From 1103b910d1a99168fd59f4eac65477a2550f96a6 Mon Sep 17 00:00:00 2001
From: Pierre Rousselin <rousselin@math.univ-paris13.fr>
Date: Thu, 30 Nov 2023 18:47:36 +0100
Subject: [PATCH] Adapt to Coq/Coq#18164

---
 Bedrock/Word.v                                |   1 -
 src/ADTRefinement/FixedPoint.v                |   5 +-
 .../Extraction/External/Lists.v               |   4 +-
 .../QueryStructures/CallRules/TupleList.v     |   4 +-
 .../QueryStructures/QueryStructures.v         |   2 +-
 src/Common.v                                  |   7 +-
 src/Common/BoundedLookup.v                    |   4 +-
 src/Common/Enumerable.v                       |   4 +-
 src/Common/Equality.v                         |   7 +-
 src/Common/FixedPoints.v                      |   4 +-
 src/Common/Frame.v                            |  12 +-
 src/Common/Le.v                               |  11 +-
 src/Common/List/FlattenList.v                 |   2 +-
 src/Common/List/ListFacts.v                   |  34 ++---
 src/Common/List/Operations.v                  |   2 +-
 src/Common/NatFacts.v                         |  31 ++--
 src/Common/StringFacts.v                      |  35 ++---
 src/Common/Wf.v                               |  13 +-
 src/Computation/Decidable.v                   |   4 +-
 src/Computation/ListComputations.v            |  10 +-
 src/Narcissus/Automation/Decision.v           |   8 +-
 .../AbstractInterpretation/NonTerminalMap.v   |   2 +-
 src/Parsers/BaseTypesLemmas.v                 |   6 +-
 src/Parsers/BooleanRecognizerOptimized.v      |  57 ++++----
 src/Parsers/BooleanRecognizerTests.v          |   2 +-
 src/Parsers/ContextFreeGrammar/Carriers.v     |   2 +-
 .../ContextFreeGrammar/Fix/Definitions.v      |   2 +-
 src/Parsers/ContextFreeGrammar/Fix/Fix.v      |   6 +-
 src/Parsers/ContextFreeGrammar/Fold.v         |   2 +-
 src/Parsers/ContextFreeGrammar/Properties.v   |   2 +-
 .../ContextFreeGrammar/ValidReflective.v      |   3 +-
 src/Parsers/ExtrOcamlParsers.v                |   2 +-
 src/Parsers/GenericCorrectnessBaseTypes.v     |   2 +-
 src/Parsers/GenericRecognizer.v               |  17 +--
 src/Parsers/GenericRecognizerMin.v            | 130 ++++++++---------
 src/Parsers/GenericRecognizerOptimized.v      |  19 ++-
 .../GenericRecognizerOptimizedTactics.v       |  16 +-
 src/Parsers/MinimalParseOfParse.v             |  66 ++++-----
 src/Parsers/ParserImplementation.v            |   4 +-
 .../All/MinimalReachableOfReachable.v         |  36 ++---
 .../Reachable/MaybeEmpty/MinimalOfCore.v      |  32 ++--
 src/Parsers/Reachable/MaybeEmpty/OfParse.v    |   4 +-
 .../OnlyFirst/MinimalReachableOfReachable.v   |  36 ++---
 .../OnlyLast/MinimalReachableOfReachable.v    |  36 ++---
 src/Parsers/Reachable/ParenBalanced/Core.v    |   4 +-
 .../Reachable/ParenBalanced/MinimalOfCore.v   |  24 +--
 src/Parsers/Reachable/ParenBalanced/OfParse.v |   2 +-
 .../ParenBalancedHiding/MinimalOfCore.v       |  28 ++--
 src/Parsers/RecognizerPreOptimized.v          |  31 ++--
 .../Refinement/BinOpBrackets/BinOpRules.v     |  42 +++---
 .../Refinement/BinOpBrackets/MakeBinOpTable.v |  16 +-
 .../Refinement/BinOpBrackets/ParenBalanced.v  |  12 +-
 .../BinOpBrackets/ParenBalancedGrammar.v      |   4 +-
 .../BinOpBrackets/ParenBalancedLemmas.v       |   4 +-
 src/Parsers/Refinement/DisjointLemmas.v       |   5 +-
 src/Parsers/Refinement/DisjointRules.v        |  20 +--
 src/Parsers/Refinement/DisjointRulesRev.v     |   4 +-
 src/Parsers/Refinement/EmptyLemmas.v          |   6 +-
 src/Parsers/Refinement/FixedLengthLemmas.v    |  18 +--
 ...IndexedAndAtMostOneNonTerminalReflective.v | 137 +++++++++---------
 ...exedAndAtMostOneNonTerminalReflectiveOpt.v |   4 +-
 src/Parsers/Refinement/PreTactics.v           |   9 +-
 src/Parsers/Reflective/LogicalRelations.v     |   2 +-
 src/Parsers/Reflective/ParserSemantics.v      |   7 +-
 src/Parsers/Reflective/PartialUnfold.v        |   2 +-
 src/Parsers/Reflective/Reify.v                |   2 +-
 src/Parsers/Reflective/Semantics.v            |   2 +-
 .../Reflective/SyntaxEquivalenceReflective.v  |   2 +-
 src/Parsers/SimpleRecognizerCorrect.v         |   6 +-
 src/Parsers/SplitterFromParserADT.v           |   6 +-
 src/Parsers/Splitters/BruteForce.v            |   6 +-
 src/Parsers/Splitters/RDPList.v               |  48 +++---
 src/Parsers/StringLike/FirstChar.v            |   7 +-
 src/Parsers/StringLike/FirstCharSuchThat.v    |   9 +-
 src/Parsers/StringLike/ForallChars.v          |   5 +-
 src/Parsers/StringLike/LastChar.v             |   5 +-
 src/Parsers/StringLike/LastCharSuchThat.v     |  17 +--
 src/Parsers/StringLike/OcamlString.v          |  29 ++--
 src/Parsers/StringLike/Properties.v           |  77 +++++-----
 src/Parsers/StringLike/String.v               |  22 +--
 .../Constraints/ConstraintChecksRefinements.v |  14 +-
 .../DataStructures/Bags/BagsProperties.v      |   2 +-
 .../DataStructures/Bags/CountingListBags.v    |   4 +-
 .../DataStructures/Bags/ListBags.v            |   2 +-
 .../DataStructures/Bags/RangeTreeBags.v       |   9 +-
 .../DataStructures/Bags/TreeBags.v            |   9 +-
 .../DataStructures/Bags/TrieBags.v            |  30 ++--
 .../Operations/General/InsertRefinements.v    |   1 -
 .../Representation/QueryStructure.v           |   2 +-
 89 files changed, 688 insertions(+), 697 deletions(-)

diff --git a/Bedrock/Word.v b/Bedrock/Word.v
index 8f5729642..0e8b8ab88 100644
--- a/Bedrock/Word.v
+++ b/Bedrock/Word.v
@@ -2,7 +2,6 @@ Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** Fixed precision machine words *)
 
 Require Import Coq.Arith.Arith
-        Coq.Arith.Div2
         Coq.NArith.NArith
         Coq.Bool.Bool
         Coq.ZArith.ZArith.
diff --git a/src/ADTRefinement/FixedPoint.v b/src/ADTRefinement/FixedPoint.v
index c32e56230..2c2138438 100644
--- a/src/ADTRefinement/FixedPoint.v
+++ b/src/ADTRefinement/FixedPoint.v
@@ -1,6 +1,8 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Fiat.ADT Fiat.ADTNotation.
 Require Export Fiat.Computation.FixComp.
+Require Import Coq.ZArith.ZArith.
+Require Import Coq.Arith.PeanoNat.
 
 Import LeastFixedPointFun.
 
@@ -113,9 +115,8 @@ Lemma length_wf' : forall A len l,
 Proof.
   induction len; simpl; intros;
     constructor; intros.
-  contradiction (NPeano.Nat.nlt_0_r _ H).
+  contradiction (Nat.nlt_0_r _ H).
   apply IHlen.
-  Require Import Coq.ZArith.ZArith.
   omega.
 Qed.
 
diff --git a/src/CertifiedExtraction/Extraction/External/Lists.v b/src/CertifiedExtraction/Extraction/External/Lists.v
index 06ac5a7ea..71f11a836 100644
--- a/src/CertifiedExtraction/Extraction/External/Lists.v
+++ b/src/CertifiedExtraction/Extraction/External/Lists.v
@@ -5,7 +5,7 @@ Require Export Bedrock.Platform.Facade.examples.QsADTs.
 Lemma CompileEmpty_helper {A}:
   forall (lst : list A) (w : W),
     Programming.propToWord (lst = @nil A) w ->
-    ret (bool2w (EqNat.beq_nat (Datatypes.length lst) 0)) ↝ w.
+    ret (bool2w (Nat.eqb (Datatypes.length lst) 0)) ↝ w.
 Proof.
   unfold Programming.propToWord, IF_then_else in *.
   destruct lst; simpl in *; destruct 1; repeat cleanup; try discriminate; fiat_t.
@@ -15,7 +15,7 @@ Hint Resolve CompileEmpty_helper : call_helpers_db.
 
 Lemma ListEmpty_helper :
   forall {A} (seq: list A),
-    (EqNat.beq_nat (Datatypes.length seq) 0) = match seq with
+    (Nat.eqb (Datatypes.length seq) 0) = match seq with
                                                | nil => true
                                                | _ :: _ => false
                                                end.
diff --git a/src/CertifiedExtraction/Extraction/QueryStructures/CallRules/TupleList.v b/src/CertifiedExtraction/Extraction/QueryStructures/CallRules/TupleList.v
index 5fbbe9308..003d2d156 100644
--- a/src/CertifiedExtraction/Extraction/QueryStructures/CallRules/TupleList.v
+++ b/src/CertifiedExtraction/Extraction/QueryStructures/CallRules/TupleList.v
@@ -123,7 +123,7 @@ Module ADTListCompilation
       GLabelMap.MapsTo fpointer (Axiomatic Empty) env ->
       {{ tenv }}
         Call vtest fpointer (vlst :: nil)
-      {{ [[`vtest ->> (bool2w (EqNat.beq_nat (Datatypes.length lst) 0)) as _]] :: (DropName vtest tenv) }} ∪ {{ ext }} // env.
+      {{ [[`vtest ->> (bool2w (Nat.eqb (Datatypes.length lst) 0)) as _]] :: (DropName vtest tenv) }} ∪ {{ ext }} // env.
   Proof.
     repeat (SameValues_Facade_t_step || facade_cleanup_call || LiftPropertyToTelescope_t || PreconditionSet_t || injections).
 
@@ -144,7 +144,7 @@ Module ADTListCompilation
       GLabelMap.MapsTo fpointer (Axiomatic Empty) env ->
       {{ tenv }}
         Call vtest fpointer (vlst :: nil)
-      {{ [[`vtest ->> (bool2w (EqNat.beq_nat (Datatypes.length (rev lst)) 0)) as _]] :: tenv }} ∪ {{ ext }} // env.
+      {{ [[`vtest ->> (bool2w (Nat.eqb (Datatypes.length (rev lst)) 0)) as _]] :: tenv }} ∪ {{ ext }} // env.
   Proof.
     intros.
     setoid_rewrite rev_length.
diff --git a/src/CertifiedExtraction/Extraction/QueryStructures/QueryStructures.v b/src/CertifiedExtraction/Extraction/QueryStructures/QueryStructures.v
index fa288da04..0d3a597c8 100644
--- a/src/CertifiedExtraction/Extraction/QueryStructures/QueryStructures.v
+++ b/src/CertifiedExtraction/Extraction/QueryStructures/QueryStructures.v
@@ -154,7 +154,7 @@ Ltac _compile_length :=
   match_ProgOk
     ltac:(fun prog pre post ext env =>
             match constr:((pre, post)) with
-            | (?pre, Cons ?k (ret (bool2w (EqNat.beq_nat (Datatypes.length (rev ?seq)) 0))) (fun _ => ?pre')) =>
+            | (?pre, Cons ?k (ret (bool2w (Nat.eqb (Datatypes.length (rev ?seq)) 0))) (fun _ => ?pre')) =>
               let vlst := find_fast (wrap (FacadeWrapper := WrapInstance (H := QS_WrapFiatWTupleList)) seq) ext in
               match vlst with
               | Some ?vlst => eapply (WTupleListCompilation.CompileEmpty_spec (idx := 3) (vlst := vlst))
diff --git a/src/Common.v b/src/Common.v
index 2d265ddeb..e3ea01bfa 100644
--- a/src/Common.v
+++ b/src/Common.v
@@ -1,5 +1,4 @@
 Require Import Coq.Lists.List.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Coq.ZArith.ZArith Coq.Lists.SetoidList.
 Require Export Coq.Setoids.Setoid Coq.Classes.RelationClasses
         Coq.Program.Program Coq.Classes.Morphisms.
@@ -1330,7 +1329,7 @@ Definition path_prod' {A B} {x x' : A} {y y' : B}
 
 Lemma lt_irrefl' {n m} (H : n = m) : ~n < m.
 Proof.
-  subst; apply Lt.lt_irrefl.
+  subst; apply Nat.lt_irrefl.
 Qed.
 
 Lemma or_not_l {A B} (H : A \/ B) (H' : ~A) : B.
@@ -1627,7 +1626,7 @@ Module opt.
   Definition snd {A B} := Eval compute in @snd A B.
   Definition andb := Eval compute in andb.
   Definition orb := Eval compute in orb.
-  Definition beq_nat := Eval compute in EqNat.beq_nat.
+  Definition beq_nat := Eval compute in Nat.eqb.
 
   Module Export Notations.
     Delimit Scope opt_bool_scope with opt_bool.
@@ -1641,7 +1640,7 @@ Module opt2.
   Definition snd {A B} := Eval compute in @snd A B.
   Definition andb := Eval compute in andb.
   Definition orb := Eval compute in orb.
-  Definition beq_nat := Eval compute in EqNat.beq_nat.
+  Definition beq_nat := Eval compute in Nat.eqb.
   Definition leb := Eval compute in Compare_dec.leb.
 
   Module Export Notations.
diff --git a/src/Common/BoundedLookup.v b/src/Common/BoundedLookup.v
index 29092d83f..cd6dd889a 100644
--- a/src/Common/BoundedLookup.v
+++ b/src/Common/BoundedLookup.v
@@ -141,7 +141,7 @@ Section BoundedIndex.
 
     Fixpoint fin_beq {m n} (p : Fin.t m) (q : Fin.t n) :=
       match p, q with
-      | @Fin.F1 m', @Fin.F1 n' => EqNat.beq_nat m' n'
+      | @Fin.F1 m', @Fin.F1 n' => Nat.eqb m' n'
       | Fin.FS _ _, Fin.F1 _ => false
       | Fin.F1 _, Fin.FS _ _ => false
       | Fin.FS _ p', Fin.FS _ q' => fin_beq p' q'
@@ -153,7 +153,7 @@ Section BoundedIndex.
     Proof.
       intros; pattern n, p, q; eapply Fin.rect2; simpl;
         intuition; try (congruence || discriminate).
-      - symmetry; eapply beq_nat_refl.
+      - eapply Nat.eqb_refl.
       - eauto using Fin.FS_inj.
     Qed.
 
diff --git a/src/Common/Enumerable.v b/src/Common/Enumerable.v
index 7dd3cee3c..df38dc10b 100644
--- a/src/Common/Enumerable.v
+++ b/src/Common/Enumerable.v
@@ -20,7 +20,7 @@ Section enum.
 
   Definition enumerate_fun_beq (enumerate : list A) (f g : A -> B)
   : bool
-    := EqNat.beq_nat (List.length (List.filter (fun x => negb (beq (f x) (g x))) enumerate)) 0.
+    := Nat.eqb (List.length (List.filter (fun x => negb (beq (f x) (g x))) enumerate)) 0.
 
   Definition enumerate_fun_bl_in
              (bl : forall x y, beq x y = true -> x = y)
@@ -29,7 +29,7 @@ Section enum.
   Proof.
     unfold enumerate_fun_beq.
     intros H x H'.
-    apply EqNat.beq_nat_true in H.
+    apply Nat.eqb_eq in H.
     apply bl.
     destruct (beq (f x) (g x)) eqn:Heq; [ reflexivity | exfalso ].
     apply Bool.negb_true_iff in Heq.
diff --git a/src/Common/Equality.v b/src/Common/Equality.v
index 450be0d26..ac9498281 100644
--- a/src/Common/Equality.v
+++ b/src/Common/Equality.v
@@ -1,5 +1,6 @@
 Require Import Coq.Lists.List Coq.Lists.SetoidList.
 Require Import Coq.Bool.Bool.
+Require Import Coq.Arith.PeanoNat.
 Require Import Coq.Strings.Ascii.
 Require Import Coq.Strings.String.
 Require Import Coq.Logic.Eqdep_dec.
@@ -379,10 +380,10 @@ Definition internal_prod_dec_lb {A B} eq_A eq_B A_lb B_lb x y
              H
              (surjective_pairing _))).
 
-Global Instance nat_BoolDecR : BoolDecR nat := EqNat.beq_nat.
-Global Instance nat_BoolDec_bl : BoolDec_bl (@eq nat) := @EqNat.beq_nat_true.
+Global Instance nat_BoolDecR : BoolDecR nat := Nat.eqb.
+Global Instance nat_BoolDec_bl : BoolDec_bl (@eq nat) := fun n m => (proj1 (Nat.eqb_eq n m)).
 Global Instance nat_BoolDec_lb : BoolDec_lb (@eq nat) :=
-  fun x y => proj2 (@EqNat.beq_nat_true_iff x y).
+  fun x y => proj2 (@Nat.eqb_eq x y).
 
 Lemma unit_bl {x y} : unit_beq x y = true -> x = y.
 Proof. apply internal_unit_dec_bl. Qed.
diff --git a/src/Common/FixedPoints.v b/src/Common/FixedPoints.v
index 3c6472f4f..2477f162b 100644
--- a/src/Common/FixedPoints.v
+++ b/src/Common/FixedPoints.v
@@ -62,7 +62,7 @@ Section fixedpoints.
              | _ => omega
              | [ H : forall a, _ -> _ = _ |- _ ] => rewrite H by omega
              | [ H : _ < _ |- _ ] => hnf in H
-             | [ H : S _ <= S _ |- _ ] => apply Le.le_S_n in H
+             | [ H : S _ <= S _ |- _ ] => apply Nat.succ_le_mono in H
              | [ H : sizeof ?x <= ?y, H' : ?y <= sizeof (step ?x) |- _ ]
                => let H'' := fresh in
                   assert (H'' : sizeof (step x) <= sizeof x) by apply step_monotonic;
@@ -95,7 +95,7 @@ Section listpair.
   Lemma step_monotonic lss : sizeof_pair (step lss) <= sizeof_pair lss.
   Proof.
     unfold step, sizeof_pair; simpl;
-      apply Plus.plus_le_compat;
+      apply Nat.add_le_mono;
       apply length_filter.
   Qed.
 
diff --git a/src/Common/Frame.v b/src/Common/Frame.v
index e2dafcb70..65681af97 100644
--- a/src/Common/Frame.v
+++ b/src/Common/Frame.v
@@ -1,7 +1,7 @@
 Require Import
         SetoidClass
-        Coq.Arith.Max
-        Coq.Classes.Morphisms.
+        Coq.Classes.Morphisms
+        Coq.Arith.PeanoNat.
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 
 Generalizable All Variables.
@@ -212,7 +212,7 @@ Module PreO.
   Qed.
 
   Definition Nat : t le.
-  Proof. constructor; [ apply Le.le_refl | apply Le.le_trans ].
+  Proof. constructor; [ apply Nat.le_refl | apply Nat.le_trans ].
   Qed.
 
   Definition discrete (A : Type) : t (@Logic.eq A).
@@ -408,7 +408,7 @@ Module PO.
     constructor; intros.
     - apply PreO.Nat.
     - solve_proper.
-    - apply Le.le_antisym; assumption.
+    - apply Nat.le_antisymm; assumption.
   Qed.
 
   Definition discrete (A : Type) : t (@Logic.eq A) Logic.eq.
@@ -585,8 +585,8 @@ Module JoinLat.
   Proof. constructor; intros.
          - apply PO.Nat.
          - solve_proper.
-         - constructor. simpl. apply Max.le_max_l. apply Max.le_max_r.
-           apply Max.max_lub.
+         - constructor. simpl. apply Nat.le_max_l. apply Nat.le_max_r.
+           apply Nat.max_lub.
   Qed.
 
   (** Max for propositions is the propositional OR, i.e., disjunction *)
diff --git a/src/Common/Le.v b/src/Common/Le.v
index 08b893fc1..980bb9c12 100644
--- a/src/Common/Le.v
+++ b/src/Common/Le.v
@@ -1,6 +1,5 @@
 (** * Common facts about [≤] *)
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
-Require Import Coq.Arith.Le.
 Require Import Coq.ZArith.ZArith.
 Require Import Coq.Classes.Morphisms.
 Require Import Coq.Program.Basics.
@@ -46,7 +45,7 @@ Proof.
   induction n; simpl; try reflexivity.
   rewrite IHn; clear IHn.
   edestruct le_canonical; [ exfalso | reflexivity ].
-  { eapply le_Sn_n; eassumption. }
+  { eapply Nat.nle_succ_diag_l; eassumption. }
 Qed.
 
 Lemma le_canonical_nS {n m pf} (H : le_canonical n m = left pf)
@@ -103,13 +102,13 @@ Proof. omega. Qed.
 Global Instance le_lt_Proper : Proper (Basics.flip le ==> le ==> impl) lt.
 Proof.
   repeat intro.
-  eapply lt_le_trans; [ | eassumption ].
-  eapply le_lt_trans; eassumption.
+  eapply Nat.lt_le_trans; [ | eassumption ].
+  eapply Nat.le_lt_trans; eassumption.
 Qed.
 
 Global Instance le_lt_Proper' : Proper (le ==> Basics.flip le ==> Basics.flip impl) lt.
 Proof.
   repeat intro.
-  eapply lt_le_trans; [ | eassumption ].
-  eapply le_lt_trans; eassumption.
+  eapply Nat.lt_le_trans; [ | eassumption ].
+  eapply Nat.le_lt_trans; eassumption.
 Qed.
diff --git a/src/Common/List/FlattenList.v b/src/Common/List/FlattenList.v
index 42eebb587..74c46cd3e 100644
--- a/src/Common/List/FlattenList.v
+++ b/src/Common/List/FlattenList.v
@@ -92,7 +92,7 @@ Section FlattenList.
     - auto with arith.
     - unfold compose;
       rewrite app_length, <- IHseq.
-      rewrite plus_comm, <- plus_assoc; auto with arith.
+      rewrite Nat.add_comm, <- Nat.add_assoc; auto with arith.
   Qed.
 
   Lemma length_flatten :
diff --git a/src/Common/List/ListFacts.v b/src/Common/List/ListFacts.v
index e54018e02..09c578432 100644
--- a/src/Common/List/ListFacts.v
+++ b/src/Common/List/ListFacts.v
@@ -351,8 +351,8 @@ Section ListFacts.
     clear IHseq; revert a default; induction seq;
       simpl; intros; auto with arith.
     rewrite <- IHseq.
-    rewrite Plus.plus_comm, <- Plus.plus_assoc; f_equal.
-    rewrite Plus.plus_comm; reflexivity.
+    rewrite Nat.add_comm, <- Nat.add_assoc; f_equal.
+    rewrite Nat.add_comm; reflexivity.
   Qed.
 
   Lemma map_snd {A B C} :
@@ -433,7 +433,7 @@ Section ListFacts.
   Proof.
     induction seq; simpl; intros; eauto with arith.
     rewrite IHseq; f_equal; eauto with arith.
-    repeat rewrite <- Plus.plus_assoc; f_equal; auto with arith.
+    repeat rewrite <- Nat.add_assoc; f_equal; auto with arith.
   Qed.
 
   Lemma length_flat_map :
@@ -603,7 +603,7 @@ Section ListFacts.
     { destruct y; simpl.
       { destruct x; simpl; repeat (f_equal; []); try reflexivity; omega. }
       { rewrite IHls; destruct x; simpl; repeat (f_equal; []); try reflexivity.
-        rewrite NPeano.Nat.add_succ_r; reflexivity. } }
+        rewrite Nat.add_succ_r; reflexivity. } }
   Qed.
 
   Lemma in_map_iffT' {A B}
@@ -639,7 +639,7 @@ Section ListFacts.
         (f : A -> nat) (ls : list A) (y : nat)
     : iffT (In y (map f ls)) { x : A | f x = y /\ In x ls }.
   Proof.
-    apply in_map_iffT, NPeano.Nat.eq_dec.
+    apply in_map_iffT, Nat.eq_dec.
   Defined.
 
   Lemma nth_take_1_drop {A} (ls : list A) n a
@@ -670,7 +670,7 @@ Section ListFacts.
     : drop 1 (drop y ls) = drop (S y) ls.
   Proof.
     rewrite drop_drop.
-    rewrite NPeano.Nat.add_1_r.
+    rewrite Nat.add_1_r.
     reflexivity.
   Qed.
 
@@ -967,7 +967,7 @@ Section ListFacts.
        | [ H : context[option_map _ ?x] |- _ ] => destruct x eqn:?; unfold option_map in H
        | _ => solve [ repeat (esplit || eassumption) ]
        | [ H : context[nth_error (_::_) ?x] |- _ ] => is_var x; destruct x; simpl nth_error in H
-       | [ H : S _ < S _ |- _ ] => apply lt_S_n in H
+       | [ H : S _ < S _ |- _ ] => apply (proj2 (Nat.succ_lt_mono _ _)) in H
        | _ => solve [ eauto with nocore ]
        | [ |- context[if ?b then _ else _] ] => destruct b eqn:?
        | [ H : ?A -> ?B |- _ ] => let H' := fresh in assert (H' : A) by (assumption || omega); specialize (H H'); clear H'
@@ -975,7 +975,7 @@ Section ListFacts.
        | _ => progress simpl in *
        | [ H : forall x, ?f x = ?f ?y -> _ |- _ ] => specialize (H _ eq_refl)
        | [ H : forall x, ?f ?y = ?f x -> _ |- _ ] => specialize (H _ eq_refl)
-       | [ H : forall n, S n < S _ -> _ |- _ ] => specialize (fun n pf => H n (lt_n_S _ _ pf))
+       | [ H : forall n, S n < S _ -> _ |- _ ] => specialize (fun n pf => H n (proj1 (Nat.succ_lt_mono _ _) pf))
        | [ H : nth_error nil ?x = Some _ |- _ ] => is_var x; destruct x
        | [ H : forall m x, nth_error (_::_) m = Some _ -> _ |- _ ] => pose proof (H 0); specialize (fun m => H (S m))
        | [ H : or _ _ |- _ ] => destruct H
@@ -1106,11 +1106,11 @@ Section ListFacts.
     induction ls as [|x xs IHxs].
     { simpl; intros; destruct (n - offset); reflexivity. }
     { simpl; intros.
-      destruct (beq_nat n offset) eqn:H';
-        [ apply beq_nat_true in H'
-        | apply beq_nat_false in H' ];
+      destruct (Nat.eqb n offset) eqn:H';
+        [ apply Nat.eqb_eq in H'
+        | apply Nat.eqb_neq in H' ];
         subst;
-        rewrite ?minus_diag; trivial.
+        rewrite ?Nat.sub_diag; trivial.
       destruct (n - offset) eqn:H''.
       { omega. }
       { rewrite IHxs by omega.
@@ -1997,7 +1997,7 @@ Section ListFacts.
       List.In n ns -> f n <= fold_right (fun n acc => max (f n) acc) 0 ns.
   Proof.
     induction ns; intros; inversion H; subst; simpl;
-      apply NPeano.Nat.max_le_iff; [ left | right ]; auto.
+      apply Nat.max_le_iff; [ left | right ]; auto.
   Qed.
 
