *)
+(*    | | '__/ _ \ / __/ _` | *       Enzo Crance <[email protected]>     *)
+(*    | | | | (_) | (_| (_| | *   Assia Mahboubi <[email protected]>   *)
+(*    |_|_|  \___/ \___\__, | ************************************************)
+(*                        | | * This file is distributed under the terms of  *)
+(*                        |_| * GNU Lesser General Public License Version 3  *)
+(*                            * see LICENSE file for the text of the license *)
+(*****************************************************************************)
+
+From Coq Require Import ssreflect.
+From HoTT Require Import HoTT.
+Require Import HoTT_additions Hierarchy.
+
+Set Universe Polymorphism.
+Unset Universe Minimization ToSet.
+
+Notation bool := Datatypes.bool.
+Notation false := Datatypes.false.
+Notation true := Datatypes.true.
+
+Inductive boolR : bool -> bool -> Type :=
+  | falseR : boolR false false
+  | trueR : boolR true true.
+
+Definition map_bool : bool -> bool := id.
+
+Definition map_in_R_bool {b b' : bool} (e : map_bool b = b') : boolR b b' :=
+  match e with
+  | idpath =>
+    match b with
+    | false => falseR
+    | true => trueR
+    end
+  end.
+
+Definition R_in_map_bool {b b' : bool} (bR : boolR b b') : map_bool b = b' :=
+  match bR with
+  | falseR => idpath
+  | trueR => idpath
+  end.
+
+Definition R_in_mapK_bool {b b' : bool} (bR : boolR b b') :
+  map_in_R_bool (R_in_map_bool bR) = bR :=
+    match bR with
+    | falseR => idpath
+    | trueR => idpath
+    end.
+
+Definition Param_bool_sym {b b' : bool} (bR : boolR b b') : boolR b' b :=
+  match bR with
+  | falseR => falseR
+  | trueR => trueR
+  end.
+
+Definition Param_bool_sym_inv {b b' : bool} (bR : boolR b b') :
+  Param_bool_sym (Param_bool_sym bR) = bR :=
+    match bR with
+    | falseR => idpath
+    | trueR => idpath
+    end.
+
+Definition boolR_sym : forall (b b' : bool), sym_rel boolR b b' <~> boolR b b'.
+Proof.
+  intros b b'.
+  unshelve eapply equiv_adjointify.
+  - apply Param_bool_sym.
+  - apply Param_bool_sym.
+  - intro bR. apply Param_bool_sym_inv.
+  - intro bR. apply Param_bool_sym_inv.
+Defined.
+
+Definition Map4_bool : Map4.Has boolR.
+Proof.
+  unshelve econstructor.
+  - exact map_bool.
+  - exact @map_in_R_bool.
+  - exact @R_in_map_bool.
+  - exact @R_in_mapK_bool.
+Defined.
+
+Definition Param44_bool : Param44.Rel bool bool.
+Proof.
+  unshelve econstructor.
+  - exact boolR.
+  - exact Map4_bool.
+  - apply (fun e => @eq_Map4 _ _ (sym_rel boolR) boolR e Map4_bool).
+    apply boolR_sym.
+Defined.