From 5e21f60e07be116a33cb0d160b0e6940c481993d Mon Sep 17 00:00:00 2001
From: Yann Herklotz <git@yannherklotz.com>
Date: Thu, 28 Jul 2022 16:29:07 +0100
Subject: Remove NonEmpty.v

---
 NonEmpty.v | 96 --------------------------------------------------------------
 main.org   | 26 +++++++++++++++++
 2 files changed, 26 insertions(+), 96 deletions(-)
 delete mode 100644 NonEmpty.v

diff --git a/NonEmpty.v b/NonEmpty.v
deleted file mode 100644
index 2bc5234..0000000
--- a/NonEmpty.v
+++ /dev/null
@@ -1,96 +0,0 @@
-(*
- * Vericert: Verified high-level synthesis.
- * Copyright (C) 2021-2022 Yann Herklotz <yann@yannherklotz.com>
- *
- * This program is free software: you can redistribute it and/or modify
- * it under the terms of the GNU General Public License as published by
- * the Free Software Foundation, either version 3 of the License, or
- * (at your option) any later version.
- *
- * This program is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
- * GNU General Public License for more details.
- *
- * You should have received a copy of the GNU General Public License
- * along with this program.  If not, see <https://www.gnu.org/licenses/>.
- *)
-
-Inductive non_empty (A: Type) :=
-| singleton : A -> non_empty A
-| cons : A -> non_empty A -> non_empty A
-.
-
-Arguments singleton [A].
-Arguments cons [A].
-
-Declare Scope non_empty_scope.
-Delimit Scope non_empty_scope with non_empty.
-
-Module NonEmptyNotation.
-  Infix "::|" := cons (at level 60, right associativity) : non_empty_scope.
-End NonEmptyNotation.
-Import NonEmptyNotation.
-
-#[local] Open Scope non_empty_scope.
-
-Fixpoint map {A B} (f: A -> B) (l: non_empty A): non_empty B :=
-  match l with
-  | singleton a => singleton (f a)
-  | a ::| b => f a ::| map f b
-  end.
-
-Fixpoint to_list {A} (l: non_empty A): list A :=
-  match l with
-  | singleton a => a::nil
-  | a ::| b => a :: to_list b
-  end.
-
-Fixpoint app {A} (l1 l2: non_empty A) :=
-  match l1 with
-  | singleton a => a ::| l2
-  | a ::| b => a ::| app b l2
-  end.
-
-Fixpoint non_empty_prod {A B} (l: non_empty A) (l': non_empty B) :=
-  match l with
-  | singleton a => map (fun x => (a, x)) l'
-  | a ::| b => app (map (fun x => (a, x)) l') (non_empty_prod b l')
-  end.
-
-Fixpoint of_list {A} (l: list A): option (non_empty A) :=
-  match l with
-  | a::b =>
-      match of_list b with
-      | Some b' => Some (a ::| b')
-      | _ => None
-      end
-  | nil => None
-  end.
-
-Fixpoint replace {A} (f: A -> A -> bool) (a b: A) (l: non_empty A) :=
-  match l with
-  | a' ::| l' => if f a a' then b ::| replace f a b l' else a' ::| replace f a b l'
-  | singleton a' => if f a a' then singleton b else singleton a'
-  end.
-
-Inductive In {A: Type} (x: A) : non_empty A -> Prop :=
-| In_cons : forall a b, x = a \/ In x b -> In x (a ::| b)
-| In_single : In x (singleton x).
-
-Lemma in_dec:
-  forall A (a: A) (l: non_empty A),
-    (forall a b: A, {a = b} + {a <> b}) ->
-    {In a l} + {~ In a l}.
-Proof.
-  induction l; intros.
-  { specialize (X a a0). inv X.
-    left. constructor.
-    right. unfold not. intros. apply H. inv H0. auto. }
-  { pose proof X as X2.
-    specialize (X a a0). inv X.
-    left. constructor; tauto.
-    apply IHl in X2. inv X2.
-    left. constructor; tauto.
-    right. unfold not in *; intros. apply H0. inv H1. now inv H3. }
-Qed.
diff --git a/main.org b/main.org
index e43e4b5..fe44cc6 100644
--- a/main.org
+++ b/main.org
@@ -81,6 +81,18 @@ governed by how the expressions are generated, and to what resource they are ass
 final expression tree.  Speaking of the tree, we can also define the forest that contains a mapping
 from resource to corresponding expression.
 
+#+begin_src coq
+Definition predicate := positive.
+Definition pred_op := @Predicate.pred_op predicate.
+Definition predicated A := NE.non_empty (pred_op * A).
+Definition pred_expr := predicated expression.
+
+Definition apred : Type := expression.
+Definition apred_op := @Predicate.pred_op apred.
+Definition apredicated A := NE.non_empty (apred_op * A).
+Definition apred_expr := apredicated expression.
+#+end_src
+
 #+begin_src coq
 <<R_indexed-def>>
 Module Rtree := ITree(R_indexed).
@@ -90,6 +102,13 @@ Definition forest : Type := Rtree.t apred_expr.
 where ~R_indexed~ is a proof of injectivity from resources into the positives, and is further
 defined in the [[R_indexed-def-link][next section]].
 
+Finally, lets assume that we have a correct syntactic equality check between two expressions, so
+that we can get a decidable equality check between expressions.
+
+#+begin_src coq
+Axiom expression_dec: forall e1 e2: expression, {e1 = e2} + {e1 <> e2}.
+#+end_src
+
 ** Definition of R_indexed
 :PROPERTIES:
 :CUSTOM_ID: R_indexed-def-link
@@ -127,6 +146,7 @@ Module R_indexed.
   <<R_indexed-injectivity>>
 End R_indexed.
 #+end_src
+* First Attempt at Symbolic Analysis
 
 * Top-level imports
 :PROPERTIES:
@@ -158,8 +178,14 @@ Require Import vericert.hls.HashTree.
 Require Import vericert.hls.Predicate.
 Require Import vericert.common.DecEq.
 
+Require symb.NonEmpty.
+
+Module NE := NonEmpty.
+Import NE.NonEmptyNotation.
+
 #[local] Open Scope positive.
 #[local] Open Scope pred_op.
+#[local] Open Scope non_empty_scope.
 #+end_src
 
 * Footnotes
-- 
cgit