   Lemma fold_right_higher_is_higher {A}
@@ -2006,7 +2006,7 @@ Section ListFacts.
       fold_right (fun n acc => max (f n) acc) 0 ns <= x.
   Proof.
     induction ns; simpl; intros; [ apply le_0_n | ].
-    apply NPeano.Nat.max_lub.
+    apply Nat.max_lub.
     apply H; left; auto.
     apply IHns; intros; apply H; right; auto.
   Qed.
@@ -2023,8 +2023,8 @@ Section ListFacts.
   (* if a list is empty, the result of filtering the list with anything will still be empty *)
   Lemma filter_nil_is_nil {A}
     : forall (l : list A) (pred : A -> bool),
-      beq_nat (Datatypes.length l) 0 = true
-      ->  beq_nat (Datatypes.length (filter pred l)) 0 = true.
+      Nat.eqb (Datatypes.length l) 0 = true
+      ->  Nat.eqb (Datatypes.length (filter pred l)) 0 = true.
   Proof.
     induction l; intros; simpl; try inversion H.
     reflexivity.
@@ -2374,7 +2374,7 @@ Section ListFacts.
       = option_map (fun v => (start + idx, v)) (nth_error ls idx).
   Proof.
     revert start idx; induction ls as [|l ls IHls], idx as [|idx]; try reflexivity.
-    { simpl; rewrite NPeano.Nat.add_0_r; reflexivity. }
+    { simpl; rewrite Nat.add_0_r; reflexivity. }
     { simpl; rewrite <- plus_n_Sm, IHls; reflexivity. }
   Qed.
 
diff --git a/src/Common/List/Operations.v b/src/Common/List/Operations.v
index 5669d0fae..e3100b877 100644
--- a/src/Common/List/Operations.v
+++ b/src/Common/List/Operations.v
@@ -74,7 +74,7 @@ Module Export List.
   Fixpoint nth'_helper {A} (n : nat) (ls : list A) (default : A) (offset : nat) :=
     match ls with
       | nil => default
-      | x::xs => if EqNat.beq_nat n offset
+      | x::xs => if Nat.eqb n offset
                  then x
                  else nth'_helper n xs default (S offset)
     end.
diff --git a/src/Common/NatFacts.v b/src/Common/NatFacts.v
index 1a33852cb..724dfb4cd 100644
--- a/src/Common/NatFacts.v
+++ b/src/Common/NatFacts.v
@@ -1,12 +1,11 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Coq.Classes.Morphisms.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Coq.ZArith.ZArith.
 
 Lemma min_def {x y} : min x y = x - (x - y).
-Proof. apply Min.min_case_strong; omega. Qed.
+Proof. apply Nat.min_case_strong; omega. Qed.
 Lemma max_def {x y} : max x y = x + (y - x).
-Proof. apply Max.max_case_strong; omega. Qed.
+Proof. apply Nat.max_case_strong; omega. Qed.
 Ltac coq_omega := omega.
 Ltac handle_min_max_for_omega :=
   repeat match goal with
@@ -19,8 +18,8 @@ Ltac handle_min_max_for_omega_case :=
   repeat match goal with
          | [ H : context[min _ _] |- _ ] => revert H
          | [ H : context[max _ _] |- _ ] => revert H
-         | [ |- context[min _ _] ] => apply Min.min_case_strong
-         | [ |- context[max _ _] ] => apply Max.max_case_strong
+         | [ |- context[min _ _] ] => apply Nat.min_case_strong
+         | [ |- context[max _ _] ] => apply Nat.max_case_strong
          end;
   intros.
 Ltac omega_with_min_max :=
@@ -59,9 +58,9 @@ Section NatFacts.
 
   Lemma beq_nat_eq_nat_dec :
     forall x y,
-      beq_nat x y = if eq_nat_dec x y then true else false.
+      Nat.eqb x y = if eq_nat_dec x y then true else false.
   Proof.
-    intros; destruct (eq_nat_dec _ _); [ apply beq_nat_true_iff | apply beq_nat_false_iff ]; assumption.
+    intros; destruct (eq_nat_dec _ _); [ apply Nat.eqb_eq | apply Nat.eqb_neq ]; assumption.
   Qed.
 
   Lemma min_minus_l x y
@@ -110,7 +109,7 @@ Section dec_prod.
   Proof.
     destruct max as [|max];
     [ clear dec_stabalize' | specialize (dec_stabalize' max) ].
-    { destruct (Hdec 0 (le_refl _)) as [Hd|Hd]; [ left | right ].
+    { destruct (Hdec 0 (Nat.le_refl _)) as [Hd|Hd]; [ left | right ].
       { intro n.
         induction n as [|n IHn].
         { assumption. }
@@ -140,7 +139,7 @@ Section dec_prod.
   Proof.
     destruct max as [|max];
     [ clear dec_stabalize | specialize (dec_stabalize max) ].
-    { destruct (Hdec 0 (le_refl _)) as [Hd|Hd]; [ left | right ].
+    { destruct (Hdec 0 (Nat.le_refl _)) as [Hd|Hd]; [ left | right ].
       { exists 0; split; [ reflexivity | assumption ]. }
       { intros n Pn. apply Hd.
         clear -Pn Hstable.
@@ -196,8 +195,8 @@ Lemma min_case_strong_r n m (P : nat -> Type)
 : (n <= m -> P n) -> (m < n -> P m) -> P (min n m).
 Proof.
   destruct (Compare_dec.le_lt_dec n m);
-  first [ rewrite Min.min_r by omega
-        | rewrite Min.min_l by omega ];
+  first [ rewrite Nat.min_r by omega
+        | rewrite Nat.min_l by omega ];
   auto.
 Qed.
 
@@ -205,19 +204,19 @@ Lemma min_case_strong_l n m (P : nat -> Type)
 : (n < m -> P n) -> (m <= n -> P m) -> P (min n m).
 Proof.
   destruct (Compare_dec.le_lt_dec m n);
-  first [ rewrite Min.min_r by omega
-        | rewrite Min.min_l by omega ];
+  first [ rewrite Nat.min_r by omega
+        | rewrite Nat.min_l by omega ];
   auto.
 Qed.
 
 Lemma beq_0_1_leb x
-: (EqNat.beq_nat x 1 || EqNat.beq_nat x 0)%bool = Compare_dec.leb x 1.
+: (Nat.eqb x 1 || Nat.eqb x 0)%bool = Compare_dec.leb x 1.
 Proof.
   destruct x as [|[|]]; simpl; reflexivity.
 Qed.
 
 Lemma beq_S_leb x n
-: (EqNat.beq_nat x (S n) || Compare_dec.leb x n)%bool = Compare_dec.leb x (S n).
+: (Nat.eqb x (S n) || Compare_dec.leb x n)%bool = Compare_dec.leb x (S n).
 Proof.
   revert x; induction n as [|n IHn]; simpl.
   { intros [|[|]]; reflexivity. }
@@ -246,7 +245,7 @@ Lemma min_subr_same {x y} : (min x y - x)%natr = 0.
 Proof. clear; rewrite minusr_minus; omega *. Qed.
 
 Lemma beq_nat_min_0 {x y}
-  : EqNat.beq_nat (min x y) 0 = orb (EqNat.beq_nat x 0) (EqNat.beq_nat y 0).
+  : Nat.eqb (min x y) 0 = orb (Nat.eqb x 0) (Nat.eqb y 0).
 Proof. destruct x, y; simpl; reflexivity. Qed.
 
 Lemma max_min_n {x y} : max (min x y) y = y.
diff --git a/src/Common/StringFacts.v b/src/Common/StringFacts.v
index e14e4f912..10b7aca73 100644
--- a/src/Common/StringFacts.v
+++ b/src/Common/StringFacts.v
@@ -1,8 +1,8 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Coq.ZArith.ZArith.
+Require Import Coq.Arith.PeanoNat.
 Require Import Coq.Strings.Ascii.
 Require Import Coq.Strings.String.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Fiat.Common.List.Operations.
 Require Import Fiat.Common.StringOperations.
 
@@ -23,10 +23,10 @@ Proof.
   induction s; simpl.
   { intros; destruct m; trivial. }
   { intros; destruct m.
-    { apply Lt.lt_n_0 in H.
+{ apply Nat.nlt_0_r in H.
       destruct H. }
     { rewrite IHs; trivial.
-      apply Le.le_S_n; trivial. } }
+      apply Nat.succ_le_mono; trivial. } }
 Qed.
 
 Lemma substring_correct3' {s : string}
@@ -47,7 +47,7 @@ Proof.
   revert n m H; induction s; simpl; intros n m H.
   { destruct n, m; trivial. }
   { destruct n, m; trivial.
-    { apply le_Sn_0 in H; destruct H. }
+    { apply Nat.nlt_0_r in H; destruct H. }
     { rewrite IHs; eauto with arith. }
     { rewrite IHs; eauto with arith. } }
 Qed.
@@ -60,14 +60,15 @@ Proof.
   induction s; simpl in *.
   { intros; destruct m, m', n; trivial. }
   { intros; destruct m, m', n; trivial; simpl in *;
-    try (apply Lt.lt_n_0 in H; destruct H);
-    try (apply Lt.lt_n_0 in H'; destruct H');
-    try apply Le.le_S_n in H;
-    try apply Le.le_S_n in H';
-    try solve [ try rewrite plus_comm in H;
-                try rewrite plus_comm in H';
+    try (apply Nat.nlt_0_r in H; destruct H);
+    try (apply Nat.nlt_0_r in H'; destruct H');
+    try apply le_S_n in H;
+    try apply le_S_n in H';
+    try solve [ try rewrite Nat.add_comm in H;
+                try rewrite Nat.add_comm in H';
                 simpl in *;
                 rewrite !substring_correct0' by auto with arith; trivial ].
+                
     { apply f_equal.
       apply IHs; trivial. }
     { apply IHs; trivial. } }
@@ -90,7 +91,7 @@ Proof.
   induction s; simpl; intros.
   { destruct (y + z), x, y, z; reflexivity. }
   { destruct x, y, z; try reflexivity; simpl;
-    rewrite ?plus_0_r, ?substring_correct0, ?string_concat_empty_r;
+    rewrite ?Nat.add_0_r, ?substring_correct0, ?string_concat_empty_r;
     try reflexivity.
     { apply f_equal.
       apply IHs. }
@@ -151,14 +152,14 @@ Proof.
   induction s; intros.
   { destruct n, m, n', m'; reflexivity. }
   { destruct n', m';
-    rewrite <- ?plus_n_O, <- ?minus_n_O, ?Min.min_0_r, ?Min.min_0_l;
+    rewrite <- ?Nat.add_0_r, <- ?Nat.sub_0_r, ?Nat.min_0_r, ?Nat.min_0_l;
     destruct n, m;
     trivial; simpl;
-    rewrite ?IHs, ?substring_correct0, <- ?plus_n_O, <- ?minus_n_O;
+    rewrite ?IHs, ?substring_correct0, <- ?plus_n_O, ?Nat.sub_0_r;
     simpl;
     try reflexivity.
-    rewrite (plus_comm _ (S _)); simpl.
-    rewrite (plus_comm n n').
+    rewrite (Nat.add_comm _ (S _)); simpl.
+    rewrite (Nat.add_comm n n').
     reflexivity. }
 Qed.
 
@@ -178,7 +179,7 @@ Lemma substring_min {x x' y y' z str} (H : substring x y str = substring x' y' s
 Proof.
   pose proof (fun y x => @substring_substring str 0 z x y) as H'; simpl in *.
   setoid_rewrite Nat.sub_0_r in H'.
-  setoid_rewrite Min.min_comm in H'.
+  setoid_rewrite Nat.min_comm in H'.
   rewrite <- !H', H; reflexivity.
 Qed.
 
@@ -232,7 +233,7 @@ Qed.
 Lemma substring_min_length str x y
 : substring x (min y (length str)) str = substring x y str.
 Proof.
-  apply Min.min_case_strong; try reflexivity.
+  apply Nat.min_case_strong; try reflexivity.
   intro H.
   apply substring_correct4; omega.
 Qed.
diff --git a/src/Common/Wf.v b/src/Common/Wf.v
index d097c4682..9be4543dc 100644
--- a/src/Common/Wf.v
+++ b/src/Common/Wf.v
@@ -1,6 +1,7 @@
 (** * Miscellaneous Well-Foundedness Facts *)
 Require Import Coq.Setoids.Setoid Coq.Program.Program Coq.Program.Wf Coq.Arith.Wf_nat Coq.Classes.Morphisms Coq.Init.Wf.
 Require Import Coq.Lists.SetoidList.
+Require Import Coq.Arith.PeanoNat.
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 
 Set Implicit Arguments.
@@ -447,8 +448,8 @@ Section wf.
          | S n' => prod_relationA eqA R (@iterated_prod_relationA n')
          end.
 
-    Fixpoint nat_eq_transfer (P : nat -> Type) (n m : nat) : P n -> (P m) + (EqNat.beq_nat n m = false)
-      := match n, m return P n -> (P m) + (EqNat.beq_nat n m = false) with
+    Fixpoint nat_eq_transfer (P : nat -> Type) (n m : nat) : P n -> (P m) + (Nat.eqb n m = false)
+      := match n, m return P n -> (P m) + (Nat.eqb n m = false) with
          | 0, 0 => fun x => inl x
          | S n', S m' => @nat_eq_transfer (fun v => P (S v)) n' m'
          | _, _ => fun _ => inr eq_refl
@@ -461,11 +462,11 @@ Section wf.
          end.
 
     Fixpoint nat_eq_transfer_neq (P : nat -> Type) (n m : nat)
-      : forall v : P n, (if EqNat.beq_nat n m as b return ((P m) + (b = false)) -> Prop
+      : forall v : P n, (if Nat.eqb n m as b return ((P m) + (b = false)) -> Prop
                          then fun _ => True
                          else fun v => v = inr eq_refl)
                           (nat_eq_transfer P n m v)
-      := match n, m return forall v : P n, (if EqNat.beq_nat n m as b return ((P m) + (b = false)) -> Prop
+      := match n, m return forall v : P n, (if Nat.eqb n m as b return ((P m) + (b = false)) -> Prop
                                             then fun _ => True
                                             else fun v => v = inr eq_refl)
                                              (nat_eq_transfer P n m v)
@@ -521,8 +522,8 @@ Section wf.
                             cbv beta in *;
                             generalize dependent (nat_eq_transfer iterated_prod n1' n0');
                             let Heq := fresh in
-                            destruct (EqNat.beq_nat n1' n0') eqn:Heq;
-                            [ apply EqNat.beq_nat_true_iff in Heq; subst; rewrite <- EqNat.beq_nat_refl in H;
+                            destruct (Nat.eqb n1' n0') eqn:Heq;
+                            [ apply Nat.eqb_eq in Heq; subst; rewrite Nat.eqb_refl in H;
                               exfalso; clear -H; congruence
                             | ]
                        | [ |- context[@nat_eq_transfer iterated_prod n0' n1' _] ]
diff --git a/src/Computation/Decidable.v b/src/Computation/Decidable.v
index b3f7929f7..027d6d4c6 100644
--- a/src/Computation/Decidable.v
+++ b/src/Computation/Decidable.v
@@ -150,9 +150,9 @@ Obligation 1. t; destruct (Ascii.ascii_dec n m); auto; discriminate. Qed.
 Require Import Coq.Arith.Arith.
 
 Global Program Instance nat_eq_Decidable {n m : nat} : Decidable (n = m) := {
-  Decidable_witness := beq_nat n m
+  Decidable_witness := Nat.eqb n m
 }.
-Obligation 1. t' beq_nat_true_iff. Qed.
+Obligation 1. t' Nat.eqb_eq. Qed.
 
 Global Program Instance le_Decidable {n m} : Decidable (Nat.le n m) := {
   Decidable_witness := Compare_dec.leb n m
diff --git a/src/Computation/ListComputations.v b/src/Computation/ListComputations.v
index e1a1ff0e8..8f160c4b3 100644
--- a/src/Computation/ListComputations.v
+++ b/src/Computation/ListComputations.v
@@ -107,14 +107,14 @@ Section UpperBound.
   Definition find_UpperBound (f : A -> nat) (ns : list A) : list A :=
     let max := fold_right
                  (fun n acc => max (f n) acc) O ns in
-    filter (fun n => NPeano.Nat.leb max (f n)) ns.
+    filter (fun n => Nat.leb max (f n)) ns.
 
   Lemma find_UpperBound_highest_length
     : forall (f : A -> nat) ns n,
       List.In n (find_UpperBound f ns) -> forall n', List.In n' ns -> (f n) >= (f n').
   Proof.
     unfold ge, find_UpperBound; intros.
-    apply filter_In in H; destruct H. apply NPeano.Nat.leb_le in H1.
+    apply filter_In in H; destruct H. apply Nat.leb_le in H1.
     rewrite <- H1; clear H1 H n.
     apply fold_right_max_is_max; auto.
   Qed.
@@ -127,9 +127,9 @@ Instance DecideableEnsembleUpperBound {A}
          (ns : list A)
   : DecideableEnsemble (UpperBound (fun a a' => f a <= f a') ns).
 Proof.
-refine {| dec n := NPeano.Nat.leb (fold_right (fun n acc => max (f n) acc) O ns) (f n) |}.
-  unfold UpperBound, ge; intros; rewrite NPeano.Nat.leb_le; intuition.
-  - remember (f a); clear Heqn; subst; eapply le_trans;
+refine {| dec n := Nat.leb (fold_right (fun n acc => max (f n) acc) O ns) (f n) |}.
+  unfold UpperBound, ge; intros; rewrite Nat.leb_le; intuition.
+  - remember (f a); clear Heqn; subst; eapply Nat.le_trans;
       [ apply fold_right_max_is_max; apply H0 | assumption ].
   - eapply fold_right_higher_is_higher; eauto.
 Defined.
diff --git a/src/Narcissus/Automation/Decision.v b/src/Narcissus/Automation/Decision.v
index f0c9dfb07..583b75148 100644
--- a/src/Narcissus/Automation/Decision.v
+++ b/src/Narcissus/Automation/Decision.v
@@ -50,12 +50,12 @@ Qed.
 
 Lemma decides_nat_eq :
   forall (n n' : nat),
-    decides (EqNat.beq_nat n n') (n = n').
+    decides (Nat.eqb n n') (n = n').
 Proof.
   unfold decides, If_Then_Else; intros.
-  destruct (EqNat.beq_nat n n') eqn: ? ;
-    try eapply EqNat.beq_nat_true_iff;
-    try eapply EqNat.beq_nat_false_iff; eauto.
+  destruct (Nat.eqb n n') eqn: ? ;
+    try eapply Nat.eqb_true_iff;
+    try eapply Nat.eqb_false_iff; eauto.
 Qed.
 
 Lemma decides_word_eq {sz}:
diff --git a/src/Parsers/AbstractInterpretation/NonTerminalMap.v b/src/Parsers/AbstractInterpretation/NonTerminalMap.v
index 6696da88f..5887df386 100644
--- a/src/Parsers/AbstractInterpretation/NonTerminalMap.v
+++ b/src/Parsers/AbstractInterpretation/NonTerminalMap.v
@@ -16,7 +16,7 @@ Definition positive_to_nonterminal (nt : positive) : default_nonterminal_carrier
 Lemma positive_to_nonterminal_to_positive nt : nonterminal_to_positive (positive_to_nonterminal nt) = nt.
 Proof.
   unfold nonterminal_to_positive, positive_to_nonterminal.
-  erewrite <- S_pred by apply Pos2Nat.is_pos.
+  erewrite Nat.lt_succ_pred by apply Pos2Nat.is_pos.
   rewrite Pos2Nat.id.
   reflexivity.
 Qed.
diff --git a/src/Parsers/BaseTypesLemmas.v b/src/Parsers/BaseTypesLemmas.v
index da64a787c..340a95630 100644
--- a/src/Parsers/BaseTypesLemmas.v
+++ b/src/Parsers/BaseTypesLemmas.v
@@ -112,13 +112,13 @@ Section recursive_descent_parser.
   Lemma nonempty_nonterminals {ls nt} (H : is_valid_nonterminal ls nt)
   : 0 < nonterminals_length ls.
   Proof.
-    eapply Lt.le_lt_trans;
-    [ apply Le.le_0_n
+    eapply Nat.le_lt_trans;
+    [ apply Nat.le_0_l
     | exact (remove_nonterminal_dec ls nt H) ].
   Qed.
 
   Lemma nonempty_nonterminals' {ls nt} (H : is_valid_nonterminal ls nt)
-  : negb (EqNat.beq_nat (nonterminals_length ls) 0).
+  : negb (Nat.eqb (nonterminals_length ls) 0).
   Proof.
     pose proof (nonempty_nonterminals H).
     destruct (nonterminals_length ls); simpl; try reflexivity; try omega.
diff --git a/src/Parsers/BooleanRecognizerOptimized.v b/src/Parsers/BooleanRecognizerOptimized.v
index eb562e417..c22d1c655 100644
--- a/src/Parsers/BooleanRecognizerOptimized.v
+++ b/src/Parsers/BooleanRecognizerOptimized.v
@@ -1,7 +1,7 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Definition of a boolean-returning CFG parser-recognizer *)
 Require Import Coq.Lists.List Coq.Strings.String.
-Require Import Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Compare_dec Coq.Arith.Wf_nat.
+Require Import Coq.Arith.Compare_dec Coq.Arith.Wf_nat Coq.Arith.PeanoNat.
 Require Import Fiat.Common.List.Operations.
 Require Import Fiat.Common.List.ListMorphisms.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
@@ -33,7 +33,6 @@ Require Export Fiat.Common.NatFacts.
 Require Export Fiat.Common.Sigma.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
-Import NPeano.
 
