Adjust statement of relate_fold_add lemma,

used by the prune_terms tactic, to match the statement of
relate_fold_add that appears in my recent pull request to coq-mmaps.
coq-community/mmaps#14

That way, when coq-mmaps is (soon) made part of the coq platform,
we can easily switch vcfloat from using FMaps (the old, obsolete version
of functional maps) to the new MMaps (which is better).

doc/VCFloat-Manual.tex

@@ -286,7 +286,7 @@ \chapter{Reification}
 The first argument of \lstinline{HO_reify_float_expr}
 should be list of identifiers, to associate with those
 parameters of the functional model.  In this case the list is
-simpliy \lstinline{[_x;_v]}.
+simply \lstinline{[_x;_v]}.
 
 \chapter{Boundsmap}
 

vcfloat/FMap_lemmas.v

@@ -74,7 +74,7 @@ split; intro H0; inversion H0; clear H0; subst.
 - apply Table.Raw.InRight; rewrite IHt2; auto.
 Qed.
 
-Lemma relate_fold_add:
+Lemma relate_fold_add':
  forall [elt A: Type]
      [eqv: A -> A -> Prop] 
      (eqv_rel: Equivalence eqv)
@@ -199,6 +199,53 @@ rewrite u_unit.
 reflexivity.
 Qed.
 
+
+Lemma relate_fold_add:
+ forall [elt A: Type]
+     [eqv: A -> A -> Prop] 
+     (eqv_rel: Equivalence eqv)
+     (lift: Table.key -> elt -> A)
+     (lift_prop: forall k k' x, Keys.eq k k' -> eqv (lift k x) (lift k' x))
+    (f:  A -> A -> A)
+    (f_mor: forall x1 y1, eqv x1 y1 ->
+              forall x2 y2, eqv x2 y2 ->
+              eqv (f x1 x2) (f y1 y2))
+    (f_assoc: forall x y z : A, eqv (f x (f y z)) (f (f x y) z))
+    (f_commut: forall x y : A, eqv (f x y) (f y x))
+    (u: A)
+    (u_unit: forall x, eqv (f u x) x)
+    (g: Table.key -> elt -> A -> A)
+    (g_eqv: forall k x a, eqv (g k x a) (f (lift k x) a))
+    (tab: Table.t elt)
+    (k: Table.key),
+    eqv (Table.fold g tab u)
+      (f (match Table.find k tab with Some x => lift k x | None => u end)
+       (Table.fold (fun k' x a => if Keys.eq_dec k k' then a else g k' x a) tab u)).
+Proof.
+intros.
+rewrite (relate_fold_add' eqv_rel lift lift_prop f f_mor f_assoc f_commut u u_unit g g_eqv tab k).
+apply f_mor.
+reflexivity.
+rewrite !Table.fold_1.
+clear u_unit.
+revert u.
+induction (Table.elements (elt:=elt) tab); intro.
+simpl. reflexivity.
+simpl.
+rewrite IHl.
+set (ff := fold_left _).
+clearbody ff.
+match goal with |- eqv ?A ?B => replace B with A end.
+reflexivity.
+f_equal.
+set (j := fst a). clearbody j.
+clear.
+destruct (Keys.compare k j); try apply Keys.lt_not_eq in l;
+destruct (Keys.eq_dec k j); try contradiction; auto.
+symmetry in e. contradiction.
+Qed.
+
+
 Lemma fold_add_ignore:
   forall [elt A]
    (f: Table.key -> elt -> A -> A)
@@ -224,6 +271,51 @@ rewrite ?H; auto.
 rewrite IHis_bst2. auto.
 Qed.
 
+
+
+Lemma relate_fold_add_alt:
+ forall [elt A: Type]
+     [eqv: A -> A -> Prop] 
+     (eqv_rel: Equivalence eqv)
+     (lift: Table.key -> elt -> A)
+     (lift_prop: forall k k' x, Keys.eq k k' -> eqv (lift k x) (lift k' x))
+    (f:  A -> A -> A)
+    (f_mor: forall x1 y1, eqv x1 y1 ->
+              forall x2 y2, eqv x2 y2 ->
+              eqv (f x1 x2) (f y1 y2))
+    (f_assoc: forall x y z : A, eqv (f x (f y z)) (f (f x y) z))
+    (f_commut: forall x y : A, eqv (f x y) (f y x))
+    (u: A)
+    (u_unit: forall x, eqv (f u x) x)
+    (g: Table.key -> elt -> A -> A)
+    (g_eqv: forall k x a, eqv (g k x a) (f (lift k x) a))
+    (tab: Table.t elt)
+    (k: Table.key) (new oldnew : elt),
+    eqv (f (lift k new) (match Table.find k tab with Some x => lift k x | None => u end)) (lift k oldnew) ->
+    eqv (f (lift k new) (Table.fold g tab u)) (Table.fold g (Table.add k oldnew tab) u).
+Proof.
+intros.
+pose proof relate_fold_add eqv_rel lift lift_prop f f_mor f_assoc f_commut u u_unit g g_eqv.
+etransitivity.
+apply f_mor.
+reflexivity.
+apply H0 with (k:=k).
+rewrite f_assoc.
+etransitivity.
+apply f_mor.
+apply H.
+reflexivity.
+rewrite H0 with (k:=k).
+apply f_mor.
+erewrite Table.find_1 by (apply Table.add_1; reflexivity).
+reflexivity.
+rewrite fold_add_ignore.
+reflexivity.
+intros.
+destruct (Keys.eq_dec k k'); try contradiction.
+auto.
+Qed.
+
 End FMapAVL_extra.
 