 Set Implicit Arguments.
 Local Open Scope string_like_scope.
@@ -58,7 +57,7 @@ Module Export opt.
     Definition hd {A} := Eval compute in @hd A.
     Definition tl {A} := Eval compute in @tl A.
     Definition nth {A} := Eval compute in @nth A.
-    Definition nth' {A} := Eval cbv beta iota zeta delta -[EqNat.beq_nat] in @nth' A.
+    Definition nth' {A} := Eval cbv beta iota zeta delta -[Nat.eqb] in @nth' A.
     Definition fst {A B} := Eval compute in @fst A B.
     Definition snd {A B} := Eval compute in @snd A B.
     Definition list_caset {A} := Eval compute in @list_caset A.
@@ -68,7 +67,7 @@ Module Export opt.
     Definition pred := Eval compute in pred.
     Definition minusr := Eval compute in minusr.
     Definition id {A} := Eval compute in @id A.
-    Definition beq_nat := Eval compute in EqNat.beq_nat.
+    Definition beq_nat := Eval compute in Nat.eqb.
     Definition leb := Eval compute in Compare_dec.leb.
     Definition interp_RCharExpr {Char idata} := Eval cbv -[andb orb negb Compare_dec.leb] in @interp_RCharExpr Char idata.
   End opt.
@@ -94,7 +93,7 @@ Module Export opt2.
     Definition hd {A} := Eval compute in @hd A.
     Definition tl {A} := Eval compute in @tl A.
     Definition nth {A} := Eval compute in @nth A.
-    Definition nth' {A} := Eval cbv beta iota zeta delta -[EqNat.beq_nat] in @nth' A.
+    Definition nth' {A} := Eval cbv beta iota zeta delta -[Nat.eqb] in @nth' A.
     Definition fst {A B} := Eval compute in @fst A B.
     Definition snd {A B} := Eval compute in @snd A B.
     Definition list_caset {A} := Eval compute in @list_caset A.
@@ -104,7 +103,7 @@ Module Export opt2.
     Definition pred := Eval compute in pred.
     Definition minusr := Eval compute in minusr.
     Definition id {A} := Eval compute in @id A.
-    Definition beq_nat := Eval compute in EqNat.beq_nat.
+    Definition beq_nat := Eval compute in Nat.eqb.
     Definition leb := Eval compute in Compare_dec.leb.
     Definition interp_RCharExpr {Char idata} := Eval cbv -[andb orb negb Compare_dec.leb] in @interp_RCharExpr Char idata.
   End opt2.
@@ -130,7 +129,7 @@ Module Export opt3.
     Definition hd {A} := Eval compute in @hd A.
     Definition tl {A} := Eval compute in @tl A.
     Definition nth {A} := Eval compute in @nth A.
-    Definition nth' {A} := Eval cbv beta iota zeta delta -[EqNat.beq_nat] in @nth' A.
+    Definition nth' {A} := Eval cbv beta iota zeta delta -[Nat.eqb] in @nth' A.
     Definition fst {A B} := Eval compute in @fst A B.
     Definition snd {A B} := Eval compute in @snd A B.
     Definition list_caset {A} := Eval compute in @list_caset A.
@@ -140,7 +139,7 @@ Module Export opt3.
     Definition pred := Eval compute in pred.
     Definition minusr := Eval compute in minusr.
     Definition id {A} := Eval compute in @id A.
-    Definition beq_nat := Eval compute in EqNat.beq_nat.
+    Definition beq_nat := Eval compute in Nat.eqb.
     Definition leb := Eval compute in Compare_dec.leb.
     Definition interp_RCharExpr {Char idata} := Eval cbv -[andb orb negb Compare_dec.leb] in @interp_RCharExpr Char idata.
   End opt3.
@@ -271,9 +270,9 @@ Section recursive_descent_parser.
              | [ H : (_ && _)%bool = true |- _ ] => apply Bool.andb_true_iff in H
              | [ H : _ = in_left |- _ ] => clear H
              | [ H : _ /\ _ |- _ ] => destruct H
-             | [ H : context[negb (EqNat.beq_nat ?x ?y)] |- _ ] => destruct (EqNat.beq_nat x y) eqn:?
-             | [ H : EqNat.beq_nat _ _ = false |- _ ] => apply EqNat.beq_nat_false in H
-             | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
+             | [ H : context[negb (Nat.eqb ?x ?y)] |- _ ] => destruct (Nat.eqb x y) eqn:?
+             | [ H : Nat.eqb _ _ = false |- _ ] => apply (fun n m => proj1 (Nat.eqb_neq n m)) in H
+             | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
              | [ H : snd ?x = _ |- _ ] => is_var x; destruct x
              | _ => progress simpl negb in *
              | [ H : false = true |- _ ] => inversion H
@@ -425,7 +424,7 @@ Section recursive_descent_parser.
              | [ |- _ = 0 ] => reflexivity
              | [ |- _ = 1 ] => reflexivity
              | [ |- _ = None ] => reflexivity
-             | [ |- _ = EqNat.beq_nat _ _ ] => apply f_equal2
+             | [ |- _ = Nat.eqb _ _ ] => apply f_equal2
              | [ |- _ = Compare_dec.leb _ _ ] => apply f_equal2
              | [ |- _ = S _ ] => apply f_equal
              | [ |- _ = string_beq _ _ ] => apply f_equal2
@@ -447,9 +446,9 @@ Section recursive_descent_parser.
   Local Ltac misc_opt' :=
     idtac;
     match goal with
-    | _ => progress rewrite ?max_min_n, ?Minus.minus_diag, ?Nat.sub_0_r, ?uneta_bool, ?beq_nat_min_0(*, ?bool_rect_flatten*)
-    | _ => rewrite Min.min_l by assumption
-    | _ => rewrite Min.min_r by assumption
+    | _ => progress rewrite ?max_min_n, ?Nat.sub_diag, ?Nat.sub_0_r, ?uneta_bool, ?beq_nat_min_0(*, ?bool_rect_flatten*)
+    | _ => rewrite Nat.min_l by assumption
+    | _ => rewrite Nat.min_r by assumption
     | [ |- context[if ?ltb ?x ?y then _ else _] ] => rewrite if_to_min
     | [ |- context[min ?x ?y - ?x] ] => rewrite min_sub_same
     | [ |- context[(min ?x ?y - ?x)%natr] ] => rewrite min_subr_same
@@ -1159,7 +1158,7 @@ Section recursive_descent_parser.
                                               (fun _ => _)
                                               (N0 a' b' c')
                                               (list_rect P0 (fun _ _ _ => true) C0 ls' a' b' c')
-                                              (EqNat.beq_nat (List.length ls') 0))
+                                              (Nat.eqb (List.length ls') 0))
                                            = list_rect P0 N0 C0 ls' a' b' c')
                              _
                              _
@@ -1192,12 +1191,12 @@ Section recursive_descent_parser.
                        => is_var e; destruct e
                      | _ => progress simpl
                      | [ |- ?x = ?x ] => reflexivity
-                     | _ => progress rewrite ?Bool.andb_true_r, ?Min.min_idempotent, ?Minus.minus_diag
-                     | [ H : EqNat.beq_nat _ _ = true |- _ ]
-                       => apply EqNat.beq_nat_true in H
+                     | _ => progress rewrite ?Bool.andb_true_r, ?Nat.min_idempotent, ?Nat.sub_diag
+                     | [ H : Nat.eqb _ _ = true |- _ ]
+                       => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
                      | _ => progress subst
-                     | [ |- context[EqNat.beq_nat ?x ?y] ]
-                       => is_var x; destruct (EqNat.beq_nat x y) eqn:?
+                     | [ |- context[Nat.eqb ?x ?y] ]
+                       => is_var x; destruct (Nat.eqb x y) eqn:?
                      | [ H := _ |- _ ] => subst H
                      | [ |- context[orb _ false] ] => rewrite Bool.orb_false_r
                      | _ => rewrite minusr_minus
@@ -1246,13 +1245,13 @@ Section recursive_descent_parser.
               { misc_opt.
                 rewrite <- andbr_andb.
                 apply (f_equal2 andbr); [ | reflexivity ].
-                rewrite Min.min_idempotent.
+                rewrite Nat.min_idempotent.
                 reflexivity. }
               { apply (f_equal2 andb); [ | reflexivity ].
                 step_opt'; [].
                 apply (f_equal2 andb); [ | reflexivity ].
                 match goal with
-                | [ |- _ = EqNat.beq_nat (min ?v ?x) ?v ]
+                | [ |- _ = Nat.eqb (min ?v ?x) ?v ]
                   => refine (_ : Compare_dec.leb v x = _)
                 end.
                 match goal with
@@ -1418,13 +1417,13 @@ Section recursive_descent_parser.
             rewrite bool_rect_andb; simpl.
             rewrite Bool.andb_true_r.
             match goal with
-            | [ |- _ = (orb (negb (EqNat.beq_nat ?x 0)) (andb (EqNat.beq_nat ?x 0) ?y)) ]
+            | [ |- _ = (orb (negb (Nat.eqb ?x 0)) (andb (Nat.eqb ?x 0) ?y)) ]
               => let z := fresh in
                  let y' := fresh in
                  set (z := x);
                    set (y' := y);
                    refine (_ : orb (Compare_dec.leb 1 x) y = _);
-                   change (orb (Compare_dec.leb 1 z) y' = orb (negb (EqNat.beq_nat z 0)) (andb (EqNat.beq_nat z 0) y'));
+                   change (orb (Compare_dec.leb 1 z) y' = orb (negb (Nat.eqb z 0)) (andb (Nat.eqb z 0) y'));
                    destruct z, y'; reflexivity
             end. }
           etransitivity_rev _.
@@ -1627,13 +1626,13 @@ Section recursive_descent_parser.
           { rewrite bool_rect_andb.
             rewrite Bool.andb_true_r.
             match goal with
-            | [ |- _ = (orb (negb (EqNat.beq_nat ?x 0)) (andb (EqNat.beq_nat ?x 0) ?y)) ]
+            | [ |- _ = (orb (negb (Nat.eqb ?x 0)) (andb (Nat.eqb ?x 0) ?y)) ]
               => let z := fresh in
                  let y' := fresh in
                  set (z := x);
                    set (y' := y);
                    refine (_ : orb (Compare_dec.leb 1 x) y = _);
-                   change (orb (Compare_dec.leb 1 z) y' = orb (negb (EqNat.beq_nat z 0)) (andb (EqNat.beq_nat z 0) y'));
+                   change (orb (Compare_dec.leb 1 z) y' = orb (negb (Nat.eqb z 0)) (andb (Nat.eqb z 0) y'));
                    destruct z, y'; reflexivity
             end. }
           apply (f_equal2 orb); fin_step_opt; [].
@@ -1749,7 +1748,7 @@ Section recursive_descent_parser.
     | [ |- context G[snd (of_string ?str')] ]
       => let G' := context G[opt.id (opt.snd (of_string str'))] in
          change G'
-    | [ |- context G[EqNat.beq_nat (opt2.id ?x) 0] ]
+    | [ |- context G[Nat.eqb (opt2.id ?x) 0] ]
       => let G' := context G[opt2.id (opt2.beq_nat x 0)] in
          change G'
     | [ |- context G[(opt2.id ?x, 0)] ]
@@ -1758,7 +1757,7 @@ Section recursive_descent_parser.
     | [ |- context G[(opt2.id ?x, opt2.id ?y)] ]
       => let G' := context G[opt2.id (x, y)] in
          change G'
-    | [ |- context G[EqNat.beq_nat (opt.id ?x) 0] ]
+    | [ |- context G[Nat.eqb (opt.id ?x) 0] ]
       => let G' := context G[opt.id (opt.beq_nat x 0)] in
          change G'
     | [ |- context G[S (opt2.id ?x)] ]
diff --git a/src/Parsers/BooleanRecognizerTests.v b/src/Parsers/BooleanRecognizerTests.v
index 469abe443..db21575e7 100644
--- a/src/Parsers/BooleanRecognizerTests.v
+++ b/src/Parsers/BooleanRecognizerTests.v
@@ -1,6 +1,6 @@
 (** * Some simple examples with the boolean-returning CFG parser-recognizer *)
 Require Import Coq.Lists.List Coq.Strings.String.
-Require Import Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Compare_dec Coq.Arith.Wf_nat.
+Require Import Coq.Arith.PeanoNat Coq.Arith.Compare_dec Coq.Arith.Wf_nat.
 Require Import Fiat.Parsers.Grammars.Trivial Fiat.Parsers.Grammars.ABStar.
 Require Import Fiat.Parsers.Splitters.RDPList Fiat.Parsers.Splitters.BruteForce.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
diff --git a/src/Parsers/ContextFreeGrammar/Carriers.v b/src/Parsers/ContextFreeGrammar/Carriers.v
index c06ae52dc..f73bbb4f5 100644
--- a/src/Parsers/ContextFreeGrammar/Carriers.v
+++ b/src/Parsers/ContextFreeGrammar/Carriers.v
@@ -106,7 +106,7 @@ Section grammar.
           { simpl.
             rewrite (string_lb eq_refl); trivial. }
           { simpl; intros H'.
-            specialize (IHxs idx (Le.le_S_n _ _ H')).
+            specialize (IHxs idx (proj2 (Nat.succ_le_mono _ _) H')).
             rewrite first_index_helper_first_index_error, IHxs by omega.
             apply first_index_error_Some_correct in IHxs.
             repeat match goal with
diff --git a/src/Parsers/ContextFreeGrammar/Fix/Definitions.v b/src/Parsers/ContextFreeGrammar/Fix/Definitions.v
index dd8f542c1..ee64a2684 100644
--- a/src/Parsers/ContextFreeGrammar/Fix/Definitions.v
+++ b/src/Parsers/ContextFreeGrammar/Fix/Definitions.v
@@ -233,7 +233,7 @@ Definition positive_to_nonterminal (nt : positive) : default_nonterminal_carrier
 Lemma positive_to_nonterminal_to_positive nt : nonterminal_to_positive (positive_to_nonterminal nt) = nt.
 Proof.
   unfold nonterminal_to_positive, positive_to_nonterminal.
-  erewrite <- S_pred by apply Pos2Nat.is_pos.
+  erewrite Nat.lt_succ_pred by apply Pos2Nat.is_pos.
   rewrite Pos2Nat.id.
   reflexivity.
 Qed.
diff --git a/src/Parsers/ContextFreeGrammar/Fix/Fix.v b/src/Parsers/ContextFreeGrammar/Fix/Fix.v
index f37c3ac93..08296970e 100644
--- a/src/Parsers/ContextFreeGrammar/Fix/Fix.v
+++ b/src/Parsers/ContextFreeGrammar/Fix/Fix.v
@@ -80,8 +80,8 @@ Section grammar_fixedpoint.
   Qed.
 
   Create HintDb aggregate_step_db discriminated.
-  Hint Rewrite PositiveMap.fold_1 PositiveMap.gmapi nonterminal_to_positive_to_nonterminal positive_to_nonterminal_to_positive PositiveMap.gempty PositiveMapAdditionalFacts.gsspec (@state_beq_refl _ gdata) orb_true_iff orb_true_r orb_false_iff (@state_le_bottom_eq_bottom _ gdata) (@no_state_lt_bottom _ gdata) (@state_le_bottom_eq_bottom _ gdata) (@state_ge_top_eq_top _ gdata) (@bottom_lub_r _ gdata) (@bottom_lub_l _ gdata) (@top_lub_r _ gdata) (@top_lub_l _ gdata) (fun a b => @least_upper_bound_correct_l _ gdata a b : _ = true) (fun a b => @least_upper_bound_correct_r _ gdata a b : _ = true) (fun s => @bottom_bottom _ gdata s : _ = true) (fun s => @top_top _ gdata s : _ = true) beq_nat_true_iff @PositiveMapExtensions.lift_brelation_iff : aggregate_step_db.
-  Hint Rewrite <- beq_nat_refl : aggregate_step_db.
+  Hint Rewrite PositiveMap.fold_1 PositiveMap.gmapi nonterminal_to_positive_to_nonterminal positive_to_nonterminal_to_positive PositiveMap.gempty PositiveMapAdditionalFacts.gsspec (@state_beq_refl _ gdata) orb_true_iff orb_true_r orb_false_iff (@state_le_bottom_eq_bottom _ gdata) (@no_state_lt_bottom _ gdata) (@state_le_bottom_eq_bottom _ gdata) (@state_ge_top_eq_top _ gdata) (@bottom_lub_r _ gdata) (@bottom_lub_l _ gdata) (@top_lub_r _ gdata) (@top_lub_l _ gdata) (fun a b => @least_upper_bound_correct_l _ gdata a b : _ = true) (fun a b => @least_upper_bound_correct_r _ gdata a b : _ = true) (fun s => @bottom_bottom _ gdata s : _ = true) (fun s => @top_top _ gdata s : _ = true) Nat.eqb_eq @PositiveMapExtensions.lift_brelation_iff : aggregate_step_db.
+  Hint Rewrite Nat.eqb_refl : aggregate_step_db.
   Hint Rewrite PositiveMapExtensions.map2_1bis_for_rewrite using reflexivity : aggregate_step_db.
   Hint Rewrite PositiveMapExtensions.fold_andb_true : aggregate_step_db.
 
@@ -486,7 +486,7 @@ Section grammar_fixedpoint.
           { apply orb_true_iff; intuition. }
           { do 2 match goal with
                  | [ H : is_true (orb _ _) |- _ ] => apply orb_true_iff in H
-                 | [ H : _ |- _ ] => setoid_rewrite beq_nat_true_iff in H
+                 | [ H : _ |- _ ] => setoid_rewrite Nat.eqb_eq in H
                  end.
             repeat intuition (congruence || subst || (autorewrite with aggregate_step_db in * ) || eauto). } } }
     Qed.
diff --git a/src/Parsers/ContextFreeGrammar/Fold.v b/src/Parsers/ContextFreeGrammar/Fold.v
index ae285bb31..7c7c6a65c 100644
--- a/src/Parsers/ContextFreeGrammar/Fold.v
+++ b/src/Parsers/ContextFreeGrammar/Fold.v
@@ -283,7 +283,7 @@ Section fold_correctness.
         { apply sub_nonterminals_listT_remove_2; assumption. }
         { apply IH.
           { apply sub_nonterminals_listT_remove_2; assumption. }
-          { apply Le.le_S_n.
+          { apply (fun n m => proj2 (Nat.succ_le_mono n m)).
             etransitivity; [ | exact Hlen ].
             apply (remove_nonterminal_dec valid0 (of_nonterminal nt) Hvalid). } } }
       { apply Pnt_redundant; subst; assumption. } }
diff --git a/src/Parsers/ContextFreeGrammar/Properties.v b/src/Parsers/ContextFreeGrammar/Properties.v
index d82f34ab6..e28fe2688 100644
--- a/src/Parsers/ContextFreeGrammar/Properties.v
+++ b/src/Parsers/ContextFreeGrammar/Properties.v
@@ -1,5 +1,5 @@
 (** * Properties about Context Free Grammars *)
-Require Import Coq.Numbers.Natural.Peano.NPeano Coq.Lists.List.
+Require Import Coq.Lists.List Coq.Arith.PeanoNat.
 Require Import Fiat.Common Fiat.Common.UIP.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
 Require Import Fiat.Parsers.ContextFreeGrammar.Equality.
diff --git a/src/Parsers/ContextFreeGrammar/ValidReflective.v b/src/Parsers/ContextFreeGrammar/ValidReflective.v
index 95d7f36a6..0c2dec518 100644
--- a/src/Parsers/ContextFreeGrammar/ValidReflective.v
+++ b/src/Parsers/ContextFreeGrammar/ValidReflective.v
@@ -1,5 +1,6 @@
 (** * Definition of Context Free Grammars *)
 Require Import Coq.Strings.String Coq.Lists.List.
+Require Import Coq.Arith.PeanoNat.
 Require Export Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.BaseTypes.
 Require Import Fiat.Parsers.Splitters.RDPList.
@@ -59,7 +60,7 @@ Section cfg.
       | [ H : List.Forall _ (_::_) |- _ ] => inversion H; clear H
       | _ => progress subst
       | _ => congruence
-      | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
+      | [ H : Nat.eqb _ _ = true |- _ ] => apply (proj1 (Nat.eqb_eq _ _)) in H
       | [ H : Equality.string_beq _ _ = true |- _ ] => apply Equality.string_bl in H
       | [ H : or _ _ |- _ ] => destruct H
       | [ H : forall x, @?A x \/ @?B x -> _ |- _ ]
diff --git a/src/Parsers/ExtrOcamlParsers.v b/src/Parsers/ExtrOcamlParsers.v
index 579b325c0..d74f9ee2b 100644
--- a/src/Parsers/ExtrOcamlParsers.v
+++ b/src/Parsers/ExtrOcamlParsers.v
@@ -93,7 +93,7 @@ Definition sample_variance (ls : list Ocaml.float) : Ocaml.float
 Definition median (ls : list Ocaml.float) : Ocaml.float
   := let ls' := Ocaml.List.sort Pervasives.compare ls in
      let len := List.length ls in
-     ((List.nth (Div2.div2 (len - 1)) ls' 0 + List.nth (Div2.div2 len) ls' 0)
+     ((List.nth (Nat.div2 (len - 1)) ls' 0 + List.nth (Nat.div2 len) ls' 0)
         / 2)%ocaml_float.
 
 Parameter display_info : forall (sum median mean sample_variance : Ocaml.float)
diff --git a/src/Parsers/GenericCorrectnessBaseTypes.v b/src/Parsers/GenericCorrectnessBaseTypes.v
index d209c8e18..59a7eb459 100644
--- a/src/Parsers/GenericCorrectnessBaseTypes.v
+++ b/src/Parsers/GenericCorrectnessBaseTypes.v
@@ -63,7 +63,7 @@ Section correctness.
       ret_Terminal_true_correct
       : forall str offset len ch,
           len = 0 \/ offset + len <= length str
-          -> (beq_nat len 1 && char_at_matches offset str ch)%bool = true
+          -> (Nat.eqb len 1 && char_at_matches offset str ch)%bool = true
           -> parse_item_is_correct
                (substring offset len str) (Terminal ch)
                true (ret_Terminal_true (unsafe_get offset str));
diff --git a/src/Parsers/GenericRecognizer.v b/src/Parsers/GenericRecognizer.v
index e9fd7a2ec..c346bd20e 100644
--- a/src/Parsers/GenericRecognizer.v
+++ b/src/Parsers/GenericRecognizer.v
@@ -6,7 +6,6 @@ Require Import Fiat.Parsers.ContextFreeGrammar.Core.
 Require Import Fiat.Parsers.BaseTypes.
 Require Import Fiat.Parsers.GenericBaseTypes.
 Require Import Fiat.Common.Wf.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 
 Set Implicit Arguments.
 Local Open Scope string_like_scope.
@@ -25,7 +24,7 @@ Section recursive_descent_parser.
                  (it : item Char)
       : parse_item_T
         := match it with
-             | Terminal P => if EqNat.beq_nat len 1 && char_at_matches offset str P
+             | Terminal P => if Nat.eqb len 1 && char_at_matches offset str P
                              then ret_Terminal_true (unsafe_get offset str)
                              else ret_Terminal_false (unsafe_get offset str)
              | NonTerminal nt => if is_valid_nonterminal initial_nonterminals_data (of_nonterminal nt)
@@ -59,7 +58,7 @@ Section recursive_descent_parser.
                     parse_production_T)
                ((** 0-length production, only accept empty *)
                  fun _ offset len0_minus_len
-                 => if beq_nat (len0 - len0_minus_len) 0
+                 => if Nat.eqb (len0 - len0_minus_len) 0
                     then ret_production_nil_true
                     else ret_production_nil_false)
                (fun it its parse_production' idx offset len0_minus_len
@@ -117,12 +116,12 @@ Section recursive_descent_parser.
                                  (offset : nat) (len : nat),
                             len <= fst p -> nonterminal_carrierT -> parse_nt_T).
 
-          Lemma pred_lt_beq {x} : negb (beq_nat x 0) = true -> pred x < x.
+          Lemma pred_lt_beq {x} : negb (Nat.eqb x 0) = true -> pred x < x.
           Proof.
             destruct x; simpl; try congruence; try omega.
           Qed.
           Lemma pred_lt_beq_helper {valid nt x}
-            : (negb (beq_nat x 0) && is_valid_nonterminal valid nt)%bool = true -> pred x < x.
+            : (negb (Nat.eqb x 0) && is_valid_nonterminal valid nt)%bool = true -> pred x < x.
           Proof.
             intro is_valid.
             generalize (proj1 (proj1 (Bool.andb_true_iff _ _) is_valid)).
@@ -162,7 +161,7 @@ Section recursive_descent_parser.
                                   initial_nonterminals_data
                                   offset'
                                   (len - len0_minus_len')
-                                  (le_minus _ _)))
+                                  (Nat.le_sub_l _ _)))
                     (fun _ => (** [str] didn't get smaller, so we cache the fact that we've hit this nonterminal already *)
                        sumbool_rect
                          (fun _ => option (forall (offset' : nat) (len0_minus_len' : nat), nonterminal_carrierT -> parse_nt_T))
@@ -174,11 +173,11 @@ Section recursive_descent_parser.
                                        (remove_nonterminal valid nt)
                                        offset'
                                        (len0 - len0_minus_len')
-                                       (le_minus _ _)))
+                                       (Nat.le_sub_l _ _)))
 
                          (fun _ => (** oops, we already saw this nonterminal in the past.  ABORT! *)
                             None)
-                         (Sumbool.sumbool_of_bool (negb (EqNat.beq_nat valid_len 0) && is_valid_nonterminal valid nt)))
+                         (Sumbool.sumbool_of_bool (negb (Nat.eqb valid_len 0) && is_valid_nonterminal valid nt)))
                     (lt_dec len len0)));
               first [ assumption
                     | apply le_minus
@@ -211,7 +210,7 @@ Section recursive_descent_parser.
           Proof.
             refine (@parse_nonterminal_or_abort
                       p valid offset (fst p - len0_minus_len) _).
-            clear; try apply le_minus; abstract omega.
+            clear; try apply Nat.le_sub_l; abstract omega.
           Defined.
 
           Definition parse_nonterminal'
diff --git a/src/Parsers/GenericRecognizerMin.v b/src/Parsers/GenericRecognizerMin.v
index 579837c1a..e113d90f0 100644
--- a/src/Parsers/GenericRecognizerMin.v
+++ b/src/Parsers/GenericRecognizerMin.v
@@ -23,7 +23,6 @@ Require Import Fiat.Common.NatFacts.
 Require Import Fiat.Common.UIP.
 Require Import Fiat.Common.
 Import ListNotations.
-Import NPeano.
 
 Set Implicit Arguments.
 Local Open Scope string_like_scope.
@@ -182,14 +181,14 @@ Section recursive_descent_parser.
                try unfold x'; try unfold y'
         end.
 
-      Local Hint Resolve beq_nat_true : generic_parser_correctness.
+      Local Hint Resolve (fun n m => proj1 (Nat.eqb_eq n m)) : generic_parser_correctness.
 
       Local Ltac eq_t' :=
         first [ progress subst_le_proof
               | progress subst_nat_eq_proof
               | progress subst_bool_eq_proof
               | solve [ eauto with generic_parser_correctness nocore ]
-              | rewrite sub_twice, Min.min_r by assumption
+              | rewrite sub_twice, Nat.min_r by assumption
               | rewrite !@min_max_sub
               | rewrite Nat.sub_max_distr_l
               | rewrite <- Nat.sub_add_distr
@@ -313,9 +312,9 @@ Section recursive_descent_parser.
                            transitivity G'; [ | symmetry; exact H ]
                          | let G' := context G[false] in
                            transitivity G'; [ | symmetry; exact H ] ]
-                  | context G[beq_nat ?x ?y]
+                  | context G[Nat.eqb ?x ?y]
                     => let H := fresh in
-                       destruct (Utils.dec (beq_nat x y)) as [H|H];
+                       destruct (Utils.dec (Nat.eqb x y)) as [H|H];
                          [ let G' := context G[true] in
                            transitivity G'; [ | symmetry; exact H ]
                          | let G' := context G[false] in
@@ -773,7 +772,7 @@ Section recursive_descent_parser.
                                          let H1 := fresh in
                                          destruct H as [H0 H1]; try clear H
                | [ H : or _ _ |- _ ] => let H0 := fresh in destruct H as [H0|H0]; try clear H
-               | [ H : beq_nat _ _ = true |- _ ] => apply Nat.eqb_eq in H
+               | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
                | [ H : ?x = 0, H' : context[?x] |- _ ] => rewrite H in H'
                | _ => progress subst
                | _ => progress simpl in *
@@ -792,17 +791,17 @@ Section recursive_descent_parser.
                  apply length_singleton in H
                | [ H : context[length (substring _ 0 _)] |- _ ]
                  => rewrite take_length in H
-               | [ H : beq_nat ?len 1 = false,
+               | [ H : Nat.eqb ?len 1 = false,
                        H' : ?offset + ?len <= length ?str,
                             H'' : is_true (is_char (substring ?offset ?len ?str) _)
                    |- _ ]
                  => apply length_singleton in H''; rewrite substring_length in H''
-               | [ H : context[min] |- _ ] => rewrite Min.min_l in H by omega
-               | [ H : context[min] |- _ ] => rewrite Min.min_r in H by omega
+               | [ H : context[min] |- _ ] => rewrite Nat.min_l in H by omega
+               | [ H : context[min] |- _ ] => rewrite Nat.min_r in H by omega
                | [ H : _ |- _ ] => rewrite Nat.add_sub in H
-               | [ H : andb (beq_nat _ 1) (char_at_matches _ _ _) = false |- False ] => contradict H
+               | [ H : andb (Nat.eqb _ 1) (char_at_matches _ _ _) = false |- False ] => contradict H
                | [ |- _ <> false ] => apply Bool.not_false_iff_true
-               | [ |- andb (beq_nat _ 1) (char_at_matches _ _ _) = true ] => apply char_at_matches_is_char
+               | [ |- andb (Nat.eqb _ 1) (char_at_matches _ _ _) = true ] => apply char_at_matches_is_char
                | [ |- ex _ ] => eexists; split; eassumption
                | [ H : context[to_nonterminal (of_nonterminal _)] |- _ ]
                  => rewrite to_of_nonterminal in H by assumption
@@ -840,7 +839,7 @@ Section recursive_descent_parser.
           : dec (minimal_parse_of_item (G := G) len0 valid (substring offset (len0 - len0_minus_len) str) it).
           Proof.
             refine (match it return dec (minimal_parse_of_item len0 valid (substring offset _ str) it) with
-                      | Terminal P => if Sumbool.sumbool_of_bool (EqNat.beq_nat (len0 - len0_minus_len) 1 && char_at_matches offset str P)%bool
+                      | Terminal P => if Sumbool.sumbool_of_bool (Nat.eqb (len0 - len0_minus_len) 1 && char_at_matches offset str P)%bool
                                       then inl (match get offset str as g return get offset str = g -> _ with
                                                 | Some ch => fun H => MinParseTerminal _ _ _ ch _ _ _
                                                 | None => fun _ => !
@@ -904,7 +903,7 @@ Section recursive_descent_parser.
           : length (substring offset len str) = len.
         Proof.
           destruct Hlen; subst; rewrite substring_length; simpl;
-          apply Min.min_case_strong; omega.
+          apply Nat.min_case_strong; omega.
         Qed.
 
         Lemma dec_in_helper {ls it its offset len0_minus_len}
@@ -922,7 +921,7 @@ Section recursive_descent_parser.
           split; first [ intros [n [[H0 H1] H2]]
                        | intros [n [H0 [H1 H2]]] ].
           { destruct (le_lt_dec (len0 - len0_minus_len) n) as [pf|pf].
-            { rewrite Min.min_l in H0 by assumption.
+            { rewrite Nat.min_l in H0 by assumption.
               clear -H0 H1 H2 rdata cdata pf HSLP.
               induction ls as [|x xs IHxs]; destruct_head_hnf False.
               destruct (le_lt_dec (len0 - len0_minus_len) x).
@@ -931,14 +930,14 @@ Section recursive_descent_parser.
                 { left; reflexivity. }
                 { eapply expand_minimal_parse_of_item_beq; [ .. | eassumption ].
                   rewrite take_take, <- Nat.sub_min_distr_l.
-                  rewrite !Min.min_r by omega.
+                  rewrite !Nat.min_r by omega.
                   reflexivity. }
                 { eapply expand_minimal_parse_of_production_beq; [ .. | eassumption ].
                   rewrite drop_take, StringLike.drop_drop.
                   rewrite Nat.sub_add_distr.
-                  apply bool_eq_empty; rewrite substring_length; apply Min.min_case_strong; generalize dependent (len0 - len0_minus_len); intros; omega. } }
+                  apply bool_eq_empty; rewrite substring_length; apply Nat.min_case_strong; generalize dependent (len0 - len0_minus_len); intros; omega. } }
               { simpl in *.
-                rewrite Min.min_r in H0 by omega.
+                rewrite Nat.min_r in H0 by omega.
                 destruct IHxs as [n' [IH0 [IH1 IH2]]].
                 { destruct H0; try omega; assumption. }
                 { exists n'; repeat split; try assumption.
@@ -949,9 +948,9 @@ Section recursive_descent_parser.
                        | _ => progress destruct_head ex
                        | _ => progress destruct_head and
                        | [ H : context[min ?x ?y] |- _ ]
-                         => rewrite (Min.min_r x y) in H by omega
+                         => rewrite (Nat.min_r x y) in H by omega
                        | _ => progress subst
-                       | [ H : min ?x ?y < ?x |- _ ] => revert H; apply (Min.min_case_strong x y)
+                       | [ H : min ?x ?y < ?x |- _ ] => revert H; apply (Nat.min_case_strong x y)
                        | _ => intro
                        | _ => omega
                        | _ => assumption
@@ -959,21 +958,21 @@ Section recursive_descent_parser.
               { eapply expand_minimal_parse_of_item_beq; [ .. | eassumption ].
                 rewrite take_take.
                 rewrite <- Nat.sub_min_distr_l, sub_twice.
-                rewrite (Min.min_r len0) by omega.
+                rewrite (Nat.min_r len0) by omega.
                 reflexivity. }
               { eapply expand_minimal_parse_of_production_beq; [ .. | eassumption ].
                 rewrite drop_take, StringLike.drop_drop.
-                rewrite (plus_comm offset), Nat.sub_add_distr; reflexivity. } } }
+                rewrite (Nat.add_comm offset), Nat.sub_add_distr; reflexivity. } } }
           { exists n; repeat split; try assumption.
             { apply in_map; assumption. }
             { eapply expand_minimal_parse_of_item_beq; [ .. | eassumption ].
               rewrite take_take.
               rewrite <- Nat.sub_min_distr_l, sub_twice.
-              rewrite (Min.min_comm len0), <- !Min.min_assoc, (Min.min_r len0) by omega.
+              rewrite (Nat.min_comm len0), <- !Nat.min_assoc, (Nat.min_r len0) by omega.
               reflexivity. }
             { eapply expand_minimal_parse_of_production_beq; [ .. | eassumption ].
               rewrite drop_take, StringLike.drop_drop.
-              rewrite (plus_comm offset), Nat.sub_add_distr.
+              rewrite (Nat.add_comm offset), Nat.sub_add_distr.
               reflexivity. } }
         Defined.
 
@@ -997,7 +996,7 @@ Section recursive_descent_parser.
         Proof. subst; assumption. Defined.
 
         Lemma min_le_r {x y z} (H : y <= z) : min x y <= z.
-        Proof. apply Min.min_case_strong; omega. Qed.
+        Proof. apply Nat.min_case_strong; omega. Qed.
 
         Lemma lift_le {offset len n length_str} (H : len = 0 \/ offset + len <= length_str)
           : len - n = 0 \/ offset + n + (len - n) <= length_str.
@@ -1011,7 +1010,7 @@ Section recursive_descent_parser.
         Lemma lift_le_min {offset n len length_str} (H : len = 0 \/ offset + len <= length_str)
           : min n len = 0 \/ offset + min n len <= length_str.
         Proof.
-          apply Min.min_case_strong; [ | intro; assumption ].
+          apply Nat.min_case_strong; [ | intro; assumption ].
           destruct H; subst; [ left | right ]; omega.
         Qed.
 
@@ -1036,21 +1035,21 @@ Section recursive_descent_parser.
           destruct H as [pi pp]; split.
           { eapply expand_minimal_parse_of_item_beq; [ | eassumption ].
             rewrite take_take, <- Nat.sub_min_distr_l, sub_twice.
-            rewrite (Min.min_comm len0), <- !Min.min_assoc, min_minus_r.
+            rewrite (Nat.min_comm len0), <- !Nat.min_assoc, min_minus_r.
             reflexivity. }
           { eapply expand_minimal_parse_of_production_beq; [ | eassumption ].
-            rewrite drop_take, StringLike.drop_drop, (plus_comm a offset), Nat.sub_add_distr.
+            rewrite drop_take, StringLike.drop_drop, (Nat.add_comm a offset), Nat.sub_add_distr.
             reflexivity. }
         Defined.
 
         Local Ltac parse_production'_for_t' :=
           idtac;
           match goal with
-            | [ H : (beq_nat _ _) = true |- _ ] => apply EqNat.beq_nat_true in H
+            | [ H : (Nat.eqb _ _) = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
             | _ => progress subst
             | _ => solve [ constructor; assumption
                          | constructor;
-                           rewrite substring_length; apply Min.min_case_strong; omega ]
+                           rewrite substring_length; apply Nat.min_case_strong; omega ]
             | [ H : minimal_parse_of_production _ _ _ nil |- _ ] => (inversion H; clear H)
             | [ H : minimal_parse_of_production _ _ _ (_::_) |- _ ] => (inversion H; clear H)
             | [ H : ?x = 0, H' : context[?x] |- _ ] => rewrite H in H'
@@ -1058,12 +1057,12 @@ Section recursive_descent_parser.
             | _ => discriminate
             | [ H : forall x, (_ * _)%type -> _ |- _ ] => specialize (fun x y z => H x (y, z))
             | _ => solve [ eauto with nocore ]
-            | _ => solve [ apply Min.min_case_strong; omega ]
+            | _ => solve [ apply Nat.min_case_strong; omega ]
             | _ => omega
             | [ H : or _ _ |- _ ] => let H0 := fresh in destruct H as [H0|H0]; try clear H
             | [ H : length (substring _ _ _) = 0 |- _ ] => rewrite substring_length in H
-            | [ H : context[min] |- _ ] => rewrite Min.min_l in H by omega
-            | [ H : context[min] |- _ ] => rewrite Min.min_r in H by omega
+            | [ H : context[min] |- _ ] => rewrite Nat.min_l in H by omega
+            | [ H : context[min] |- _ ] => rewrite Nat.min_r in H by omega
             | [ H : _ |- _ ] => rewrite Nat.add_sub in H
           end.
         Local Ltac parse_production'_for_t := repeat parse_production'_for_t'.
@@ -1124,7 +1123,7 @@ Section recursive_descent_parser.
         Lemma substring_length_le_helper {offset len0_minus_len}
           : length (substring offset (len0 - len0_minus_len) str) <= len0.
         Proof.
-          rewrite substring_length; apply Min.min_case_strong; omega.
+          rewrite substring_length; apply Nat.min_case_strong; omega.
         Qed.
 
         Lemma Hlen_sub_more {offset n len0_minus_len}
@@ -1132,14 +1131,14 @@ Section recursive_descent_parser.
             -> len0 - max (len0 - n) len0_minus_len = 0 \/
                offset + (len0 - max (len0 - n) len0_minus_len) <= length str.
         Proof.
-          clear; intros [Hlen|Hlen]; [ left | right ]; apply Max.max_case_strong; omega.
+          clear; intros [Hlen|Hlen]; [ left | right ]; apply Nat.max_case_strong; omega.
         Qed.
 
         Lemma Hlen_sub_some {n len0_minus_len offset}
           : len0 - len0_minus_len = 0 \/ offset + (len0 - len0_minus_len) <= length str
             -> len0 - max (len0 - n) len0_minus_len <= len0.
         Proof.
-          apply Max.max_case_strong; omega.
+          apply Nat.max_case_strong; omega.
         Qed.
 
         Lemma Hlen_sub_helper {offset n len0_minus_len}
@@ -1183,7 +1182,7 @@ Section recursive_descent_parser.
                     dec (minimal_parse_of_production (G := G) len0 valid (substring offset (len0 - len0_minus_len) str) ps))
                ((** 0-length production, only accept empty *)
                  fun idx Hidx Hreachable Hvalid offset len0_minus_len Hlen
-                 => match Utils.dec (beq_nat (len0 - len0_minus_len) 0) with
+                 => match Utils.dec (Nat.eqb (len0 - len0_minus_len) 0) with
                       | left H => inl _
                       | right H => inr (fun p => _)
                     end)
@@ -1529,9 +1528,9 @@ Section recursive_descent_parser.
               | [ H : is_true ?e, H' : context[?e] |- _ ] => rewrite H in H'
               | [ H : context[andb _ true] |- _ ] => rewrite Bool.andb_true_r in H
               | [ H : negb _ = false |- _ ] => apply Bool.negb_false_iff in H
-              | [ H : beq_nat _ _ = true |- _ ] => apply beq_nat_true in H
-              | [ H : context[beq_nat ?x 0] |- context[pred ?x] ] => is_var x; destruct x
-              | [ H : _ <= 0 |- _ ] => apply le_n_0_eq in H
+              | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
+              | [ H : context[Nat.eqb ?x 0] |- context[pred ?x] ] => is_var x; destruct x
+              | [ H : _ <= 0 |- _ ] => apply (fun n Hle => eq_sym (proj1 (Nat.le_0_r n) Hle)) in H
               | [ H : 0 = _ |- _ ] => symmetry in H
               | [ H : nonterminals_length ?v = 0, H' : context[is_valid_nonterminal ?v ?nt] |- _ ]
                 => rewrite nonterminals_length_zero in H' by assumption
@@ -1555,24 +1554,24 @@ Section recursive_descent_parser.
               | [ H : or _ _ |- _ ] => let H0 := fresh in destruct H as [H0|H0]; try clear H
               | [ |- context[length (substring _ _ _)] ]
                 => rewrite substring_length
-              | _ => apply Min.min_case_strong; omega
+              | _ => apply Nat.min_case_strong; omega
               | [ H : ?x = 0 \/ ?T |- _ ]
                 => destruct (Compare_dec.zerop x);
                   [ clear H | assert T by (destruct H; try assumption; omega); clear H ]
               | [ |- context[min ?x ?y - ?y] ]
-                => rewrite <- Nat.sub_min_distr_r, minus_diag, Min.min_0_r
+                => rewrite <- Nat.sub_min_distr_r, Nat.sub_diag, Nat.min_0_r
               | _ => rewrite Nat.add_sub
-              | _ => rewrite Min.min_r by omega
-              | _ => rewrite Min.min_l by omega
+              | _ => rewrite Nat.min_r by omega
+              | _ => rewrite Nat.min_l by omega
               | [ H : context[length (substring _ 0 _)] |- _ ]
                 => rewrite take_length in H
               | [ H : context[length (substring _ _ _)] |- _ ]
-                => rewrite substring_length, Min.min_r, Nat.add_sub in H by omega
+                => rewrite substring_length, Nat.min_r, Nat.add_sub in H by omega
               | [ H : context[?x - (?x - _)] |- _ ] => rewrite sub_twice in H
-              | [ H : context[min ?x ?y] |- _ ] => rewrite (Min.min_r x y) in H by assumption
-              | [ H : context[min ?x ?y] |- _ ] => rewrite (Min.min_l x y) in H by assumption
-              | [ H : context[min ?x ?x] |- _ ] => rewrite Min.min_idempotent in H
-              | [ H : context[?x - ?x] |- _ ] => rewrite minus_diag in H
+              | [ H : context[min ?x ?y] |- _ ] => rewrite (Nat.min_r x y) in H by assumption
+              | [ H : context[min ?x ?y] |- _ ] => rewrite (Nat.min_l x y) in H by assumption
+              | [ H : context[min ?x ?x] |- _ ] => rewrite Nat.min_idempotent in H
+              | [ H : context[?x - ?x] |- _ ] => rewrite Nat.sub_diag in H
               | [ H : context[?x - 0] |- _ ] => rewrite Nat.sub_0_r in H
             end.
           Local Ltac p_step := repeat p_step_t'.
@@ -1612,7 +1611,7 @@ Section recursive_descent_parser.
                                 (or_introl pf')
                                 initial_nonterminals_data
                                 (reflexivity _)
-                                offset (len - len0_minus_len) Hlen (le_minus _ _) nt)
+                                offset (len - len0_minus_len) Hlen (Nat.le_sub_l _ _) nt)
                           offset (len - len)
                           (Hlen_helper_sub_sub Hlen)
                           (nonterminal_to_production nt))
@@ -1628,7 +1627,7 @@ Section recursive_descent_parser.
                         (fun _ => _)
                         (fun is_valid => _)
                         (fun is_valid => _)
-                        (Sumbool.sumbool_of_bool (negb (EqNat.beq_nat valid_len 0) && is_valid_nonterminal valid nt)));
+                        (Sumbool.sumbool_of_bool (negb (Nat.eqb valid_len 0) && is_valid_nonterminal valid nt)));
               [ ((** It was valid, so we can remove it *)
                   edestruct (fun pf'' pf'''
                             => @parse_productions'
@@ -1640,7 +1639,7 @@ Section recursive_descent_parser.
                                        (or_intror (conj eq_refl pf''))
                                        (remove_nonterminal valid nt)
                                        pf''' offset (len0 - len0_minus_len)
-                                       Hlen (le_minus _ _))
+                                       Hlen (Nat.le_sub_l _ _))
                                  offset (len0 - len)
                                  (Hlen_helper_sub_sub Hlen)
                                  (nonterminal_to_production nt))
@@ -1794,7 +1793,7 @@ Section recursive_descent_parser.
           : dec (minimal_parse_of_nonterminal (G := G) (length str) initial_nonterminals_data (substring 0 (length str - 0) str) (to_nonterminal nt)).
           Proof.
             destruct (parse_nonterminal'_substring nt) as [p|p]; [ left | right ];
-              rewrite <- minus_n_O;
+              rewrite Nat.sub_0_r;
               exact p.
           Defined.
 
@@ -1819,7 +1818,7 @@ Section recursive_descent_parser.
           Proof.
             rewrite <- drop_0 at 1.
             erewrite <- take_long at 1 by reflexivity.
-            rewrite drop_length, <- minus_n_O.
+            rewrite drop_length, Nat.sub_0_r.
             expand_once.
             destruct (Utils.dec (is_valid_nonterminal initial_nonterminals_data nt)) as [H|H];
               repeat eq_t'.
@@ -1842,8 +1841,7 @@ Section recursive_descent_parser.
           Proof.
             R_etransitivity_eq; [ eapply parse_nonterminal'_substring_correct | ].
             unfold parse_nonterminal'_substring_minus.
-            edestruct parse_nonterminal'_substring;
-              destruct (minus_n_O (length str)); reflexivity.
+            edestruct parse_nonterminal'_substring; reflexivity.
           Qed.
 
           Lemma parse_nonterminal'_correct
@@ -1969,7 +1967,7 @@ Section recursive_descent_parser.
       erewrite <- take_long at 1 by reflexivity;
       rewrite drop_length(*, <- minus_n_O at 1*).
     Local Ltac substring_to_str :=
-      repeat rewrite <- minus_n_O at 1; rewrite drop_0, take_long at 1 by reflexivity.
+      repeat rewrite Nat.sub_0_r at 1; rewrite drop_0, take_long at 1 by reflexivity.
 
     Lemma Hlen0 {lenstr} : lenstr - 0 = 0 \/ 0 + (lenstr - 0) <= lenstr.
     Proof. omega. Qed.
@@ -1987,7 +1985,7 @@ Section recursive_descent_parser.
         destruct parse_item_substring as [p|np];
         [ left | right; intro p; apply np; clear np ];
         (eapply expand_minimal_parse_of_item_beq; [ | eassumption ]);
-        clear -HSLP; abstract (rewrite <- minus_n_O, substring_correct3'; reflexivity).
+        clear -HSLP; abstract (rewrite Nat.sub_0_r, substring_correct3'; reflexivity).
       Defined.
 
       Lemma parse_item_substring_correct
@@ -1998,7 +1996,7 @@ Section recursive_descent_parser.
       Proof.
         str_to_substring.
         unfold GenericRecognizer.parse_item.
-        rewrite (minus_n_O (length str)) at 6;
+        rewrite <-(Nat.sub_0_r (length str)) at 6;
           apply parse_item'_all_correct; intro; substring_to_str.
         apply parse_nonterminal'_substring_minus_correct.
       Qed.
@@ -2024,17 +2022,17 @@ Section recursive_descent_parser.
       Definition parse_production_substring_minus
         : dec (minimal_parse_of_production (G := G) (length str) initial_nonterminals_data (substring 0 (length str - 0) str) (to_production p)).
       Proof.
-        eapply parse_production'; [ | right; clear; apply le_minus | reflexivity.. | assumption | assumption ].
+        eapply parse_production'; [ | right; clear; apply Nat.le_sub_l | reflexivity.. | assumption | assumption ].
         intros.
         eapply (@parse_nonterminal_or_abort (length str, _));
-          simpl; try reflexivity; subst; try assumption; apply le_minus.
+          simpl; try reflexivity; subst; try assumption; apply Nat.le_sub_l.
       Defined.
 
       Definition parse_production_substring
         : dec (minimal_parse_of_production (G := G) (length str) initial_nonterminals_data (substring 0 (length str) str) (to_production p)).
       Proof.
         destruct parse_production_substring_minus as [p'|p']; [ left | right ];
-          rewrite <- minus_n_O in p';
+          rewrite Nat.sub_0_r in p';
           exact p'.
       Defined.
 
@@ -2070,7 +2068,7 @@ Section recursive_descent_parser.
         R_etransitivity_eq; [ eapply parse_production_substring_minus_correct | ].
         unfold parse_production_substring.
         destruct parse_production_substring_minus;
-          destruct (minus_n_O (length str)); reflexivity.
+          destruct (Nat.sub_0_r (length str)); reflexivity.
       Qed.
 
       Lemma parse_production_correct
@@ -2094,17 +2092,17 @@ Section recursive_descent_parser.
       Definition parse_productions_substring_minus
         : dec (minimal_parse_of (G := G) (length str) initial_nonterminals_data (substring 0 (length str - 0) str) (List.map to_production ps)).
       Proof.
-        eapply parse_productions'; [ | right; apply le_minus | reflexivity.. | assumption | assumption ].
+        eapply parse_productions'; [ | right; apply Nat.le_sub_l | reflexivity.. | assumption | assumption ].
         intros.
         eapply (@parse_nonterminal_or_abort (length str, _));
-          simpl; try reflexivity; subst; try apply le_minus; assumption.
+          simpl; try reflexivity; subst; try apply Nat.le_sub_l; assumption.
       Defined.
 
       Definition parse_productions_substring
         : dec (minimal_parse_of (G := G) (length str) initial_nonterminals_data (substring 0 (length str) str) (List.map to_production ps)).
       Proof.
         destruct parse_productions_substring_minus as [p'|p']; [ left | right ];
-          rewrite <- minus_n_O in p';
+          rewrite Nat.sub_0_r in p';
           exact p'.
       Defined.
 
@@ -2137,7 +2135,7 @@ Section recursive_descent_parser.
         R_etransitivity_eq; [ eapply parse_productions_substring_minus_correct | ].
         unfold parse_productions_substring.
         destruct parse_productions_substring_minus;
-          destruct (minus_n_O (length str)); reflexivity.
+          destruct (Nat.sub_0_r (length str)); reflexivity.
       Qed.
 
       Lemma parse_productions_correct
diff --git a/src/Parsers/GenericRecognizerOptimized.v b/src/Parsers/GenericRecognizerOptimized.v
index 0727dae96..c483af3c3 100644
--- a/src/Parsers/GenericRecognizerOptimized.v
+++ b/src/Parsers/GenericRecognizerOptimized.v
@@ -1,7 +1,7 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Definition of an optimized CFG parser-recognizer *)
 Require Import Coq.Lists.List Coq.Strings.String.
-Require Import Coq.Numbers.Natural.Peano.NPeano Coq.Arith.Compare_dec Coq.Arith.Wf_nat.
+Require Import Coq.Arith.PeanoNat Coq.Arith.Compare_dec Coq.Arith.Wf_nat.
 Require Import Fiat.Common.List.Operations.
 Require Import Fiat.Common.List.ListMorphisms.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
@@ -36,7 +36,6 @@ Require Export Fiat.Common.Sigma.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
 Require Import Fiat.Parsers.GenericRecognizerOptimizedTactics.
-Import NPeano.
 
 Set Implicit Arguments.
 Local Open Scope string_like_scope.
@@ -333,7 +332,7 @@ Section recursive_descent_parser.
                                               (fun _ => _)
                                               (N0 a' b' c')
                                               (list_rect P0 (fun _ _ _ => ret_production_nil_true) C0 ls' a' b' c')
-                                              (EqNat.beq_nat (List.length ls') 0))
+                                              (Nat.eqb (List.length ls') 0))
                                            = list_rect P0 N0 C0 ls' a' b' c')
                              _
                              _
@@ -360,7 +359,7 @@ Section recursive_descent_parser.
             end. }
           { simpl.
             match goal with
-            | [ |- context[EqNat.beq_nat (List.length ?ls) 0] ]
+            | [ |- context[Nat.eqb (List.length ?ls) 0] ]
               => is_var ls; destruct ls; simpl; try reflexivity
             end; [].
             repeat match goal with
@@ -416,13 +415,13 @@ Section recursive_descent_parser.
             repeat optsplit_t'.
             { apply (f_equal2 ret_production_cons); [ | reflexivity ].
               repeat misc_opt'.
-              rewrite Min.min_idempotent.
+              rewrite Nat.min_idempotent.
               reflexivity. }
             { apply (f_equal2 ret_production_cons).
               { apply (f_equal3 (fun (b : bool) x y => if b then x else y)); [ | reflexivity | reflexivity ].
                 apply (f_equal2 andb); [ | reflexivity ].
                 match goal with
-                | [ |- _ = EqNat.beq_nat (min ?v ?x) ?v ]
+                | [ |- _ = Nat.eqb (min ?v ?x) ?v ]
                   => refine (_ : Compare_dec.leb v x = _)
                 end.
                 match goal with
@@ -592,8 +591,8 @@ Section recursive_descent_parser.
               => transitivity (if (orb (negb b') b) then t else f); [ | destruct b, b'; reflexivity ]
             end; fin_step_opt.
             match goal with
-            | [ |- context[negb (EqNat.beq_nat ?x 0)] ]
-              => replace (negb (EqNat.beq_nat x 0)) with (Compare_dec.leb 1 x)
+            | [ |- context[negb (Nat.eqb ?x 0)] ]
+              => replace (negb (Nat.eqb x 0)) with (Compare_dec.leb 1 x)
                 by (destruct x as [|[]]; reflexivity)
             end.
             reflexivity. }
@@ -615,8 +614,8 @@ Section recursive_descent_parser.
               => transitivity (if (orb (negb b') b) then t else f); [ | destruct b, b'; reflexivity ]
             end; fin_step_opt.
             match goal with
-            | [ |- context[negb (EqNat.beq_nat ?x 0)] ]
-              => replace (negb (EqNat.beq_nat x 0)) with (Compare_dec.leb 1 x)
+            | [ |- context[negb (Nat.eqb ?x 0)] ]
+              => replace (negb (Nat.eqb x 0)) with (Compare_dec.leb 1 x)
                 by (destruct x as [|[]]; reflexivity)
             end.
             reflexivity. }
diff --git a/src/Parsers/GenericRecognizerOptimizedTactics.v b/src/Parsers/GenericRecognizerOptimizedTactics.v
index 111104d2e..475c51791 100644
--- a/src/Parsers/GenericRecognizerOptimizedTactics.v
+++ b/src/Parsers/GenericRecognizerOptimizedTactics.v
@@ -1,6 +1,7 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Coq.Lists.List.
 Require Import Coq.Arith.Compare_dec.
+Require Import Coq.Arith.PeanoNat.
 Require Import Fiat.Parsers.BaseTypes.
 Require Import Fiat.Parsers.BaseTypesLemmas.
 Require Import Fiat.Parsers.GenericBaseTypes.
@@ -14,7 +15,6 @@ Require Import Fiat.Common.BoolFacts.
 Require Import Fiat.Common.NatFacts.
 Require Import Fiat.Common.List.ListFacts.
 Require Import Fiat.Common Fiat.Common.Wf Fiat.Common.Wf2 Fiat.Common.Equality.
-Import NPeano.
 
 Ltac t_reduce_fix :=
   repeat match goal with
@@ -66,9 +66,9 @@ Ltac t_reduce_fix :=
          | [ H : (_ && _)%bool = true |- _ ] => apply Bool.andb_true_iff in H
          | [ H : _ = in_left |- _ ] => clear H
          | [ H : _ /\ _ |- _ ] => destruct H
-         | [ H : context[negb (EqNat.beq_nat ?x ?y)] |- _ ] => destruct (EqNat.beq_nat x y) eqn:?
-         | [ H : EqNat.beq_nat _ _ = false |- _ ] => apply EqNat.beq_nat_false in H
-         | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
+         | [ H : context[negb (Nat.eqb ?x ?y)] |- _ ] => destruct (Nat.eqb x y) eqn:?
+         | [ H : Nat.eqb _ _ = false |- _ ] => apply (fun n m => proj1 (Nat.eqb_neq n m)) in H
+         | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
          | [ H : snd ?x = _ |- _ ] => is_var x; destruct x
          | _ => progress simpl negb in *
          | [ H : false = true |- _ ] => inversion H
@@ -200,7 +200,7 @@ Ltac fin_step_opt :=
          | [ |- _ = 0 ] => reflexivity
          | [ |- _ = 1 ] => reflexivity
          | [ |- _ = None ] => reflexivity
-         | [ |- _ = EqNat.beq_nat _ _ ] => apply f_equal2
+         | [ |- _ = Nat.eqb _ _ ] => apply f_equal2
          | [ |- _ = Compare_dec.leb _ _ ] => apply f_equal2
          | [ |- _ = S _ ] => apply f_equal
          | [ |- _ = string_beq _ _ ] => apply f_equal2
@@ -226,9 +226,9 @@ Ltac fin_step_opt :=
 Ltac misc_opt' :=
   idtac;
   match goal with
-  | _ => progress rewrite ?max_min_n, ?Minus.minus_diag, ?Nat.sub_0_r, ?uneta_bool, ?beq_nat_min_0(*, ?bool_rect_flatten*)
-  | _ => rewrite Min.min_l by assumption
-  | _ => rewrite Min.min_r by assumption
+  | _ => progress rewrite ?max_min_n, ?Nat.sub_diag, ?Nat.sub_0_r, ?uneta_bool, ?beq_nat_min_0(*, ?bool_rect_flatten*)
+  | _ => rewrite Nat.min_l by assumption
+  | _ => rewrite Nat.min_r by assumption
   | [ |- context[if ?ltb ?x ?y then _ else _] ] => rewrite if_to_min
   | [ |- context[min ?x ?y - ?x] ] => rewrite min_sub_same
   | [ |- context[(min ?x ?y - ?x)%natr] ] => rewrite min_subr_same
diff --git a/src/Parsers/MinimalParseOfParse.v b/src/Parsers/MinimalParseOfParse.v
index e95246021..f98ece70b 100644
--- a/src/Parsers/MinimalParseOfParse.v
+++ b/src/Parsers/MinimalParseOfParse.v
@@ -259,7 +259,7 @@ Section cfg.
     Proof.
       setoid_subst str'.
       rewrite take_length.
-      apply Min.min_case_strong; omega.
+      apply Nat.min_case_strong; omega.
     Qed.
 