 Module Demonstration.
@@ -276,34 +368,39 @@ Lemma add_to_table_correct:
 Proof.
 intros.
 pose (lift (k: Table.key) p := Z.pos p).
-pose proof relate_fold_add Z.eq_equiv lift
+pose (oldnew := Z.to_pos (lift k p + match Table.find (elt:=positive) k tab with
+                | Some x => lift k x
+                | None => 0
+                end)).
+pose proof relate_fold_add_alt Z.eq_equiv lift
    ltac:(intros; rewrite H; auto)
    Z.add
    ltac:(intros; subst; auto)
    Z.add_assoc Z.add_comm
    Z0
    Z.add_0_l
-   (fun k p x => Z.add (Z.pos p) x)
-   ltac:(intros; subst; reflexivity).
-unfold addup_table.
-rewrite (H (add_to_table k p tab) k).
-rewrite (H tab k).
-clear H.
-unfold add_to_table.
-destruct (Table.find k tab) eqn:?H;
-rewrite (Table.find_1 (Table.add_1 tab _ (eq_refl k)));
-rewrite fold_add_ignore 
- by (intros; rewrite <- H0;
-     destruct (Keys.compare k k); auto; contradiction (Z.lt_irrefl k l));
+   (fun k p x => Z.add (lift k p) x)
+   ltac:(intros; subst; reflexivity)
+   tab k p oldnew.
+unfold addup_table, add_to_table. 
+set (f := fun (k : Table.key) (p : positive) (x : Z) => (lift k p + x)%Z) in *.
+change (fun (_ : Table.key) (p1 : positive) (i : Z) => (Z.pos p1 + i)%Z) with f in *.
+rewrite Z.add_comm.
+change (Z.pos p) with (lift k p).
+rewrite H; clear H.
+f_equal.
+destruct (Table.find k tab) eqn:?H; auto.
+unfold oldnew.
+set (b := match Table.find (elt:=positive) k tab with
+                        | Some x => lift k x
+                        | None => 0%Z
+                        end).
+assert (0 <= b)%Z.
+subst b.
 unfold lift.
-rewrite Pos.add_comm.
-rewrite Pos2Z.inj_add.
-rewrite <- !Z.add_assoc.
-rewrite (Z.add_comm (Z.pos p)).
-auto.
-set (u := Table.fold _ _ _).
-rewrite Z.add_0_l.
-apply Z.add_comm.
+destruct (Table.find (elt:=positive) k tab); simpl; Lia.lia.
+unfold lift.
+Lia.lia.
 Qed.
 
 End Demonstration.

vcfloat/Prune.v

@@ -2694,41 +2694,51 @@ unfold add_to_table.
 destruct nt as [k it].
 set (j := cancel1_intable (find_default [] k tab) it).
 pose (lift k x := reflect_intable_simple k x zeroexpr).
-pose proof relate_fold_add expr_equiv_rel lift
+set (f := Ebinary Add).
+set (u := zeroexpr).
+(*
+pose (oldnew := 
+   Ebinary Add (lift k j) match Table.find (elt:=intable_t) k tab with
+      | Some x => lift k x
+      | None => zeroexpr
+      end).
+*)
+pose proof relate_fold_add_alt expr_equiv_rel lift
       ltac:(intros; apply reflect_intable_simple_mor; auto; reflexivity)
      (Ebinary Add) 
     (Ebinary_mor Add Add (eq_refl _))
     ltac:(intros; intro; simpl; ring)
     ltac:(intros; intro; simpl; ring)
-    zeroexpr
+    u
     ltac:(intros; intro; simpl; ring)
     reflect_intable_simple
-     reflect_intable_simple_untangle.
-etransitivity.
-apply (H (Table.add k j tab) k).
-change (lift k []) with zeroexpr.
-pose proof  (H tab k).
-change (lift k []) with zeroexpr in H0.
-rewrite H0; clear H H0.
-rewrite fold_add_ignore.
-2:{ intros. rewrite cmp_compare in H.
-   destruct (Keys.compare k k'); auto; discriminate.
-}
-set (u := Table.fold _ _ _); clearbody u. clear.
-rewrite (Table.find_1 (Table.add_1 tab j (Keys.eq_refl k))).
-subst j.
+     reflect_intable_simple_untangle 
+   tab k [it] j.
+rewrite <- H; clear H.
+-
 unfold lift at 1.
+unfold f.
+intro; simpl.
+rewrite Rplus_comm.
+f_equal.
+symmetry.
+rewrite reflect_powers_untangle.
+simpl.
+rewrite Rplus_0_r.
+f_equal.
+apply reflect_coeff_spec.
+-
+unfold lift.
+subst j.
 rewrite cancel1_intable_correct.
-unfold reflect_intable_simple at 1.
-simpl fold_right.
+rewrite !reflect_intable_cons.
+intro.
+simpl.
+f_equal.
+apply Rplus_0_r.
 unfold find_default.
 fold intable_t.
-rewrite reflect_normterm_spec.
-destruct (Table.find k tab); subst lift; cbv beta.
-fold (reflect_intable_simple k i zeroexpr).
-set (v := reflect_intable_simple _ _ _). clearbody v.
-intro; simpl; ring.
-intro; simpl; ring.
+destruct (Table.find (elt:=intable_t) k tab); simpl; auto.
 Qed.
 
 Definition reflect_intable_aux (al: intable_t) : expr :=