     Lemma Hlen0_drop {len0 len0' str str' n}
@@ -526,8 +526,8 @@ Section cfg.
                    intro; apply H
                | _ => eapply H'; eassumption
                | _ => assumption
-               | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-               | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
              end.
     Defined.
 
@@ -536,7 +536,7 @@ Section cfg.
     : alt_option h valid str -> alt_option h' valid' str'.
     Proof.
       apply expand_alt_option'; try assumption.
-      apply Lt.lt_le_weak; assumption.
+      apply Nat.lt_le_incl; assumption.
     Defined.
 
     Section wf_parts.
@@ -606,7 +606,7 @@ Section cfg.
             eexists (MinimalParse.MinParseTerminal _ _ _ _ _ pf0 pf0');
               simpl; constructor. }
           { edestruct (fun pf => @minimal_parse_of_nonterminal__of__parse_of_nonterminal (S h') pf len0 _ valid' _ p') as [ [p'' H''] | p'' ];
-            try solve [ repeat (apply Lt.lt_n_Sn || apply Lt.lt_S)
+            try solve [ repeat (apply Nat.lt_succ_diag_r || apply Nat.lt_lt_succ_r)
                       | exact Hinit'
                       | exact H_h
                       | exact H_forall
@@ -693,8 +693,8 @@ Section cfg.
             | [ |- (_ * _)%type ]
               => split
             | [ H : _ <= _ |- _ <= _ ] => apply H
-            | _ => apply Le.le_n_S
-            | _ => apply Plus.plus_le_compat
+            | _ => apply le_n_S
+            | _ => apply Nat.add_le_mono
             | [ H0 : Forall_parse_of_item _ _,
                      H1 : Forall_parse_of_production _ _
                 |- Forall_parse_of_production _ _ ]
@@ -721,7 +721,7 @@ Section cfg.
             | [ H : context[min _ _] |- _ ] => rewrite min_r in H by omega
             | _ => rewrite drop_length
             | _
-              => solve [ eauto using le_S, Le.le_trans, Plus.le_plus_l, Plus.le_plus_r, drop_0, take_long, NPeano.Nat.eq_le_incl, bool_eq_empty, drop_length, (fun x y => proj2 (NPeano.Nat.sub_0_le x y)) with nocore ]
+              => solve [ eauto using le_S, Nat.le_trans, Nat.le_add_l, Nat.le_add_r, drop_0, take_long, Nat.eq_le_incl, bool_eq_empty, drop_length, (fun x y => proj2 (Nat.sub_0_le x y)) with nocore ]
           end.
         Local Ltac min_parse_prod_t := repeat min_parse_prod_t'.
 
@@ -765,24 +765,24 @@ Section cfg.
               repeat (reflexivity || esplit). }
           { specialize (fun h' pf
                         => @minimal_parse_of_nonterminal__of__parse_of_nonterminal
-                             h' (transitivity pf (Lt.lt_n_Sn _))).
+                             h' (transitivity pf (Nat.lt_succ_diag_r _))).
             change (S ((size_of_parse_item p0')
                        + (size_of_parse_production p1'))
                     < S h') in H_h.
-            apply Lt.lt_S_n in H_h.
-            pose proof (Lt.le_lt_trans _ _ _ (Plus.le_plus_l _ _) H_h) as H_h0.
-            pose proof (Lt.le_lt_trans _ _ _ (Plus.le_plus_r _ _) H_h) as H_h1.
+            apply Nat.succ_lt_mono in H_h.
+            pose proof (Nat.le_lt_trans _ _ _ (Nat.le_add_l _ _) H_h) as H_h0.
+            pose proof (Nat.le_lt_trans _ _ _ (Nat.le_add_r _ _) H_h) as H_h1.
             clear H_h.
             assert (pf0' : length (take n str') <= length str')
-              by (rewrite take_length; apply Min.min_case_strong; omega).
+              by (rewrite take_length; apply Nat.min_case_strong; omega).
             assert (pf1' : length (drop n str') <= length str')
               by (rewrite drop_length; omega).
-            pose proof (fun valid Hinit => @minimal_parse_of_item__of__parse_of_item' _ h'  minimal_parse_of_nonterminal__of__parse_of_nonterminal _ (transitivity pf0' pf) valid _ p0' H_h0 Hinit) as p_it.
-            pose proof (fun valid Hinit => @minimal_parse_of_production__of__parse_of_production' h' minimal_parse_of_nonterminal__of__parse_of_nonterminal _ (transitivity pf1' pf) valid _ p1' H_h1 Hinit) as p_prod.
+            pose proof (fun valid Hinit => @minimal_parse_of_item__of__parse_of_item' _ h'  minimal_parse_of_nonterminal__of__parse_of_nonterminal _ (transitivity pf0' pf) valid _ p0' H_h1 Hinit) as p_it.
+            pose proof (fun valid Hinit => @minimal_parse_of_production__of__parse_of_production' h' minimal_parse_of_nonterminal__of__parse_of_nonterminal _ (transitivity pf1' pf) valid _ p1' H_h0 Hinit) as p_prod.
             clear pf0' pf1'.
             destruct (le_lt_dec (length str') n) as [ Hle | Hle ], (zerop (min n (length str'))) as [Hstr' | Hstr' ].
             { (* empty, empty *)
-              rewrite Min.min_r in Hstr' by assumption.
+              rewrite Nat.min_r in Hstr' by assumption.
               specialize (p_it valid' Hinit'); specialize (p_prod valid' Hinit').
               destruct p_it as [ [ p0'' H0''] |], p_prod as [ [ p1'' H1'' ] |];
                 [ | | | ];
@@ -822,40 +822,40 @@ Section cfg.
           destruct p as [pat pats p' | pat pats p'].
           { clear minimal_parse_of_productions__of__parse_of_productions'.
             edestruct (@minimal_parse_of_production__of__parse_of_production' _ h' minimal_parse_of_nonterminal__of__parse_of_nonterminal _ pf valid' _ p') as [ [p'' p''H] | [nonterminal' H'] ];
-            try solve [ exact (Lt.lt_S_n _ _ H_h)
+            try solve [ exact (proj2 (Nat.succ_lt_mono _ _) H_h)
                       | exact H_forall
                       | exact Hinit' ];
             [|].
             { left.
               exists (MinParseHead pats p'').
               simpl.
-              exact (Le.le_n_S _ _ p''H). }
+              exact (le_n_S _ _ p''H). }
             { right.
               exists nonterminal'.
               split;
                 try solve [ exact (fst H') ];
                 [].
               exists (proj1_sig (snd H')).
-              exact (Lt.lt_S _ _ (proj2_sig (snd H'))). } }
+              exact (Nat.lt_lt_succ_r _ _ (proj2_sig (snd H'))). } }
           { specialize (fun h' pf
                         => @minimal_parse_of_nonterminal__of__parse_of_nonterminal
-                             h' (transitivity pf (Lt.lt_n_Sn _))).
+                             h' (transitivity pf (Nat.lt_succ_diag_r _))).
             edestruct (minimal_parse_of_productions__of__parse_of_productions' h'  minimal_parse_of_nonterminal__of__parse_of_nonterminal _ pf valid' _ p') as [ [p'' p''H] | [nonterminal' H'] ];
-            try solve [ exact (Lt.lt_S_n _ _ H_h)
+            try solve [ exact (proj2 (Nat.succ_lt_mono _ _) H_h)
                       | exact Hinit'
                       | exact H_forall ];
             [|].
             { left.
               exists (MinParseTail pat p'').
               simpl.
-              exact (Le.le_n_S _ _ p''H). }
+              exact (le_n_S _ _ p''H). }
             { right.
               exists nonterminal'.
               split;
                 try solve [ exact (fst H') ];
                 [].
               exists (proj1_sig (snd H')).
-              exact (Lt.lt_S _ _ (proj2_sig (snd H'))). } }
+              exact (Nat.lt_lt_succ_r _ _ (proj2_sig (snd H'))). } }
         Defined.
       End productions.
 
@@ -877,29 +877,29 @@ Section cfg.
 
             destruct (le_lt_eq_dec _ _ H) as [pf_lt|pf_eq].
             { (** [str] got smaller, so we reset the valid nonterminals list *)
-              destruct (@minimal_parse_of_productions__of__parse_of_productions' (length str) h minimal_parse_of_nonterminal__of__parse_of_nonterminal str (reflexivity _) initial_nonterminals_data (Lookup G nonterminal) p (Lt.lt_S_n _ _ pf) (reflexivity _)) as [p'|p'].
+              destruct (@minimal_parse_of_productions__of__parse_of_productions' (length str) h minimal_parse_of_nonterminal__of__parse_of_nonterminal str (reflexivity _) initial_nonterminals_data (Lookup G nonterminal) p (proj2 (Nat.succ_lt_mono _ _) pf) (reflexivity _)) as [p'|p'].
               { left.
                 exists (MinParseNonTerminalStrLt valid _ pf_lt (proj2 (initial_nonterminals_correct _) Hvalid') (proj1_sig p'));
                   simpl.
                 simpl in *.
-                exact (Le.le_n_S _ _ (proj2_sig p')). }
+                exact (le_n_S _ _ (proj2_sig p')). }
               { simpl.
                 right; eapply expand_alt_option; [..| exact p' ];
-                solve [ apply Lt.lt_n_Sn
+                solve [ apply Nat.lt_succ_diag_r
                       | assumption
                       | reflexivity ]. } }
             { (** [str] didn't get smaller, so we cache the fact that we've hit this nonterminal already *)
               destruct (Sumbool.sumbool_of_bool (is_valid_nonterminal valid (of_nonterminal nonterminal))) as [ Hvalid | Hinvalid ].
-              { destruct (@minimal_parse_of_productions__of__parse_of_productions' len0 h minimal_parse_of_nonterminal__of__parse_of_nonterminal str Hstr (remove_nonterminal valid (of_nonterminal nonterminal)) (Lookup G nonterminal) p (Lt.lt_S_n _ _ pf) (transitivity (R := sub_nonterminals_listT) (@sub_nonterminals_listT_remove _ _ _ _ _ _) Hinit')) as [p'|p'].
+              { destruct (@minimal_parse_of_productions__of__parse_of_productions' len0 h minimal_parse_of_nonterminal__of__parse_of_nonterminal str Hstr (remove_nonterminal valid (of_nonterminal nonterminal)) (Lookup G nonterminal) p (proj2 (Nat.succ_lt_mono _ _) pf) (transitivity (R := sub_nonterminals_listT) (@sub_nonterminals_listT_remove _ _ _ _ _ _) Hinit')) as [p'|p'].
                 { left.
                   exists (@MinimalParse.MinParseNonTerminalStrEq _ _ _ _ _ _ _ _ _ pf_eq (proj2 (initial_nonterminals_correct _) Hvalid') Hvalid (proj1_sig p')).
                   simpl in *.
-                  exact (Le.le_n_S _ _ (proj2_sig p')). }
+                  exact (le_n_S _ _ (proj2_sig p')). }
                 { destruct p' as [nonterminal' p'].
                   destruct (string_dec nonterminal nonterminal') as [|n].
                   { subst nonterminal; simpl in *.
                     edestruct (@minimal_parse_of_nonterminal__of__parse_of_nonterminal (S (size_of_parse p)) pf len0 _ valid nonterminal' (proj1_sig (snd p'))) as [p''|p''];
-                    try solve [ apply Lt.lt_n_S, (proj2_sig (snd p'))
+                    try solve [ apply (proj1 (Nat.succ_lt_mono _ _)), (proj2_sig (snd p'))
                               | subst; reflexivity
                               | assumption
                               | split; [ exact (proj2 (fst p'))
@@ -909,7 +909,7 @@ Section cfg.
                       exists (proj1_sig p'').
                       etransitivity;
                         [ exact (proj2_sig p'')
-                        | exact (Lt.lt_le_weak _ _ (Lt.lt_n_S _ _ (proj2_sig (snd p')))) ]. }
+                        | exact (Nat.lt_le_incl _ _ ((proj1 (Nat.succ_lt_mono _ _)) (proj2_sig (snd p')))) ]. }
                     { right.
                       exists (projT1 p'').
                       split;
@@ -918,7 +918,7 @@ Section cfg.
                       exists (proj1_sig (snd (projT2 p''))).
                       etransitivity;
                         [ exact (proj2_sig (snd (projT2 p'')))
-                        | exact (Lt.lt_n_S _ _ (proj2_sig (snd p'))) ]. } }
+                        | exact ((proj1 (Nat.succ_lt_mono _ _)) (proj2_sig (snd p'))) ]. } }
                   { right.
                     exists nonterminal'.
                     destruct p' as [p'H p'p].
@@ -945,7 +945,7 @@ Section cfg.
                                   end.
                       exact p'H. }
                     { exists (proj1_sig p'p).
-                      exact (Lt.lt_S _ _ (proj2_sig p'p)). } } } }
+                      exact (Nat.lt_lt_succ_r _ _ (proj2_sig p'p)). } } } }
               { (** oops, we already saw this nonterminal in the past.  ABORT! *)
                 right.
                 exists nonterminal.
@@ -953,7 +953,7 @@ Section cfg.
                 split; [ split; assumption
                        | ].
                 exists p.
-                apply Lt.lt_n_Sn. } }
+                apply Nat.lt_succ_diag_r. } }
           Defined.
         End step.
 
diff --git a/src/Parsers/ParserImplementation.v b/src/Parsers/ParserImplementation.v
index da2a00c7d..da6259600 100644
--- a/src/Parsers/ParserImplementation.v
+++ b/src/Parsers/ParserImplementation.v
@@ -87,11 +87,11 @@ Section implementation.
                            (parse_of_item__of__minimal_parse_of_item pit)
                            (parse_of_production__of__minimal_parse_of_production pits)).
           specialize_by assumption.
-          rewrite Min.min_r by assumption.
+          rewrite Nat.min_r by assumption.
           apply H'; eauto. }
         { exists (length (substring offset len str)).
           specialize (H' _ (reflexivity _)).
-          rewrite Min.min_idempotent.
+          rewrite Nat.min_idempotent.
           rewrite !substring_length_no_min in * by assumption.
           repeat match goal with
                    | [ H : context[length (substring _ _ _)] |- _ ] => rewrite !substring_length_no_min in H by assumption
diff --git a/src/Parsers/Reachable/All/MinimalReachableOfReachable.v b/src/Parsers/Reachable/All/MinimalReachableOfReachable.v
index a70d1ccf3..0aa10217c 100644
--- a/src/Parsers/Reachable/All/MinimalReachableOfReachable.v
+++ b/src/Parsers/Reachable/All/MinimalReachableOfReachable.v
@@ -160,8 +160,8 @@ Section cfg.
                    intro; apply H
                | _ => eapply H'; eassumption
                | _ => assumption
-               | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-               | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
              end.
     Defined.
 
@@ -170,7 +170,7 @@ Section cfg.
     : alt_option h valid -> alt_option h' valid'.
     Proof.
       apply expand_alt_option'; try assumption.
-      apply Lt.lt_le_weak; assumption.
+      apply Nat.lt_le_incl; assumption.
     Defined.
 
     Section wf_parts.
@@ -221,7 +221,7 @@ Section cfg.
           intros valid' pats p H_h Hinit'.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_reachable_from_production__of__reachable_from_production' h' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))).
-          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Le.le_n_Sn _)).
+          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ? ? p' | ? ? p' ].
           { destruct (fun k => minimal_reachable_from_item__of__reachable_from_item valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -229,18 +229,18 @@ Section cfg.
             | left | right ].
             { eexists (MinReachableProductionHead _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_reachable_from_production__of__reachable_from_production' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_reachable_from_production__of__reachable_from_production' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { eexists (MinReachableProductionTail _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End production.
@@ -257,7 +257,7 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_reachable_from_productions__of__reachable_from_productions' h' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))).
           pose proof (minimal_reachable_from_production__of__reachable_from_production' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))) as minimal_reachable_from_production__of__reachable_from_production''.
-          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Le.le_n_Sn _)).
+          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ? ? p' | ? ? p' ].
           { destruct (fun k => minimal_reachable_from_production__of__reachable_from_production'' valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -265,18 +265,18 @@ Section cfg.
             | left | right ].
             { eexists (MinReachableHead _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_reachable_from_productions__of__reachable_from_productions' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_reachable_from_productions__of__reachable_from_productions' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { eexists (MinReachableTail _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End productions.
@@ -294,12 +294,12 @@ Section cfg.
           { left.
             eexists (MinReachableTerminal _ _ _ Pch); simpl; constructor. }
           { case_eq (is_valid_nonterminal valid' (of_nonterminal nonterminal')); intro H'''.
-            { edestruct (fun k => @minimal_reachable_from_productions__of__reachable_from_productions' _ (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (Le.le_n_S _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
+            { edestruct (fun k => @minimal_reachable_from_productions__of__reachable_from_productions' _ (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ ((fun n m => proj1 (Nat.succ_le_mono n m)) _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
               as [ [ p'' H'' ] | [ nt'' H'' ] ];
             [ solve [ auto with arith ]
             | left | ].
             { eexists (MinReachableNonTerminal _ _ H''' p'').
-              apply Le.le_n_S; eassumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); eassumption. }
             { destruct (string_dec nonterminal' nt''); subst.
               { destruct H'' as [ H'' [ p'' H'''' ] ].
                 simpl in *.
@@ -341,7 +341,7 @@ Section cfg.
                   eassumption. }
                 { destruct_head sigT.
                   eexists.
-                  apply Lt.lt_S; eassumption. } } } }
+                  apply Nat.lt_lt_succ_r; eassumption. } } } }
             { right.
               exists nonterminal'; repeat split; trivial; [].
               exists p'.
diff --git a/src/Parsers/Reachable/MaybeEmpty/MinimalOfCore.v b/src/Parsers/Reachable/MaybeEmpty/MinimalOfCore.v
index c28f26f0a..bb40bee5c 100644
--- a/src/Parsers/Reachable/MaybeEmpty/MinimalOfCore.v
+++ b/src/Parsers/Reachable/MaybeEmpty/MinimalOfCore.v
@@ -154,8 +154,8 @@ Section cfg.
                    intro; apply H
                | _ => eapply H'; eassumption
                | _ => assumption
-               | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-               | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
              end.
     Defined.
 
@@ -164,7 +164,7 @@ Section cfg.
     : alt_option h valid -> alt_option h' valid'.
     Proof.
       apply expand_alt_option'; try assumption.
-      apply Lt.lt_le_weak; assumption.
+      apply Nat.lt_le_incl; assumption.
     Defined.
 
     Section wf_parts.
@@ -227,7 +227,7 @@ Section cfg.
           intros valid' pats p H_h Hinit'.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_maybe_empty_production__of__maybe_empty_production' h' (fun h'' pf => minimal_maybe_empty_item__of__maybe_empty_item _ (le_S _ _ pf))).
-          specialize (minimal_maybe_empty_item__of__maybe_empty_item h' (Le.le_n_Sn _)).
+          specialize (minimal_maybe_empty_item__of__maybe_empty_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ | ?? p0' p1' ].
           { left; eexists (MinMaybeEmptyProductionNil _); simpl; reflexivity. }
           { assert (size_of_maybe_empty_item p0' < h') by exact (lt_helper_1 H_h).
@@ -242,14 +242,14 @@ Section cfg.
               | right ].
             { eexists (MinMaybeEmptyProductionCons p0'' p1'').
               simpl in *.
-              apply Le.le_n_S, Plus.plus_le_compat; assumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)), Nat.add_le_mono; assumption. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | apply lt_helper_1'
                         | apply lt_helper_2'
                         | reflexivity ]. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | apply lt_helper_1'
                         | apply lt_helper_2'
                         | reflexivity ]. } }
@@ -268,7 +268,7 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_maybe_empty_productions__of__maybe_empty_productions' h' (fun h'' pf => minimal_maybe_empty_item__of__maybe_empty_item _ (le_S _ _ pf))).
           pose proof (minimal_maybe_empty_production__of__maybe_empty_production' (fun h'' pf => minimal_maybe_empty_item__of__maybe_empty_item _ (le_S _ _ pf))) as minimal_maybe_empty_production__of__maybe_empty_production''.
-          specialize (minimal_maybe_empty_item__of__maybe_empty_item h' (Le.le_n_Sn _)).
+          specialize (minimal_maybe_empty_item__of__maybe_empty_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ? ? p' | ? ? p' ].
           { destruct (fun k => minimal_maybe_empty_production__of__maybe_empty_production'' valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -276,18 +276,18 @@ Section cfg.
             | left | right ].
             { eexists (MinMaybeEmptyHead _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_maybe_empty_productions__of__maybe_empty_productions' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_maybe_empty_productions__of__maybe_empty_productions' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { eexists (MinMaybeEmptyTail _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End productions.
@@ -303,12 +303,12 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           destruct p as [nonterminal' H' p'].
           { case_eq (is_valid_nonterminal valid' (of_nonterminal nonterminal')); intro H'''.
-            { edestruct (fun k => @minimal_maybe_empty_productions__of__maybe_empty_productions' _ (fun h'' pf => minimal_maybe_empty_item__of__maybe_empty_item _ (Le.le_n_S _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
+            { edestruct (fun k => @minimal_maybe_empty_productions__of__maybe_empty_productions' _ (fun h'' pf => minimal_maybe_empty_item__of__maybe_empty_item _ ((fun n m => proj1 (Nat.succ_le_mono n m)) _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
               as [ [ p'' H'' ] | [ nt'' H'' ] ];
             [ solve [ auto with arith ]
             | left | ].
             { eexists (MinMaybeEmptyNonTerminal _ _ H''' p'').
-              apply Le.le_n_S; eassumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); eassumption. }
             { destruct (string_dec nonterminal' nt''); subst.
               { destruct H'' as [ H'' [ p'' H'''' ] ].
                 simpl in *.
@@ -350,7 +350,7 @@ Section cfg.
                   eassumption. }
                 { destruct_head sigT.
                   eexists.
-                  apply Lt.lt_S; eassumption. } } } }
+                  apply Nat.lt_lt_succ_r; eassumption. } } } }
             { right.
               exists nonterminal'; repeat split; trivial; [].
               exists p'.
diff --git a/src/Parsers/Reachable/MaybeEmpty/OfParse.v b/src/Parsers/Reachable/MaybeEmpty/OfParse.v
index bf1304f67..9255b36bc 100644
--- a/src/Parsers/Reachable/MaybeEmpty/OfParse.v
+++ b/src/Parsers/Reachable/MaybeEmpty/OfParse.v
@@ -63,7 +63,7 @@ Section cfg.
       { constructor. }
       { constructor.
         { apply (fun k => @parse_empty_maybe_empty_parse_of_item' parse_empty_maybe_empty_parse_of_productions valid0 _ Hsub _ k _ (fst Hforall)).
-          rewrite take_length, Hlen, Min.min_0_r; reflexivity. }
+          rewrite take_length, Hlen, Nat.min_0_r; reflexivity. }
         { apply (fun k => @parse_empty_maybe_empty_parse_of_production valid0 _ Hsub _ k _ (snd Hforall)).
           rewrite drop_length, Hlen; reflexivity. } } }
   Defined.
@@ -163,7 +163,7 @@ Section cfg.
         { eexists (ParseProductionCons _ 0 (projT1 p0) (projT1 p1)).
           exact (projT2 p0, projT2 p1). }
         { rewrite ?take_length, ?drop_length, Hlen; reflexivity. }
-        { rewrite ?take_length, ?drop_length, Hlen, Min.min_0_r; reflexivity. } } }
+        { rewrite ?take_length, ?drop_length, Hlen, Nat.min_0_r; reflexivity. } } }
   Defined.
 
   Definition parse_empty_from_maybe_empty_parse_of_item
diff --git a/src/Parsers/Reachable/OnlyFirst/MinimalReachableOfReachable.v b/src/Parsers/Reachable/OnlyFirst/MinimalReachableOfReachable.v
index f463a7a1c..e51400d72 100644
--- a/src/Parsers/Reachable/OnlyFirst/MinimalReachableOfReachable.v
+++ b/src/Parsers/Reachable/OnlyFirst/MinimalReachableOfReachable.v
@@ -162,8 +162,8 @@ Section cfg.
                    intro; apply H
                | _ => eapply H'; eassumption
                | _ => assumption
-               | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-               | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
              end.
     Defined.
 
@@ -172,7 +172,7 @@ Section cfg.
     : alt_option h valid -> alt_option h' valid'.
     Proof.
       apply expand_alt_option'; try assumption.
-      apply Lt.lt_le_weak; assumption.
+      apply Nat.lt_le_incl; assumption.
     Defined.
 
     Section wf_parts.
@@ -223,7 +223,7 @@ Section cfg.
           intros valid' pats p H_h Hinit'.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_reachable_from_production__of__reachable_from_production' h' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))).
-          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Le.le_n_Sn _)).
+          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ?? p' | ?? p_empty p' ].
           { destruct (fun k => minimal_reachable_from_item__of__reachable_from_item valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -231,18 +231,18 @@ Section cfg.
             | left | right ].
             { eexists (MinReachableProductionHead _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_reachable_from_production__of__reachable_from_production' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_reachable_from_production__of__reachable_from_production' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { exists (MinReachableProductionTail (minimal_maybe_empty_item__of__maybe_empty_item Hsub0 p_empty) p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End production.
@@ -259,7 +259,7 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_reachable_from_productions__of__reachable_from_productions' h' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))).
           pose proof (minimal_reachable_from_production__of__reachable_from_production' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))) as minimal_reachable_from_production__of__reachable_from_production''.
-          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Le.le_n_Sn _)).
+          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ? ? p' | ? ? p' ].
           { destruct (fun k => minimal_reachable_from_production__of__reachable_from_production'' valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -267,18 +267,18 @@ Section cfg.
             | left | right ].
             { eexists (MinReachableHead _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_reachable_from_productions__of__reachable_from_productions' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_reachable_from_productions__of__reachable_from_productions' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { eexists (MinReachableTail _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End productions.
@@ -296,12 +296,12 @@ Section cfg.
           { left.
             eexists (MinReachableTerminal _ _ _ _ Pch); simpl; constructor. }
           { case_eq (is_valid_nonterminal valid' (of_nonterminal nonterminal')); intro H'''.
-            { edestruct (fun k => @minimal_reachable_from_productions__of__reachable_from_productions' _ (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (Le.le_n_S _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
+            { edestruct (fun k => @minimal_reachable_from_productions__of__reachable_from_productions' _ (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ ((fun n m => proj1 (Nat.succ_le_mono n m)) _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
               as [ [ p'' H'' ] | [ nt'' H'' ] ];
             [ solve [ auto with arith ]
             | left | ].
             { eexists (MinReachableNonTerminal _ _ H''' p'').
-              apply Le.le_n_S; eassumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); eassumption. }
             { destruct (string_dec nonterminal' nt''); subst.
               { destruct H'' as [ H'' [ p'' H'''' ] ].
                 simpl in *.
@@ -343,7 +343,7 @@ Section cfg.
                   eassumption. }
                 { destruct_head sigT.
                   eexists.
-                  apply Lt.lt_S; eassumption. } } } }
+                  apply Nat.lt_lt_succ_r; eassumption. } } } }
             { right.
               exists nonterminal'; repeat split; trivial; [].
               exists p'.
diff --git a/src/Parsers/Reachable/OnlyLast/MinimalReachableOfReachable.v b/src/Parsers/Reachable/OnlyLast/MinimalReachableOfReachable.v
index 1e89d6988..f61c2c19f 100644
--- a/src/Parsers/Reachable/OnlyLast/MinimalReachableOfReachable.v
+++ b/src/Parsers/Reachable/OnlyLast/MinimalReachableOfReachable.v
@@ -162,8 +162,8 @@ Section cfg.
                    intro; apply H
                | _ => eapply H'; eassumption
                | _ => assumption
-               | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-               | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+               | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
              end.
     Defined.
 
@@ -172,7 +172,7 @@ Section cfg.
     : alt_option h valid -> alt_option h' valid'.
     Proof.
       apply expand_alt_option'; try assumption.
-      apply Lt.lt_le_weak; assumption.
+      apply Nat.lt_le_incl; assumption.
     Defined.
 
     Section wf_parts.
@@ -223,7 +223,7 @@ Section cfg.
           intros valid' pats p H_h Hinit'.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_reachable_from_production__of__reachable_from_production' h' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))).
-          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Le.le_n_Sn _)).
+          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ?? p_empty p' | ?? p' ].
           { destruct (fun k => minimal_reachable_from_item__of__reachable_from_item valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -231,18 +231,18 @@ Section cfg.
             | left | right ].
             { exists (MinReachableProductionHead (minimal_maybe_empty_production__of__maybe_empty_production Hsub0 p_empty) p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_reachable_from_production__of__reachable_from_production' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_reachable_from_production__of__reachable_from_production' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { eexists (MinReachableProductionTail _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End production.
@@ -259,7 +259,7 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_reachable_from_productions__of__reachable_from_productions' h' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))).
           pose proof (minimal_reachable_from_production__of__reachable_from_production' (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (le_S _ _ pf))) as minimal_reachable_from_production__of__reachable_from_production''.
-          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Le.le_n_Sn _)).
+          specialize (minimal_reachable_from_item__of__reachable_from_item h' (Nat.le_succ_diag_r _)).
           destruct p as [ ? ? p' | ? ? p' ].
           { destruct (fun k => minimal_reachable_from_production__of__reachable_from_production'' valid' _ p' k Hinit')
               as [ [p'' H''] | p'' ];
@@ -267,18 +267,18 @@ Section cfg.
             | left | right ].
             { eexists (MinReachableHead _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
-          { destruct (minimal_reachable_from_productions__of__reachable_from_productions' valid' _ p' (Lt.lt_S_n _ _ H_h) Hinit')
+          { destruct (minimal_reachable_from_productions__of__reachable_from_productions' valid' _ p' ((fun n m => proj2 (Nat.succ_lt_mono n m)) _ _ H_h) Hinit')
               as [ [p'' H''] | p'' ];
             [ left | right ].
             { eexists (MinReachableTail _ p'').
               simpl in *.
-              apply Le.le_n_S; exact H''. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); exact H''. }
             { eapply expand_alt_option; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | reflexivity ]. } }
         Defined.
       End productions.
@@ -296,12 +296,12 @@ Section cfg.
           { left.
             eexists (MinReachableTerminal _ _ _ _ Pch); simpl; constructor. }
           { case_eq (is_valid_nonterminal valid' (of_nonterminal nonterminal')); intro H'''.
-            { edestruct (fun k => @minimal_reachable_from_productions__of__reachable_from_productions' _ (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ (Le.le_n_S _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
+            { edestruct (fun k => @minimal_reachable_from_productions__of__reachable_from_productions' _ (fun h'' pf => minimal_reachable_from_item__of__reachable_from_item _ ((fun n m => proj1 (Nat.succ_le_mono n m)) _ _ pf)) (remove_nonterminal valid' (of_nonterminal nonterminal')) _ p' k)
               as [ [ p'' H'' ] | [ nt'' H'' ] ];
             [ solve [ auto with arith ]
             | left | ].
             { eexists (MinReachableNonTerminal _ _ H''' p'').
-              apply Le.le_n_S; eassumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); eassumption. }
             { destruct (string_dec nonterminal' nt''); subst.
               { destruct H'' as [ H'' [ p'' H'''' ] ].
                 simpl in *.
@@ -343,7 +343,7 @@ Section cfg.
                   eassumption. }
                 { destruct_head sigT.
                   eexists.
-                  apply Lt.lt_S; eassumption. } } } }
+                  apply Nat.lt_lt_succ_r; eassumption. } } } }
             { right.
               exists nonterminal'; repeat split; trivial; [].
               exists p'.
diff --git a/src/Parsers/Reachable/ParenBalanced/Core.v b/src/Parsers/Reachable/ParenBalanced/Core.v
index f9d9c6858..3b03ff8a5 100644
--- a/src/Parsers/Reachable/ParenBalanced/Core.v
+++ b/src/Parsers/Reachable/ParenBalanced/Core.v
@@ -84,7 +84,7 @@ pb = pb' '+' 0
 
   Definition pb_new_level_fun (P : Char -> bool) (start_level : nat) : list nat
     := List.uniquize
-         EqNat.beq_nat
+         Nat.eqb
          (List.map
             (fun ch => pb_new_level ch start_level)
             (List.filter
@@ -96,7 +96,7 @@ pb = pb' '+' 0
     <-> ((exists ch, P ch) /\ forall ch, P ch -> new_level = pb_new_level ch start_level).
   Proof.
     unfold pb_new_level_fun.
-    rewrite (uniquize_singleton (beq := EqNat.beq_nat) bl lb).
+    rewrite (uniquize_singleton (beq := Nat.eqb) bl lb).
     setoid_rewrite in_map_iff.
     setoid_rewrite filter_In.
     setoid_rewrite (fun x => @and_TrueP_L (In x (enumerate Char)) (P x = true) (enumerate_correct _)).
diff --git a/src/Parsers/Reachable/ParenBalanced/MinimalOfCore.v b/src/Parsers/Reachable/ParenBalanced/MinimalOfCore.v
index 851db61fa..8a1a42d5d 100644
--- a/src/Parsers/Reachable/ParenBalanced/MinimalOfCore.v
+++ b/src/Parsers/Reachable/ParenBalanced/MinimalOfCore.v
@@ -142,8 +142,8 @@ Section cfg.
                  | _ => eapply H'; eassumption
                  | _ => assumption
                  | _ => eassumption
-                 | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-                 | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+                 | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+                 | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
                end.
         Unshelve.
         intros ??????; subst; assumption.
@@ -155,7 +155,7 @@ Section cfg.
       : alt_option h valid -> alt_option h' valid'.
       Proof.
         apply expand_alt_option'; try assumption.
-        apply Lt.lt_le_weak; assumption.
+        apply Nat.lt_le_incl; assumption.
       Defined.
     End alt_option.
 
@@ -222,7 +222,7 @@ Section cfg.
           intros valid' n pat p H_h Hinit'.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_pb'_production__of__pb'_production' h' (fun h'' pf => minimal_pb'_item__of__pb'_item' _ (le_S _ _ pf))).
-          specialize (minimal_pb'_item__of__pb'_item' h' (Lt.lt_n_Sn _)).
+          specialize (minimal_pb'_item__of__pb'_item' h' (Nat.lt_succ_diag_r _)).
           subst_body.
           destruct p as [ | ?? nt' ?? p0' p1' | valid ?????? p' ]; simpl_size_of.
           { left; eexists (PBProductionNil _ _ _); reflexivity. }
@@ -244,15 +244,15 @@ Section cfg.
               { eexists (PBProductionConsNonTerminal _ Hnt' p0'' p1'').
                 rewrite ?expand_size_of_pb'_production, ?expand_size_of_pb'_productions in *.
                 simpl_size_of.
-                apply Le.le_n_S, Plus.plus_le_compat; assumption. }
+                apply (fun n m => proj1 (Nat.succ_le_mono n m)), Nat.add_le_mono; assumption. }
               { eapply expand_alt_option'; [ .. | eassumption ];
-                try solve [ apply Lt.lt_n_Sn
+                try solve [ apply Nat.lt_succ_diag_r
                           | apply lt_helper_1'
                           | apply lt_helper_2'
                           | reflexivity
                           | omega ]. }
               { eapply expand_alt_option'; [ .. | eassumption ];
-                try solve [ apply Lt.lt_n_Sn
+                try solve [ apply Nat.lt_succ_diag_r
                           | apply lt_helper_1'
                           | apply lt_helper_2'
                           | reflexivity
@@ -274,10 +274,10 @@ Section cfg.
             { eexists (PBProductionConsTerminal _ _ _ _ p0'').
               rewrite ?expand_size_of_pb'_production, ?expand_size_of_pb'_productions in *.
               simpl_size_of.
-              apply Le.le_n_S; assumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); assumption. }
             { unfold pb_new_level in *.
               eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | apply lt_helper_1'
                         | apply lt_helper_2'
                         | reflexivity
@@ -300,7 +300,7 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_pb'_productions__of__pb'_productions' h' (fun h'' pf => minimal_pb'_item__of__pb'_item _ (le_S _ _ pf))).
           pose proof (minimal_pb'_production__of__pb'_production' (fun h'' pf => minimal_pb'_item__of__pb'_item _ (le_S _ _ pf))) as minimal_pb'_production__of__pb'_production''.
-          specialize (minimal_pb'_item__of__pb'_item h' (Lt.lt_n_Sn _)).
+          specialize (minimal_pb'_item__of__pb'_item h' (Nat.lt_succ_diag_r _)).
           destruct p as [ | ??? p0' p1' ].
           { left.
             eexists (PBNil _ _ _).
@@ -327,11 +327,11 @@ Section cfg.
               simpl_size_of.
               omega. }
             { eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | simpl_size_of; omega
                         | reflexivity ]. }
             { eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | simpl_size_of; omega
                         | reflexivity ]. } }
         Defined.
diff --git a/src/Parsers/Reachable/ParenBalanced/OfParse.v b/src/Parsers/Reachable/ParenBalanced/OfParse.v
index 77fd9e96e..272f0661d 100644
--- a/src/Parsers/Reachable/ParenBalanced/OfParse.v
+++ b/src/Parsers/Reachable/ParenBalanced/OfParse.v
@@ -74,7 +74,7 @@ Section cfg.
         rewrite take_length.
         unfold paren_balanced'_step, pb_check_level, pb_new_level in *.
         split_iff.
-        apply Min.min_case_strong;
+        apply Nat.min_case_strong;
           repeat match goal with
                    | _ => progress subst
                    | _ => progress intros
diff --git a/src/Parsers/Reachable/ParenBalancedHiding/MinimalOfCore.v b/src/Parsers/Reachable/ParenBalancedHiding/MinimalOfCore.v
index 346215696..5adecc422 100644
--- a/src/Parsers/Reachable/ParenBalancedHiding/MinimalOfCore.v
+++ b/src/Parsers/Reachable/ParenBalancedHiding/MinimalOfCore.v
@@ -156,8 +156,8 @@ Section cfg.
                  | _ => eapply H'; eassumption
                  | _ => assumption
                  | _ => eassumption
-                 | [ |- _ < _ ] => eapply Lt.lt_trans; eassumption
-                 | [ |- _ < _ ] => eapply Lt.lt_le_trans; eassumption
+                 | [ |- _ < _ ] => eapply Nat.lt_trans; eassumption
+                 | [ |- _ < _ ] => eapply Nat.lt_le_trans; eassumption
                end.
         Unshelve.
         unfold respectful; intros; subst; assumption.
@@ -170,7 +170,7 @@ Section cfg.
       : alt_option h valid -> alt_option h' valid'.
       Proof.
         apply expand_alt_option'; try assumption.
-        apply Lt.lt_le_weak; assumption.
+        apply Nat.lt_le_incl; assumption.
       Defined.
     End alt_option.
 
@@ -237,7 +237,7 @@ Section cfg.
           intros valid' n pat p H_h Hinit'.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_pbh'_production__of__pbh'_production' h' (fun h'' pf => minimal_pbh'_item__of__pbh'_item' _ (le_S _ _ pf))).
-          specialize (minimal_pbh'_item__of__pbh'_item' h' (Lt.lt_n_Sn _)).
+          specialize (minimal_pbh'_item__of__pbh'_item' h' (Nat.lt_succ_diag_r _)).
           destruct p as [ | ? nt' ?? p0' p1' | ?? nt' ? p0' p' | ??????? p' ]; simpl_size_of.
           { left; eexists (PBHProductionNil _ _ _ _); reflexivity. }
           { assert (size_of_pbh'_productions p0' < h') by exact (lt_helper_1 H_h).
@@ -260,15 +260,15 @@ Section cfg.
               { eexists (PBHProductionConsNonTerminal0 _ Hnt' p0'' p1'').
                 rewrite ?expand_size_of_pbh'_production, ?expand_size_of_pbh'_productions in *.
                 simpl_size_of.
-                apply Le.le_n_S, Plus.plus_le_compat; assumption. }
+                apply (fun n m => proj1 (Nat.succ_le_mono n m)), Nat.add_le_mono; assumption. }
               { eapply expand_alt_option'; [ .. | eassumption ];
-                try solve [ apply Lt.lt_n_Sn
+                try solve [ apply Nat.lt_succ_diag_r
                           | apply lt_helper_1'
                           | apply lt_helper_2'
                           | reflexivity
                           | omega ]. }
               { eapply expand_alt_option'; [ .. | eassumption ];
-                try solve [ apply Lt.lt_n_Sn
+                try solve [ apply Nat.lt_succ_diag_r
                           | apply lt_helper_1'
                           | apply lt_helper_2'
                           | reflexivity
@@ -293,9 +293,9 @@ Section cfg.
             { eexists (PBHProductionConsNonTerminalS p0' p1'').
               rewrite ?expand_size_of_pbh'_production, ?expand_size_of_pbh'_productions in *.
               simpl_size_of.
-              apply Le.le_n_S; assumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); assumption. }
             { eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | apply lt_helper_1'
                         | apply lt_helper_2'
                         | reflexivity
@@ -311,10 +311,10 @@ Section cfg.
             { eexists (PBHProductionConsTerminal _ _ _ _ p0'').
               rewrite ?expand_size_of_pbh'_production, ?expand_size_of_pbh'_productions in *.
               simpl_size_of.
-              apply Le.le_n_S; assumption. }
+              apply (fun n m => proj1 (Nat.succ_le_mono n m)); assumption. }
             { unfold pb_new_level in *.
               eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | apply lt_helper_1'
                         | apply lt_helper_2'
                         | reflexivity
@@ -337,7 +337,7 @@ Section cfg.
           destruct h as [|h']; [ exfalso; omega | ].
           specialize (minimal_pbh'_productions__of__pbh'_productions' h' (fun h'' pf => minimal_pbh'_item__of__pbh'_item _ (le_S _ _ pf))).
           pose proof (minimal_pbh'_production__of__pbh'_production' (fun h'' pf => minimal_pbh'_item__of__pbh'_item _ (le_S _ _ pf))) as minimal_pbh'_production__of__pbh'_production''.
-          specialize (minimal_pbh'_item__of__pbh'_item h' (Lt.lt_n_Sn _)).
+          specialize (minimal_pbh'_item__of__pbh'_item h' (Nat.lt_succ_diag_r _)).
           destruct p as [ | ??? p0' p1' ].
           { left.
             eexists (PBHNil _ _ _ _).
@@ -364,11 +364,11 @@ Section cfg.
               simpl_size_of.
               omega. }
             { eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | simpl_size_of; omega
                         | reflexivity ]. }
             { eapply expand_alt_option'; [ .. | eassumption ];
-              try solve [ apply Lt.lt_n_Sn
+              try solve [ apply Nat.lt_succ_diag_r
                         | simpl_size_of; omega
                         | reflexivity ]. } }
         Defined.
diff --git a/src/Parsers/RecognizerPreOptimized.v b/src/Parsers/RecognizerPreOptimized.v
index 247f724a5..43fc5334f 100644
--- a/src/Parsers/RecognizerPreOptimized.v
+++ b/src/Parsers/RecognizerPreOptimized.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Coq.Lists.List.
 Require Import Coq.ZArith.ZArith.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
 Require Import Fiat.Parsers.ContextFreeGrammar.PreNotations.
 Require Import Fiat.Parsers.BaseTypes.
@@ -49,6 +48,8 @@ Section recursive_descent_parser.
   Local Arguments minus !_ !_.
   Local Arguments min !_ !_.
 
+  Local Notation minus_plus := (fun n m : nat => eq_ind_r (fun n0 : nat => n0 - n = m) (Nat.add_sub m n) (Nat.add_comm n m)).
+
   Local Program Instance optsplitdata_correct : @boolean_parser_completeness_dataT' _ _ _ G optdata
     := { split_string_for_production_complete := _ }.
   Next Obligation.
@@ -77,7 +78,7 @@ Section recursive_descent_parser.
            | [ |- context[match ?e with nil => _ | _ => _ end] ]
              => destruct e eqn:?
            | _ => progress simpl
-           | _ => rewrite Min.min_0_r
+           | _ => rewrite Nat.min_0_r
            | _ => intro
            | [ |- context[0 = min _ _] ] => exists 0
            | [ |- ?x = ?x ] => reflexivity
@@ -89,7 +90,7 @@ Section recursive_descent_parser.
            | [ H : is_true (take _ (substring _ 0 _) ~= [_]) |- _ ]
              => apply length_singleton in H
            | [ H : length (take _ (substring _ 0 _)) = S _ |- _ ]
-             => rewrite take_length, substring_length, <- Nat.sub_min_distr_r, Nat.add_sub, !Min.min_0_r in H
+             => rewrite take_length, substring_length, <- Nat.sub_min_distr_r, Nat.add_sub, !Nat.min_0_r in H
            | [ H : MinimalParse.minimal_parse_of_production _ _ _ nil |- _ ] => inversion H; clear H
            | [ |- MinimalParse.minimal_parse_of_production _ _ _ nil ] => constructor
            | [ H : MinimalParse.minimal_parse_of_item _ _ _ (Terminal _) |- _ ]
@@ -106,22 +107,22 @@ Section recursive_descent_parser.
            | [ |- (_ * _)%type ] => split
            | [ |- { _ : nat & _ } ] => eexists; repeat split; [ left; reflexivity | .. ]
            | [ H : ?x + ?y <= _ |- context[(?y + ?x)%nat] ]
-             => not constr_eq x y; rewrite (Plus.plus_comm y x)
+             => not constr_eq x y; rewrite (Nat.add_comm y x)
            | [ H : ?x + ?y <= _, H' : context[(?y + ?x)%nat] |- _ ]
-             => not constr_eq x y; rewrite (Plus.plus_comm y x) in H'
-           | [ H : context[(?x + 1)%nat] |- _ ] => rewrite (Plus.plus_comm x 1) in H; simpl plus in H
-           | [ H : context[min ?x ?y], H' : ?y <= ?x |- _ ] => rewrite (Min.min_r x y) in H by assumption
-           | [ H' : ?y <= ?x |- context[min ?x ?y] ] => rewrite (Min.min_r x y) by assumption
+             => not constr_eq x y; rewrite (Nat.add_comm y x) in H'
+           | [ H : context[(?x + 1)%nat] |- _ ] => rewrite (Nat.add_comm x 1) in H; simpl plus in H
+           | [ H : context[min ?x ?y], H' : ?y <= ?x |- _ ] => rewrite (Nat.min_r x y) in H by assumption
+           | [ H' : ?y <= ?x |- context[min ?x ?y] ] => rewrite (Nat.min_r x y) by assumption
            | [ H : _ - _ = 0 |- _ ] => apply Nat.sub_0_le in H
            | [ |- _ - _ = 0 ] => apply Nat.sub_0_le
-           | [ H : _ |- _ ] => progress rewrite ?Nat.add_sub, ?Minus.minus_plus in H
-           | _ => progress rewrite ?Nat.add_sub, ?Minus.minus_plus, ?Minus.minus_diag, ?Min.min_idempotent
+           | [ H : _ |- _ ] => progress rewrite ?Nat.add_sub, ?minus_plus in H
+           | _ => progress rewrite ?Nat.add_sub, ?minus_plus, ?Nat.sub_diag, ?Nat.min_idempotent
            | [ |- is_true (is_char (take ?x (take ?x _)) _) ]
              => rewrite take_take
            | [ H : is_true (is_char (take ?n ?str) ?ch) |- is_true (is_char ?str ?ch) ]
              => rewrite (take_long str)
                in H
-               by (rewrite substring_length, Plus.plus_comm, Min.min_r by assumption; omega)
+               by (rewrite substring_length, Nat.add_comm, Nat.min_r by assumption; omega)
            | [ H : is_true (is_char (take ?n ?str) ?ch) |- is_true (is_char (take 1 ?str) ?ch) ]
              => apply take_n_1_singleton in H
            | [ |- MinimalParse.minimal_parse_of_production _ _ _ (_::_) ]
@@ -137,21 +138,21 @@ Section recursive_descent_parser.
              => apply length_singleton in H; rewrite take_length, substring_length in H
            | [ H : min ?x ?y = 1
                |- ?R (drop ?x (substring _ ?y _)) (drop 1 (substring _ ?y _)) ]
-             => revert H; apply Min.min_case_strong; intros; subst;
+             => revert H; apply Nat.min_case_strong; intros; subst;
                 try reflexivity;
                 apply bool_eq_empty
            | [ |- context[S ?x - ?x] ]
-             => rewrite <- Nat.add_1_r, Minus.minus_plus
+             => rewrite <- Nat.add_1_r, minus_plus
            | [ |- context[take ?x (take 0 _)] ]
              => rewrite take_take
            | [ |- context[min _ ?x - ?x] ]
              => rewrite <- Nat.sub_min_distr_r
            | [ H : ?y <= ?x |- context[take ?x (substring ?z ?y ?str)] ]
              => rewrite (take_long (substring z y str))
-               by (rewrite substring_length, Plus.plus_comm, Min.min_r by assumption; omega)
+               by (rewrite substring_length, Nat.add_comm, Nat.min_r by assumption; omega)
            | [ |- context[take ?x (substring ?z ?x ?str)] ]
              => rewrite (take_long (substring z x str))
-               by (rewrite substring_length, Plus.plus_comm, Min.min_r by assumption; omega)
+               by (rewrite substring_length, Nat.add_comm, Nat.min_r by assumption; omega)
            end.
   Qed.
 End recursive_descent_parser.
diff --git a/src/Parsers/Refinement/BinOpBrackets/BinOpRules.v b/src/Parsers/Refinement/BinOpBrackets/BinOpRules.v
index 4eda0fca7..75130af5e 100644
--- a/src/Parsers/Refinement/BinOpBrackets/BinOpRules.v
+++ b/src/Parsers/Refinement/BinOpBrackets/BinOpRules.v
@@ -70,7 +70,7 @@ Section helper_lemmas.
                => apply FirstChar.for_first_char_exists in H;
                  [
                      | rewrite !drop_length; clear -H' H''; omega ]
-             | [ H : _ < min _ _ |- _ ] => apply NPeano.Nat.min_glb_lt_iff in H
+             | [ H : _ < min _ _ |- _ ] => apply Nat.min_glb_lt_iff in H
            end.
     eapply (paren_balanced_hiding'_prefix);
       [ exact H_hiding
@@ -122,7 +122,7 @@ Section refine_rules.
     destruct (List.nth n' table None); try reflexivity.
     rewrite substring_length.
     destruct Hlen; subst; simpl;
-    apply Min.min_case_strong; omega.
+    apply Nat.min_case_strong; omega.
   Qed.
 
   Section general_table.
@@ -157,7 +157,7 @@ Section refine_rules.
         destruct (nth n table None) eqn:nth_equals.
         {
           destruct (Compare_dec.le_lt_dec (StringLike.length s_str) n0) as [|g].
-          { rewrite Min.min_l by assumption.
+          { rewrite Nat.min_l by assumption.
             left.
             reflexivity. }
           { exfalso.
@@ -168,21 +168,21 @@ Section refine_rules.
             subst s_str.
             rewrite take_take in H_nt_hiding.
             refine (paren_balanced_hiding'_prefix__index_points_to_binop _ H_nt_hiding Hn0_binop Hn0_balanced _).
-            rewrite <- Min.min_assoc.
+            rewrite <- Nat.min_assoc.
             rewrite <- take_length.
-            rewrite Min.min_idempotent.
+            rewrite Nat.min_idempotent.
             assumption.
           }
         }
         {
-          rewrite Min.min_idempotent.
+          rewrite Nat.min_idempotent.
           tauto.
         }
       }
 
       destruct Hlen as [Hlen|Hlen].
       { subst s_str m; exfalso; revert isnonzero.
-        rewrite drop_length, substring_length; apply Min.min_case_strong; simpl; intros; omega. }
+        rewrite drop_length, substring_length; apply Nat.min_case_strong; simpl; intros; omega. }
 
       subst s_str.
       rewrite take_take in H_nt_hiding.
@@ -238,7 +238,7 @@ Section refine_rules.
                | [ H : _ <= StringLike.length _ |- _ ] => rewrite take_length in H
                | [ H : context[min ?x ?y], H' : ?x <= min ?y _ |- _ ]
                  => replace (min x y) with x in H
-                                             by (revert H'; clear; abstract (repeat apply Min.min_case_strong; intros; omega))
+                                             by (revert H'; clear; abstract (repeat apply Nat.min_case_strong; intros; omega))
                | _ => progress subst
              end. *)
       destruct (List.nth n table None) as [idx|].
@@ -246,7 +246,7 @@ Section refine_rules.
         left.
         destruct (Compare_dec.lt_eq_lt_dec idx idx') as [ [?|?]|?];
           [
-          | subst; apply Min.min_r; omega
+          | subst; apply Nat.min_r; omega
           | ];
           exfalso.
         { (** idx < idx'; this contradicts the paren-balanced-hiding
@@ -257,7 +257,7 @@ Section refine_rules.
           apply paren_balanced_hiding_impl_paren_balanced' in Htable1; [ | exact _ .. ].
           eapply paren_balanced_hiding'_prefix__index_points_to_binop; try eassumption.
           rewrite <- take_length, <- take_take, take_length.
-          apply Min.min_case_strong; intros; try assumption; omega.
+          apply Nat.min_case_strong; intros; try assumption; omega.
 
  }
         { (** idx' < idx; this contradicts the paren-balanced-hiding
@@ -272,9 +272,9 @@ Section refine_rules.
             |
             |
             | eassumption ].
-          { apply Min.min_case_strong; omega. }
-          { rewrite Min.min_l
-              by (rewrite substring_length, Min.min_r in Hsmall by omega;
+          { apply Nat.min_case_strong; omega. }
+          { rewrite Nat.min_l
+              by (rewrite substring_length, Nat.min_r in Hsmall by omega;
                   omega).
             rewrite drop_take, take_take in H.
             apply take_n_1_singleton in H.
@@ -289,23 +289,23 @@ Section refine_rules.
           [
           | solve [ repeat
                       match goal with
-                        | _ => rewrite Min.min_l in Hsmall by omega
-                        | _ => rewrite Min.min_l in isnonzero by omega
-                        | _ => rewrite Min.min_l by assumption
+                        | _ => rewrite Nat.min_l in Hsmall by omega
+                        | _ => rewrite Nat.min_l in isnonzero by omega
+                        | _ => rewrite Nat.min_l by assumption
                         | [ H : is_true (_ ~= [ _ ]) |- _ ] => apply length_singleton in H
                         | [ H : _ |- _ ] => progress rewrite ?drop_length, ?take_length in H
                         | _ => progress rewrite ?drop_length, ?take_length
-                        | [ H : min _ _ = _ |- _ ] => revert H; apply Min.min_case_strong; clear; intros; omega
+                        | [ H : min _ _ = _ |- _ ] => revert H; apply Nat.min_case_strong; clear; intros; omega
                         | _ => omega
                       end ] ].
         destruct Htable' as [ch' [Ht0 Ht1] ].
-        rewrite substring_length, Min.min_r, NPeano.Nat.add_sub in Hsmall by omega.
+        rewrite substring_length, Nat.min_r, Nat.add_sub in Hsmall by omega.
         repeat match goal with
-                 | _ => rewrite Min.min_idempotent
+                 | _ => rewrite Nat.min_idempotent
                  | [ |- _ \/ False ] => left
                  | [ H : 0 < ?x - ?y |- ?x = ?y ] => exfalso
-                 | [ H : context[min] |- _ ] => rewrite Min.min_r in H by omega
-                 | [ H : context[min] |- _ ] => rewrite Min.min_l in H by omega
+                 | [ H : context[min] |- _ ] => rewrite Nat.min_r in H by omega
+                 | [ H : context[min] |- _ ] => rewrite Nat.min_l in H by omega
                  | [ H : is_true (is_char (substring _ _ (substring _ _ _)) _) |- _ ]
                    => rewrite substring_substring in H;
                      apply take_n_1_singleton in H
diff --git a/src/Parsers/Refinement/BinOpBrackets/MakeBinOpTable.v b/src/Parsers/Refinement/BinOpBrackets/MakeBinOpTable.v
index c40dabd70..d9fda10ea 100644
--- a/src/Parsers/Refinement/BinOpBrackets/MakeBinOpTable.v
+++ b/src/Parsers/Refinement/BinOpBrackets/MakeBinOpTable.v
@@ -171,7 +171,7 @@ pb = pb' '+' 0
     { specialize (IHlen (drop 1 str)).
       rewrite drop_length, H' in IHlen.
       simpl in IHlen.
-      specialize (IHlen (NPeano.Nat.sub_0_r _)).
+      specialize (IHlen (Nat.sub_0_r _)).
       rewrite list_of_next_bin_ops'_recr.
       destruct (singleton_exists (take 1 str)) as [ch H''].
       { rewrite take_length, H'; reflexivity. }
@@ -203,7 +203,7 @@ pb = pb' '+' 0
       { specialize (IHlen (drop 1 str)).
         rewrite drop_length, H' in IHlen.
         simpl in IHlen.
-        rewrite NPeano.Nat.sub_0_r in IHlen.
+        rewrite Nat.sub_0_r in IHlen.
         specialize (IHlen (pred n) eq_refl).
         rewrite list_of_next_bin_ops'_recr.
         destruct (singleton_exists (take 1 str)) as [ch H''].
@@ -228,7 +228,7 @@ pb = pb' '+' 0
   : index_points_to_binop (S offset) index str <-> index_points_to_binop offset index (drop 1 str).
   Proof.
     unfold index_points_to_binop.
-    rewrite ?drop_drop, !NPeano.Nat.add_1_r; simpl.
+    rewrite ?drop_drop, !Nat.add_1_r; simpl.
     reflexivity.
   Qed.
 
@@ -266,7 +266,7 @@ pb = pb' '+' 0
   : index_not_points_to_binop (S offset) index str <-> index_not_points_to_binop offset index (drop 1 str).
   Proof.
     unfold index_not_points_to_binop.
-    rewrite ?drop_drop, !NPeano.Nat.add_1_r; simpl.
+    rewrite ?drop_drop, !Nat.add_1_r; simpl.
     reflexivity.
   Qed.
 
@@ -339,7 +339,7 @@ pb = pb' '+' 0
     unfold cell_of_next_bin_ops_spec'' in *; simpl in *.
     rewrite !index_points_to_binop_S1.
     rewrite !index_not_points_to_binop_S1.
-    rewrite drop_drop, NPeano.Nat.add_1_r in H.
+    rewrite drop_drop, Nat.add_1_r in H.
     assumption.
   Qed.
 
@@ -386,9 +386,9 @@ pb = pb' '+' 0
              | [ H : option_map _ ?x = None |- _ ] => destruct x eqn:?; simpl in H
              | [ H : forall x, _ = _ -> @?T x |- _ ] => specialize (H _ eq_refl)
              | [ H : _ = _ -> ?T |- _ ] => specialize (H eq_refl)
-             | [ H : context[_ - 0] |- _ ] => rewrite NPeano.Nat.sub_0_r in H
-             | [ |- context[_ - 0] ] => rewrite NPeano.Nat.sub_0_r
-             | [ H : context[(_ + 1)%nat] |- _ ] => rewrite NPeano.Nat.add_1_r in H || setoid_rewrite NPeano.Nat.add_1_r in H
+             | [ H : context[_ - 0] |- _ ] => rewrite Nat.sub_0_r in H
+             | [ |- context[_ - 0] ] => rewrite Nat.sub_0_r
+             | [ H : context[(_ + 1)%nat] |- _ ] => rewrite Nat.add_1_r in H || setoid_rewrite Nat.add_1_r in H
              | [ H : ?x > 0 |- _ ] => is_var x; destruct x; [ exfalso; clear -H; omega | clear dependent H ]
              | [ H : ~ ?x > 0 |- _ ] => is_var x; destruct x; [ clear dependent H | exfalso; clear -H; omega ]
              | [ H : 0 > 0 |- _ ] => exfalso; clear -H; omega
diff --git a/src/Parsers/Refinement/BinOpBrackets/ParenBalanced.v b/src/Parsers/Refinement/BinOpBrackets/ParenBalanced.v
index 302f4f566..337adc78b 100644
--- a/src/Parsers/Refinement/BinOpBrackets/ParenBalanced.v
+++ b/src/Parsers/Refinement/BinOpBrackets/ParenBalanced.v
@@ -173,7 +173,7 @@ Section specific.
         { simpl in *.
           setoid_rewrite drop_drop in IHlen.
           setoid_rewrite take_drop in IHlen.
-          rewrite !NPeano.Nat.add_1_r in IHlen.
+          rewrite !Nat.add_1_r in IHlen.
           specialize (fun H' level1 H => IHlen level1 level2 H H').
           intros H1 H2.
           specialize_by assumption.
@@ -362,7 +362,7 @@ Section specific.
           destruct level1, level2; simpl; intros; first [ assumption | congruence ]. }
         { simpl in *.
           setoid_rewrite drop_drop in IHlen.
-          rewrite !NPeano.Nat.add_1_r in IHlen.
+          rewrite !Nat.add_1_r in IHlen.
           specialize (fun H' level1 H => IHlen level1 level2 H H').
           intros H1 H2.
           specialize_by assumption.
@@ -385,7 +385,7 @@ Section specific.
                      | _ => congruence
                      | _ => omega
                      | [ H : context[take _ (drop _ _)] |- _ ] => setoid_rewrite take_drop in H
-                     | [ H : context[(_ + 1)%nat] |- _ ] => rewrite !NPeano.Nat.add_1_r in H
+                     | [ H : context[(_ + 1)%nat] |- _ ] => rewrite !Nat.add_1_r in H
                      | [ H : context[bool_of_sumbool ?e] |- _ ] => destruct e; simpl in H
                      | [ H : context[match ?e with left _ => _ | right _ => _ end] |- _ ] => destruct e; simpl in H
                      | [ |- context[bool_of_sumbool ?e] ] => destruct e; simpl
@@ -417,7 +417,7 @@ Section specific.
       induction_str_len str.
       { rewrite paren_balanced_hiding'_nil, paren_balanced'_nil by assumption; trivial. }
       { specialize (IHlen (drop 1 str)).
-        rewrite drop_length, <- Minus.pred_of_minus in IHlen.
+        rewrite drop_length, Nat.sub_1_r in IHlen.
         specialize (IHlen (f_equal pred H)).
         rewrite paren_balanced_hiding'_recr, paren_balanced'_recr.
         edestruct get as [ch|]; trivial; [].
@@ -442,7 +442,7 @@ Section specific.
       induction_str_len str.
       { rewrite paren_balanced_hiding'_nil, paren_balanced'_nil by assumption; trivial. }
       { specialize (IHlen (drop 1 str)).
-        rewrite drop_length, <- Minus.pred_of_minus in IHlen.
+        rewrite drop_length, Nat.sub_1_r in IHlen.
         specialize (IHlen (f_equal pred H)).
         rewrite paren_balanced_hiding'_recr, paren_balanced'_recr.
         edestruct get as [ch|]; trivial; [].
@@ -468,7 +468,7 @@ Section specific.
       { simpl in *.
         intros n H'.
         apply paren_balanced_hiding'_S.
-        rewrite NPeano.Nat.add_succ_r; eauto with nocore. }
+        rewrite Nat.add_succ_r; eauto with nocore. }
     Qed.*)
   End paren_balanced_to_hiding.
 End specific.
diff --git a/src/Parsers/Refinement/BinOpBrackets/ParenBalancedGrammar.v b/src/Parsers/Refinement/BinOpBrackets/ParenBalancedGrammar.v
index 022d719b5..6b5c963c1 100644
--- a/src/Parsers/Refinement/BinOpBrackets/ParenBalancedGrammar.v
+++ b/src/Parsers/Refinement/BinOpBrackets/ParenBalancedGrammar.v
@@ -186,7 +186,7 @@ Section specific.
                        | [ H : forall x, _ = x -> _ |- _ ] => specialize (H _ eq_refl)
                        | [ H : forall x, x = _ -> _ |- _ ] => specialize (H _ eq_refl)
                        | [ H : context[In _ (uniquize _ _)] |- _ ]
-                         => setoid_rewrite <- (ListFacts.uniquize_In_refl_iff _ EqNat.beq_nat _ (lb eq_refl) bl) in H
+                         => setoid_rewrite <- (ListFacts.uniquize_In_refl_iff _ Nat.eqb _ (lb eq_refl) bl) in H
                        | [ H : context[In _ (map _ _)] |- _ ]
                          => setoid_rewrite in_map_iff in H
                        | _ => progress destruct_head ex
@@ -284,7 +284,7 @@ Section specific.
                  | _ => progress subst
                  | _ => congruence
                  | [ H : string_beq _ _ = true |- _ ] => apply string_bl in H
-                 | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
+                 | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
                  | [ H : is_true (_ || _) |- _ ] => apply Bool.orb_true_iff in H
                  | [ H : is_true (_ && _) |- _ ] => apply Bool.andb_true_iff in H
                  | [ H : _ /\ _ |- _ ] => let H1 := fresh in
diff --git a/src/Parsers/Refinement/BinOpBrackets/ParenBalancedLemmas.v b/src/Parsers/Refinement/BinOpBrackets/ParenBalancedLemmas.v
index f5a3ed9e0..adba62860 100644
--- a/src/Parsers/Refinement/BinOpBrackets/ParenBalancedLemmas.v
+++ b/src/Parsers/Refinement/BinOpBrackets/ParenBalancedLemmas.v
@@ -25,7 +25,7 @@ Section helpers.
              | [ H : is_true false |- _ ] => solve [ inversion H ]
              | [ H : false = true |- _ ] => solve [ inversion H ]
              | _ => progress rewrite ?take_length, ?drop_length
-             | [ H : _ |- _ ] => progress rewrite ?take_length, ?drop_length, ?drop_take, ?drop_0, ?NPeano.Nat.sub_0_r, ?NPeano.Nat.sub_1_r in H
+             | [ H : _ |- _ ] => progress rewrite ?take_length, ?drop_length, ?drop_take, ?drop_0, ?Nat.sub_0_r, ?Nat.sub_1_r in H
              | [ H : ?x = 0, H' : context[?x] |- _ ] => rewrite H in H'
              | [ H : length ?str = 0, H' : is_true (take _ (drop _ ?str) ~= [ _ ])%string_like |- _ ]
                => apply length_singleton in H'
@@ -42,7 +42,7 @@ Section helpers.
              | _ => solve [ eauto with nocore ]
              | [ H : context[drop _ (drop _ _)] |- _ ] => setoid_rewrite drop_drop in H
              | [ |- context[match get 0 (take _ _) with _ => _ end] ] => rewrite !get_take_lt by omega
-             | [ H : context[(_ + 1)%nat] |- _ ] => setoid_rewrite NPeano.Nat.add_1_r in H
+             | [ H : context[(_ + 1)%nat] |- _ ] => setoid_rewrite Nat.add_1_r in H
              | [ |- context[get 0 ?str] ] => erewrite (proj1 (get_0 str _)) by eassumption
              | [ |- context[get 0 (take 0 ?str)] ] => rewrite (has_first_char_nonempty (take 0 str))
                                                      by (rewrite take_length; reflexivity)
diff --git a/src/Parsers/Refinement/DisjointLemmas.v b/src/Parsers/Refinement/DisjointLemmas.v
index a3e89efd7..c5877326a 100644
--- a/src/Parsers/Refinement/DisjointLemmas.v
+++ b/src/Parsers/Refinement/DisjointLemmas.v
@@ -3,7 +3,6 @@ Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Coq.Init.Wf Coq.Arith.Wf_nat.
 Require Import Coq.ZArith.ZArith.
 Require Import Coq.Lists.List Coq.Strings.String.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
 Require Import Fiat.Parsers.ContextFreeGrammar.PreNotations.
 Require Import Fiat.Parsers.ContextFreeGrammar.Equality.
@@ -290,7 +289,7 @@ Section search_backward.
         try solve [ left; eauto
                   | right; eauto
                   | assumption
-                  | apply for_last_char_nil; rewrite ?take_length; apply Min.min_case_strong; omega ] ].
+                  | apply for_last_char_nil; rewrite ?take_length; apply Nat.min_case_strong; omega ] ].
     revert H0.
     apply forall_chars__char_in__impl__forall_chars.
     intros ch H' H''.
@@ -357,7 +356,7 @@ Section search_backward.
         try solve [ right; eauto
                   | left; split; eauto
                   | assumption
-                  | apply for_last_char_nil; rewrite ?take_length; apply Min.min_case_strong; omega ] ].
+                  | apply for_last_char_nil; rewrite ?take_length; apply Nat.min_case_strong; omega ] ].
     { revert H0.
       apply forall_chars_Proper; [ reflexivity | ].
       intros ch H' H''.
diff --git a/src/Parsers/Refinement/DisjointRules.v b/src/Parsers/Refinement/DisjointRules.v
index ff7ad8bd8..96e33892f 100644
--- a/src/Parsers/Refinement/DisjointRules.v
+++ b/src/Parsers/Refinement/DisjointRules.v
@@ -26,8 +26,8 @@ Proof.
   revert str; induction len; simpl; intros; [ reflexivity | ].
   destruct (get (length str - S len) str) eqn:H.
   { edestruct P; try omega.
-    apply Le.le_n_S, IHlen. }
-  { apply Le.le_n_S, IHlen. }
+    apply (fun n m => proj1 (Nat.succ_le_mono n m)), IHlen. }
+  { apply (fun n m => proj1 (Nat.succ_le_mono n m)), IHlen. }
 Qed.
 
 Lemma find_first_char_such_that_short {Char HSLM HSL}
@@ -46,30 +46,30 @@ Proof.
   destruct H as [n H].
   set (len := length str).
   setoid_replace str with (drop (length str - len) str) at 1
-    by (subst; rewrite Minus.minus_diag, drop_0; reflexivity).
+    by (subst; rewrite Nat.sub_diag, drop_0; reflexivity).
   setoid_replace str with (drop (length str - len) str) in H
-    by (subst; rewrite Minus.minus_diag, drop_0; reflexivity).
+    by (subst; rewrite Nat.sub_diag, drop_0; reflexivity).
   assert (len <= length str) by reflexivity.
   clearbody len.
   generalize dependent str; revert n.
   induction len; simpl; intros n str IH Hlen.
   { apply first_char_such_that_0.
     rewrite drop_length.
-    rewrite NPeano.Nat.sub_0_r in IH |- *.
-    rewrite Minus.minus_diag.
+    rewrite Nat.sub_0_r in IH |- *.
+    rewrite Nat.sub_diag.
     split.
     { apply for_first_char_nil.
       rewrite drop_length; omega. }
     { generalize dependent str.
       induction n; intros str H Hlen.
       { apply first_char_such_that_0 in H.
-        rewrite drop_length, Minus.minus_diag in H.
+        rewrite drop_length, Nat.sub_diag in H.
         destruct_head and; trivial. }
       { apply first_char_such_that_past_end in H; [ | rewrite drop_length; omega ].
         left; assumption. } } }
   { pose proof (singleton_exists (take 1 (drop (length str - S len) str))) as H'.
     rewrite take_length, drop_length in H'.
-    destruct H' as [ch H']; [ apply Min.min_case_strong; intros; omega | ].
+    destruct H' as [ch H']; [ apply Nat.min_case_strong; intros; omega | ].
     rewrite get_drop.
     rewrite (proj1 (get_0 _ _) H').
     destruct (P ch) eqn:H''.
@@ -171,7 +171,7 @@ Section with_grammar.
     specialize (H1 (ex_intro _ n H')).
     pose proof (is_first_char_such_that_eq_nat_iff H1 H') as H''.
     destruct_head or; destruct_head and; subst;
-      rewrite ?Min.min_r, ?Min.min_l by assumption;
+      rewrite ?Nat.min_r, ?Nat.min_l by assumption;
       omega.
   Qed.
 
@@ -210,7 +210,7 @@ Section with_grammar.
     specialize (H1 (ex_intro _ n H')).
     pose proof (is_first_char_such_that_eq_nat_iff H1 H') as H''.
     destruct_head or; destruct_head and; subst;
-      rewrite ?Min.min_r by assumption;
+      rewrite ?Nat.min_r by assumption;
       omega.
   Qed.
 
diff --git a/src/Parsers/Refinement/DisjointRulesRev.v b/src/Parsers/Refinement/DisjointRulesRev.v
index 7028ec03e..bbe07e686 100644
--- a/src/Parsers/Refinement/DisjointRulesRev.v
+++ b/src/Parsers/Refinement/DisjointRulesRev.v
@@ -104,7 +104,7 @@ Section with_grammar.
     unfold rev_search_for_condition in H1.
     pose proof (is_after_last_char_such_that_eq_nat_iff H1 H') as H''.
     destruct_head or; destruct_head and; subst;
-      rewrite ?Min.min_r, ?Min.min_l by assumption;
+      rewrite ?Nat.min_r, ?Nat.min_l by assumption;
       omega.
   Qed.
 
@@ -142,7 +142,7 @@ Section with_grammar.
     specialize (H1 (ex_intro _ n H')).
     pose proof (is_after_last_char_such_that_eq_nat_iff H1 H') as H''.
     destruct_head or; destruct_head and; subst;
-      rewrite ?Min.min_r by assumption;
+      rewrite ?Nat.min_r by assumption;
       omega.
   Qed.
 
diff --git a/src/Parsers/Refinement/EmptyLemmas.v b/src/Parsers/Refinement/EmptyLemmas.v
index 465813cc4..edf254431 100644
--- a/src/Parsers/Refinement/EmptyLemmas.v
+++ b/src/Parsers/Refinement/EmptyLemmas.v
@@ -118,8 +118,8 @@ Section correctness.
     | [ x : nonemptyT |- _ ] => destruct x
     | [ x : bool |- _ ] => destruct x
     | [ |- and _ _ ] => split
-    | _ => progress rewrite ?take_length, ?drop_length, ?Min.min_0_r, ?Min.min_0_l
-    | [ H : _ |- _ ] => progress rewrite ?take_length, ?drop_length, ?Min.min_0_r, ?Min.min_0_l in H
+    | _ => progress rewrite ?take_length, ?drop_length, ?Nat.min_0_r, ?Nat.min_0_l
+    | [ H : _ |- _ ] => progress rewrite ?take_length, ?drop_length, ?Nat.min_0_r, ?Nat.min_0_l in H
     | _ => omega
     | [ H : ?x = ?x -> _ |- _ ] => specialize (H eq_refl)
     | [ H : ?P ?v, H' : forall str, ?P str -> length str <> 0 |- _ ]
@@ -127,7 +127,7 @@ Section correctness.
     | [ H : ?P ?v, H' : forall str, ?P str -> length str = 0 |- _ ]
       => apply H' in H
     | [ H : ?x = 0, H' : context[?x] |- _ ] => rewrite H in H'
-    | [ H : min _ _ = 0 |- _ ] => revert H; apply Min.min_case_strong
+    | [ H : min _ _ = 0 |- _ ] => revert H; apply Nat.min_case_strong
     end.
 
   Local Ltac t_Proper := repeat t_Proper_step.
diff --git a/src/Parsers/Refinement/FixedLengthLemmas.v b/src/Parsers/Refinement/FixedLengthLemmas.v
index c4b7f8d23..d7b60c313 100644
--- a/src/Parsers/Refinement/FixedLengthLemmas.v
+++ b/src/Parsers/Refinement/FixedLengthLemmas.v
@@ -19,12 +19,12 @@ Set Implicit Arguments.
 Local Open Scope grammar_fixedpoint_scope.
 
 Definition length_result_lub (l1 l2 : nat) : lattice_for nat
-  := if beq_nat l1 l2 then constant l1 else ⊤.
+  := if Nat.eqb l1 l2 then constant l1 else ⊤.
 
 Global Instance length_result_lattice : grammar_fixedpoint_lattice_data nat.
 Proof.
   refine {| prestate_lt x y := false;
-            prestate_beq := beq_nat;
+            prestate_beq := Nat.eqb;
             preleast_upper_bound := length_result_lub |};
     try abstract (repeat match goal with
                          | [ |- is_true true ] => reflexivity
@@ -36,8 +36,8 @@ Proof.
                          | _ => progress unfold length_result_lub, is_true in *
                          | _ => progress subst
                          | [ H : constant _ = constant _ |- _ ] => inversion H; clear H
-                         | [ H : beq_nat _ _ = true |- _ ] => apply beq_nat_true_iff in H
-                         | _ => rewrite <- beq_nat_refl
+                         | [ H : Nat.eqb _ _ = true |- _ ] => apply Nat.eqb_eq in H
+                         | _ => rewrite Nat.eqb_refl
                          | [ H : context[match ?e with _ => _ end] |- _ ] => destruct e eqn:?
                          | [ |- context[match ?e with _ => _ end] ] => destruct e eqn:?
                          end).
@@ -57,8 +57,8 @@ Proof.
                | _ => progress simpl in *
                | _ => progress unfold is_true in *
                | _ => progress subst
-               | [ H : beq_nat _ _ = true |- _ ] => apply beq_nat_true_iff in H
-               | [ |- beq_nat _ _ = true ] => apply beq_nat_true_iff
+               | [ H : Nat.eqb _ _ = true |- _ ] => apply Nat.eqb_eq in H
+               | [ |- Nat.eqb _ _ = true ] => apply Nat.eqb_eq
                | _ => reflexivity
                end
       ). }
@@ -82,7 +82,7 @@ Section correctness.
     | _ => progress unfold respectful, pointwise_relation, length_result_accurate, length_result_accurate, ensemble_bottom, ensemble_top, ensemble_least_upper_bound, ensemble_on_terminal, ensemble_combine_production, lattice_for_related, not, length_result_lub, prestate_le in *
     | _ => intro
     | _ => progress subst
-    | [ H : is_true (beq_nat _ _) |- _ ] => apply beq_nat_true_iff in H
+    | [ H : is_true (Nat.eqb _ _) |- _ ] => apply Nat.eqb_eq in H
     | _ => progress destruct_head lattice_for
     | [ |- iff _ _ ] => split
     | [ H : forall x y, _ -> (_ <-> _) |- _ ] => specialize (fun x => H x x (reflexivity _))
@@ -106,8 +106,8 @@ Section correctness.
       => specialize (fun str => H (fun _ => True) str I)
     | [ H : _ |- _ ] => rewrite Bool.orb_false_r in H
     | _ => rewrite Bool.orb_false_r
-    | [ |- is_true (beq_nat _ _) ] => apply beq_nat_true_iff
-    | [ H : beq_nat _ _ = true |- _ ] => apply beq_nat_true_iff in H
+    | [ |- is_true (Nat.eqb _ _) ] => apply Nat.eqb_eq
+    | [ H : Nat.eqb _ _ = true |- _ ] => apply Nat.eqb_eq in H
     | [ H : forall P, (forall str, P str -> @?P' str) -> forall str', P str' -> @?Q' str' |- _ ]
       => specialize (H P' (fun str pf => pf))
     | [ H : forall str, length str = ?n -> length str = ?n' |- _ ]
diff --git a/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflective.v b/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflective.v
index b07518e6b..466090eef 100644
--- a/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflective.v
+++ b/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflective.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** First step of a splitter refinement; indexed representation, and handle all rules with at most one nonterminal; leave a reflective goal *)
-Require Import Coq.Strings.String Coq.Arith.Lt Coq.Lists.List.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
+Require Import Coq.Strings.String Coq.Arith.PeanoNat Coq.Lists.List.
 Require Import Fiat.Parsers.BaseTypes.
 Require Import Fiat.Parsers.Splitters.RDPList.
 Require Import Fiat.Parsers.ParserInterface.
@@ -214,14 +213,14 @@ Section helpers.
                  | [ H : context[length (drop _ _)] |- _ ] => rewrite drop_length in H
                  | [ H : min ?x ?y = 1 |- _ ] => is_var x; destruct x
                  | [ H : min (S ?x) ?y = 1 |- _ ] => is_var x; destruct x
-                 | [ H : min (S (S ?x)) ?y = 1 |- _ ] => revert H; apply (Min.min_case_strong (S (S x)) y)
-                 | [ H : context[min _ 0] |- _ ] => rewrite Min.min_0_r in H
-                 | [ H : context[min 0 _] |- _ ] => rewrite Min.min_0_l in H
+                 | [ H : min (S (S ?x)) ?y = 1 |- _ ] => revert H; apply (Nat.min_case_strong (S (S x)) y)
+                 | [ H : context[min _ 0] |- _ ] => rewrite Nat.min_0_r in H
+                 | [ H : context[min 0 _] |- _ ] => rewrite Nat.min_0_l in H
                  | [ H : 0 = 1 |- _ ] => exfalso; clear -H; discriminate
                  | [ H : S (S _) = 1 |- _ ] => exfalso; clear -H; discriminate
                  | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
                  | [ H : ?x = 1 |- context[?x] ] => rewrite H
-                 | [ H : min ?x ?y = 1 |- _ ] => revert H; apply (Min.min_case_strong x y)
+                 | [ H : min ?x ?y = 1 |- _ ] => revert H; apply (Nat.min_case_strong x y)
                  | _ => omega
                end. }
       { exfalso.
@@ -237,10 +236,10 @@ Module Export PrettyNotations.
   Infix "=p" := default_production_carrierT_beq (at level 70, no associativity).
   Infix "=ℕ" := opt.beq_nat (at level 70, no associativity).
   Infix "=ℕ" := opt2.beq_nat (at level 70, no associativity).
-  Infix "=ℕ" := EqNat.beq_nat (at level 70, no associativity).
+  Infix "=ℕ" := Nat.eqb (at level 70, no associativity).
 End PrettyNotations.
 
-Definition beq_nat_opt := Eval compute in EqNat.beq_nat.
+Definition beq_nat_opt := Eval compute in Nat.eqb.
 Definition andb_opt := Eval compute in andb.
 
 Inductive ret_cases : Set :=
@@ -257,12 +256,12 @@ Local Ltac t_ret_cases :=
          | _ => intro
          | _ => progress subst
          | _ => congruence
-         | [ H : ?f ?x ?y = ?b |- _ ] => progress change (EqNat.beq_nat x y = b) in H
-         | [ |- ?f ?x ?y = ?b ] => progress change (EqNat.beq_nat x y = b)
+         | [ H : ?f ?x ?y = ?b |- _ ] => progress change (Nat.eqb x y = b) in H
+         | [ |- ?f ?x ?y = ?b ] => progress change (Nat.eqb x y = b)
          | [ H : ?f ?x ?y = ?b |- _ ] => progress change (andb x y = b) in H
          | [ |- ?f ?x ?y = ?b ] => progress change (andb x y = b)
-         | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
-         | [ |- EqNat.beq_nat _ _ = true ] => apply EqNat.beq_nat_true_iff
+         | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
+         | [ |- Nat.eqb _ _ = true ] => apply Nat.eqb_eq
          | [ H : andb _ _ = true |- _ ] => apply Bool.andb_true_iff in H
          | [ |- andb _ _ = true ] => apply Bool.andb_true_iff
          | [ H : and _ _ |- _ ] => destruct H
@@ -557,9 +556,9 @@ Section IndexedImpl.
              | [ |- context[_ - 0] ] => rewrite Nat.sub_0_r
              | [ |- context[substring _ (min _ (length ?str)) ?str] ]
                => rewrite substring_min_length
-             | [ |- context[_ + 0] ] => rewrite Plus.plus_0_r
+             | [ |- context[_ + 0] ] => rewrite Nat.add_0_r
              | [ |- context[min ?x ?x] ]
-               => rewrite (Min.min_idempotent x)
+               => rewrite (Nat.min_idempotent x)
              | _ => reflexivity
              | _ => assumption
            end;
@@ -567,7 +566,7 @@ Section IndexedImpl.
               | repeat match goal with
                          | _ => intro
                          | [ |- context[min ?x ?x] ]
-                           => rewrite (Min.min_idempotent x)
+                           => rewrite (Nat.min_idempotent x)
                          | _ => reflexivity
                          | _ => rewrite substring_substring
                          | _ => rewrite Nat.sub_0_r
@@ -578,15 +577,15 @@ Section IndexedImpl.
                          | [ |- context[min ?x ?y] ]
                            => match goal with
                                 | [ |- context[min y x] ]
-                                  => rewrite (Min.min_comm x y)
+                                  => rewrite (Nat.min_comm x y)
                               end
                          | [ |- context[min (min _ ?x) (?x - ?y)] ]
-                           => rewrite <- (Min.min_assoc _ x (x - y)), (Min.min_r x (x - y)) by omega
+                           => rewrite <- (Nat.min_assoc _ x (x - y)), (Nat.min_r x (x - y)) by omega
                          | [ |- substring (?x + ?y) _ _ = substring (?y + ?x) _ _ ]
-                           => rewrite (Plus.plus_comm x y)
+                           => rewrite (Nat.add_comm x y)
                          | [ |- substring ?x ?y ?z = substring ?x (min ?w ?y) ?z ]
-                           => apply (@Min.min_case_strong w y)
-                         | [ H : _ |- _ ] => rewrite Min.min_assoc in H
+                           => apply (@Nat.min_case_strong w y)
+                         | [ H : _ |- _ ] => rewrite Nat.min_assoc in H
                          | _ => apply substring_correct4; omega
                        end
               | repeat match goal with
@@ -595,7 +594,7 @@ Section IndexedImpl.
                          | [ |- List.In ?x [?y] ] => left
                          | [ |- context[List.map ?f [?x] ] ] => change (List.map f [x]) with [f x]
                          | [ |- context[min ?x ?x] ]
-                           => rewrite (Min.min_idempotent x)
+                           => rewrite (Nat.min_idempotent x)
                          | _ => reflexivity
                          | [ H : parse_of_production _ _ nil |- _ ] => let H' := fresh in rename H into H'; dependent destruction H'
                          | [ H : parse_of_production _ _ (_::_) |- _ ] => let H' := fresh in rename H into H'; dependent destruction H'
@@ -604,8 +603,8 @@ Section IndexedImpl.
                          | [ H : length (substring ?n ?m ?s) = _, H' : context[length (substring ?n ?m ?s)] |- _ ] => rewrite H in H'
                          | [ H : context[length (take _ _)] |- _ ] => rewrite take_length in H
                          | _ => erewrite <- has_only_terminals_length by eassumption
-                         | [ H : _ |- _ ] => progress rewrite ?(@drop_length _ HSL HSLP), ?(@take_length _ HSL HSLP), ?substring_length, ?Nat.add_sub, <- ?plus_n_O, ?Minus.minus_diag, ?Nat.sub_0_r, ?sub_plus in H by omega
-                         | _ => progress rewrite ?drop_length, ?take_length, ?substring_length, ?Nat.add_sub, ?Minus.minus_diag, ?Nat.sub_0_r, <- ?plus_n_O, ?sub_plus by omega
+                         | [ H : _ |- _ ] => progress rewrite ?(@drop_length _ HSL HSLP), ?(@take_length _ HSL HSLP), ?substring_length, ?Nat.add_sub, <- ?plus_n_O, ?Nat.sub_diag, ?Nat.sub_0_r, ?sub_plus in H by omega
+                         | _ => progress rewrite ?drop_length, ?take_length, ?substring_length, ?Nat.add_sub, ?Nat.sub_diag, ?Nat.sub_0_r, <- ?plus_n_O, ?sub_plus by omega
                          | [ H : is_true (string_beq _ _) |- _ ] => apply string_bl in H
                          | [ |- _ \/ False ] => left
                          | [ H : String.substring _ _ _ = String.String _ _ |- _ = _ :> nat ] => apply (f_equal String.length) in H; simpl in H
@@ -615,37 +614,37 @@ Section IndexedImpl.
                            => rewrite <- (Nat.sub_min_distr_r x (y + z) z)
                          | [ H : context[min ?x (?y + ?z) - ?z] |- _ ]
                            => rewrite <- (Nat.sub_min_distr_r x (y + z) z) in H
-                         | [ |- context[min ?x (?x - 1)] ] => rewrite (Min.min_r x (x - 1)) by (clear; omega)
-                         | [ H : min ?x ?y = 1 |- _ ] => is_var x; revert H; apply (Min.min_case_strong x y)
-                         | [ H : min ?x ?y = 1 |- _ ] => is_var y; revert H; apply (Min.min_case_strong x y)
-                         | [ H : min ?x ?y = 0 |- _ ] => is_var x; revert H; apply (Min.min_case_strong x y)
-                         | [ H : min ?x ?y = 0 |- _ ] => is_var y; revert H; apply (Min.min_case_strong x y)
-                         | [ H : min ?x 1 = 0 |- _ ] => revert H; apply (Min.min_case_strong x 1)
+                         | [ |- context[min ?x (?x - 1)] ] => rewrite (Nat.min_r x (x - 1)) by (clear; omega)
+                         | [ H : min ?x ?y = 1 |- _ ] => is_var x; revert H; apply (Nat.min_case_strong x y)
+                         | [ H : min ?x ?y = 1 |- _ ] => is_var y; revert H; apply (Nat.min_case_strong x y)
+                         | [ H : min ?x ?y = 0 |- _ ] => is_var x; revert H; apply (Nat.min_case_strong x y)
+                         | [ H : min ?x ?y = 0 |- _ ] => is_var y; revert H; apply (Nat.min_case_strong x y)
+                         | [ H : min ?x 1 = 0 |- _ ] => revert H; apply (Nat.min_case_strong x 1)
                          | [ |- context[0 + ?x] ] => change (0 + x) with x
                          | [ |- context[?x - S ?y] ]
-                           => not constr_eq y 0; rewrite !(Nat.sub_succ_r x y), !Minus.pred_of_minus
+                           => not constr_eq y 0; rewrite !(Nat.sub_succ_r x y), !(fun n => eq_sym (Nat.sub_1_r n))
                          | [ H : ?x = 1 |- context[?x] ] => rewrite H
                          | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
                          | [ H : ?x <= ?y |- context[min ?x ?y] ]
-                           => rewrite (Min.min_l x y H)
+                           => rewrite (Nat.min_l x y H)
                          | [ H : ?y <= ?x |- context[min ?x ?y] ]
-                           => rewrite (Min.min_r x y H)
+                           => rewrite (Nat.min_r x y H)
                          | [ H : ?x <= ?y |- context[?x - ?y] ] => replace (x - y) with 0 by (clear -H; omega)
                          | [ H : context[?x - ?y], H' : ?x <= ?y |- _ ]
                            => rewrite (proj2 (@Nat.sub_0_le x y)) in H by exact H'
                          | [ H : context[min 0 ?x] |- _ ] => change (min 0 x) with 0 in H
                          | [ |- context[min (min _ ?x) (?x - ?y)] ]
-                           => rewrite <- (Min.min_assoc _ x (x - y)), (Min.min_r x (x - y)) by omega
+                           => rewrite <- (Nat.min_assoc _ x (x - y)), (Nat.min_r x (x - y)) by omega
                          | [ |- 1 = ?x ] => is_var x; destruct x
                          | [ |- 1 = S ?x ] => is_var x; destruct x
-                         | [ H : _ <= 0 |- _ ] => apply Le.le_n_0_eq in H; symmetry in H
+                         | [ H : _ <= 0 |- _ ] => apply (fun n Hle => eq_sym (proj1 (Nat.le_0_r n) Hle)) in H; symmetry in H
                          | [ H : context[min 1 ?x] |- _ ] => is_var x; destruct x
                          | [ H : context[min 1 (S ?x)] |- _ ] => is_var x; destruct x
                          | [ H : context[min 1 (S ?x)] |- _ ] => change (min 1 (S x)) with 1 in H
-                         | [ H : context[min ?x ?y], H' : ?x <= ?y |- _ ] => rewrite Min.min_l in H by assumption
-                         | [ H : context[min ?x ?y], H' : ?y <= ?x |- _ ] => rewrite Min.min_r in H by assumption
-                         | [ H : context[min (?x - ?y) ?x] |- _ ] => rewrite Min.min_l in H by omega
-                         | [ H : context[min ?x (?x - ?y)] |- _ ] => rewrite Min.min_r in H by omega
+                         | [ H : context[min ?x ?y], H' : ?x <= ?y |- _ ] => rewrite Nat.min_l in H by assumption
+                         | [ H : context[min ?x ?y], H' : ?y <= ?x |- _ ] => rewrite Nat.min_r in H by assumption
+                         | [ H : context[min (?x - ?y) ?x] |- _ ] => rewrite Nat.min_l in H by omega
+                         | [ H : context[min ?x (?x - ?y)] |- _ ] => rewrite Nat.min_r in H by omega
                          | _ => omega
                        end ].
 
@@ -741,7 +740,7 @@ Section IndexedImpl.
     end.
   Local Ltac pre_fin := repeat pre_fin'.
 
-  Local Arguments EqNat.beq_nat : simpl never.
+  Local Arguments Nat.eqb : simpl never.
 
   Local Ltac do_Iterate_Ensemble_BoundedIndex_equiv :=
     eapply Iterate_Ensemble_BoundedIndex_equiv;
@@ -764,7 +763,7 @@ Section IndexedImpl.
       | _ => progress subst
       | [ H : Some _ = Some _ |- _ ] => inversion H; clear H
       | [ |- context[min ?x ?x] ]
-        => rewrite (Min.min_idempotent x)
+        => rewrite (Nat.min_idempotent x)
       | [ |- List.In ?x [?y] ] => left
       | [ |- context[List.map ?f [?x] ] ] => change (List.map f [x]) with [f x]
       | [ |- _ \/ False ] => left
@@ -776,19 +775,19 @@ Section IndexedImpl.
       | [ H : context[unsafe_get] |- _ ] => erewrite unsafe_get_correct in H by eassumption
       | [ H : andb _ _ = true |- _ ] => apply Bool.andb_true_iff in H
       | [ H : andb _ _ = false |- _ ] => apply Bool.andb_false_iff in H
-      | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
-      | [ H : EqNat.beq_nat _ _ = false |- _ ] => apply EqNat.beq_nat_false in H
+      | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => proj1 (Nat.eqb_eq n m)) in H
+      | [ H : Nat.eqb _ _ = false |- _ ] => apply (fun n m => proj1 (Nat.eqb_neq n m)) in H
       | [ H : Compare_dec.leb _ _ = true |- _ ] => apply Compare_dec.leb_complete in H
       | [ H : Compare_dec.leb _ _ = false |- _ ] => apply Compare_dec.leb_complete_conv in H
-      | [ H : context[?x - ?x] |- _ ] => rewrite Minus.minus_diag in H
-      | [ H : context[min _ 0] |- _ ] => rewrite Min.min_0_r in H
-      | [ H : context[min 0 _] |- _ ] => rewrite Min.min_0_l in H
+      | [ H : context[?x - ?x] |- _ ] => rewrite Nat.sub_diag in H
+      | [ H : context[min _ 0] |- _ ] => rewrite Nat.min_0_r in H
+      | [ H : context[min 0 _] |- _ ] => rewrite Nat.min_0_l in H
       | [ H : option_beq ascii_beq _ _ = true |- _ ]
         => apply (option_bl (@ascii_bl)) in H
       | [ H : context[min ?x ?y], H' : ?x <= ?y |- _ ]
-        => rewrite (Min.min_l x y) in H by assumption
+        => rewrite (Nat.min_l x y) in H by assumption
       | [ H : context[min ?x ?y], H' : ?y <= ?x |- _ ]
-        => rewrite (Min.min_r x y) in H by assumption
+        => rewrite (Nat.min_r x y) in H by assumption
       | [ H : context[?x + ?y - ?y] |- _ ] => rewrite Nat.add_sub in H
       | [ H : context[(0 + ?x)%nat] |- _ ] => change (0 + x) with x in H
       | [ |- context[min ?x (?y + ?z) - ?z] ]
@@ -814,19 +813,19 @@ Section IndexedImpl.
       | [ H : ?x = 0, H' : context[?x] |- _ ] => rewrite H in H'
       | [ H : 1 = snd (snd ?x), H' : context[snd (snd ?x)] |- _ ]
         => is_var x; rewrite <- H in H'
-      | [ |- context[min 0 _] ] => rewrite Min.min_0_l
-      | [ |- context[min _ 0] ] => rewrite Min.min_0_r
+      | [ |- context[min 0 _] ] => rewrite Nat.min_0_l
+      | [ |- context[min _ 0] ] => rewrite Nat.min_0_r
       | [ |- context[(_ - _)%natr] ] => rewrite minusr_minus
       | [ |- context[?x - ?y + min ?y ?x] ] => rewrite minus_plus_min
-      | [ |- context[?x - ?y + (min ?y ?x + _)] ] => rewrite !Plus.plus_assoc, minus_plus_min
+      | [ |- context[?x - ?y + (min ?y ?x + _)] ] => rewrite !Nat.add_assoc, minus_plus_min
       | [ H : ?x < S ?y |- context[?x - ?y] ]
         => rewrite (proj2 (Nat.sub_0_le x y)) by omega
       | [ H : ?x + ?y <= ?z, H' : context[min ?x (?z - ?y)] |- _ ]
-        => rewrite (Min.min_l x (z - y)) in H' by (clear -H; omega)
+        => rewrite (Nat.min_l x (z - y)) in H' by (clear -H; omega)
       | [ H : ?x <= ?y |- context[min ?x ?y] ]
-        => rewrite (Min.min_l x y H)
+        => rewrite (Nat.min_l x y H)
       | [ H : ?y <= ?x |- context[min ?x ?y] ]
-        => rewrite (Min.min_r x y H)
+        => rewrite (Nat.min_r x y H)
       | [ |- context[min ?x (?x - _)] ] => rewrite min_minus_r
       | _ => progress destruct_head' and
       | _ => progress destruct_head' or
@@ -838,9 +837,9 @@ Section IndexedImpl.
       | [ |- context[_ - 0] ] => rewrite Nat.sub_0_r
       | [ H : forall n, n <= 0 -> _ |- _ ] => specialize (H 0 (reflexivity _))
       | [ H : min _ ?x = ?v |- min ?v ?x = ?v ]
-        => revert H; clear; repeat apply Min.min_case_strong; intros; omega
+        => revert H; clear; repeat apply Nat.min_case_strong; intros; omega
       | [ |- context[min (min ?x ?y) ?y] ]
-        => rewrite <- (Min.min_assoc x y y), Min.min_idempotent
+        => rewrite <- (Nat.min_assoc x y y), Nat.min_idempotent
       | [ H : ret_length_less _ = ret_length_less _ |- _ ] => inversion H; clear H
       | [ H : ret_nat _ = ret_nat _ |- _ ] => inversion H; clear H
       | [ H : ret_pick _ = ret_pick _ |- _ ] => inversion H; clear H
@@ -874,26 +873,26 @@ Section IndexedImpl.
         => rewrite substring_length, H in H'
       | [ H : length ?x = _ |- context[length (drop _ (drop _ ?x))] ]
         => (do 2 rewrite drop_length; rewrite H)
-      | _ => rewrite Minus.pred_of_minus
+      | _ => rewrite (fun n => eq_sym (Nat.sub_1_r n))
       | [ H : ?x = 1 |- context[?x] ] => rewrite H
       | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
       | [ H : min ?x ?y = 1 |- context[?x] ]
-        => (revert H; apply (Min.min_case_strong x y))
+        => (revert H; apply (Nat.min_case_strong x y))
       | _ => intro
       | [ |- context[?x - ?y] ]
         => rewrite (proj2 (Nat.sub_0_le x y)) by assumption
       | [ |- context[min ?x (pred ?x)] ]
-        => rewrite (Min.min_r x (pred x)) by omega
+        => rewrite (Nat.min_r x (pred x)) by omega
       | [ |- context[?x - ?y - (?x - ?z - ?y)] ]
-        => rewrite <- (Nat.sub_add_distr x z y), (Plus.plus_comm z y), (Nat.sub_add_distr x y z)
+        => rewrite <- (Nat.sub_add_distr x z y), (Nat.add_comm z y), (Nat.sub_add_distr x y z)
       | [ |- context[?x - (?x - _)] ]
         => rewrite sub_twice
       | [ |- context[min ?x (min (?x - ?z) ?y)] ]
-        => rewrite (Min.min_assoc x (x - z)), (Min.min_r x (x - z)) by omega
-      | [ |- min ?x ?y = ?y ] => rewrite Min.min_r by omega
+        => rewrite (Nat.min_assoc x (x - z)), (Nat.min_r x (x - z)) by omega
+      | [ |- min ?x ?y = ?y ] => rewrite Nat.min_r by omega
       | [ H : ?x - ?y = 0 |- _ ] => apply Nat.sub_0_le in H
       | [ H : ?x <= ?y, H' : ?y <= ?x |- _ ]
-        => assert (y = x) by (apply Le.le_antisym; assumption);
+        => assert (y = x) by (apply Nat.le_antisymm; assumption);
           clear H H'
       | [ H : length (substring _ ?x _) = ?v |- min ?v ?x = ?v ]
         => rewrite substring_length in H; clear -H
@@ -903,7 +902,7 @@ Section IndexedImpl.
       | [ H : context[length (substring _ _ _)] |- _ ]
         => rewrite substring_length_no_min in H by first [ assumption | left; reflexivity | right; omega ]
       | [ H : context[min (?x - ?y) ?z], H' : ?y + ?z <= ?x |- _ ]
-        => rewrite (Min.min_r (x - y) z) in H by (clear -H'; omega)
+        => rewrite (Nat.min_r (x - y) z) in H by (clear -H'; omega)
     end.
   Local Ltac fin2 := repeat first [ progress fin_common
                                   | progress string_from_parse
@@ -1017,7 +1016,7 @@ Section IndexedImpl.
       | [ H : context[length (substring ?n ?m ?s)] |- _ ]
         => let H' := fresh in
            assert (H' : length (substring n m s) = m)
-             by (rewrite substring_length; fin_common; rewrite Min.min_r by omega; omega);
+             by (rewrite substring_length; fin_common; rewrite Nat.min_r by omega; omega);
              rewrite !H' in H |- *
     end.
 
@@ -1054,7 +1053,7 @@ Section IndexedImpl.
              | _ => progress destruct_head or; try solve [ fin2 ]; []
              | [ H : ?x = ?y |- _ ] => is_var y; subst y
              | [ H : ?x = ?y |- _ ] => is_var x; subst x
-             | _ => rewrite Min.min_r by omega
+             | _ => rewrite Nat.min_r by omega
              | _ => progress fin2
              | [ H : ?x = ?x |- _ ] => clear H
              end;
@@ -1063,9 +1062,9 @@ Section IndexedImpl.
                | [ H : _ <= _ |- _ ] => revert H
                end;
         clear -HSLP; intros;
-          rewrite !drop_length, !substring_length, Min.min_r, Nat.add_sub by omega;
+          rewrite !drop_length, !substring_length, Nat.min_r, Nat.add_sub by omega;
           repeat match goal with
-                 | [ H : context[min] |- _ ] => revert H; apply Min.min_case_strong
+                 | [ H : context[min] |- _ ] => revert H; apply Nat.min_case_strong
                  | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
                  | [ H : ?x = 1 |- context[?x] ] => rewrite H
                  | [ H : ?x <= ?y |- context[?x - ?y] ] => replace (x - y) with 0 by omega
@@ -1078,8 +1077,8 @@ Section IndexedImpl.
                    => rewrite (Nat.sub_succ_r x y)
                  | _ => progress rewrite ?sub_twice, ?Nat.sub_0_r
                  | _ => erewrite <- S_pred by eassumption
-                 | _ => rewrite Min.min_r by omega
-                 | _ => rewrite Min.min_l by omega
+                 | _ => rewrite Nat.min_r by omega
+                 | _ => rewrite Nat.min_l by omega
                  | _ => reflexivity
                  end. }
     { repeat match goal with
diff --git a/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflectiveOpt.v b/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflectiveOpt.v
index 446833ae9..ea14bfa14 100644
--- a/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflectiveOpt.v
+++ b/src/Parsers/Refinement/IndexedAndAtMostOneNonTerminalReflectiveOpt.v
@@ -519,7 +519,7 @@ Section IndexedImpl_opt.
                | [ |- opt.minus _ _ = _ ] => apply f_equal2
                | [ |- opt.combine _ _ = _ ] => apply f_equal2
                | [ |- opt2.ret_cases_BoolDecR _ _ = _ ] => apply f_equal2
-               | [ |- EqNat.beq_nat _ _ = _ ] => apply f_equal2
+               | [ |- Nat.eqb _ _ = _ ] => apply f_equal2
                | [ |- opt.nth _ _ _ = _ ] => apply f_equal3
                | [ |- 0 = _ ] => reflexivity
                | [ |- opt.length (pregrammar_productions G) = _ ] => reflexivity
@@ -572,7 +572,7 @@ Section IndexedImpl_opt.
       change (snd d) with (Common.opt2.snd d);
       change (fst (Common.opt2.snd d)) with (Common.opt2.fst (Common.opt2.snd d));
       change (snd (Common.opt2.snd d)) with (Common.opt2.snd (Common.opt2.snd d)).
-      change EqNat.beq_nat with Common.opt2.beq_nat.
+      change Nat.eqb with Common.opt2.beq_nat.
       change andb with Common.opt2.andb.
       reflexivity.
     }
diff --git a/src/Parsers/Refinement/PreTactics.v b/src/Parsers/Refinement/PreTactics.v
index 6d49acc33..0f0f590bb 100644
--- a/src/Parsers/Refinement/PreTactics.v
+++ b/src/Parsers/Refinement/PreTactics.v
@@ -2,6 +2,7 @@ Require Export Fiat.Parsers.ContextFreeGrammar.Core.
 Require Export Fiat.Parsers.ContextFreeGrammar.PreNotations.
 Require Export Fiat.Parsers.StringLike.FirstCharSuchThat.
 Require Export Coq.Strings.String.
+Require Export Coq.Arith.PeanoNat.
 Require Export Fiat.Computation.Core.
 Require Export Coq.Program.Program.
 Require Export Fiat.Computation.ApplyMonad.
@@ -123,14 +124,14 @@ Ltac solve_prod_beq :=
   repeat match goal with
            | [ |- context[true = false] ] => congruence
            | [ |- context[false = true] ] => congruence
-           | [ |- context[EqNat.beq_nat ?x ?y] ]
+           | [ |- context[Nat.eqb ?x ?y] ]
              => is_var x;
                let H := fresh in
-               destruct (EqNat.beq_nat x y) eqn:H;
-                 [ apply EqNat.beq_nat_true in H; subst x
+               destruct (Nat.eqb x y) eqn:H;
+                 [ apply (fun n m => proj1 (Nat.eqb_eq n m)) in H; subst x
                  | ];
                  simpl
-           | [ |- context[EqNat.beq_nat ?x ?y] ]
+           | [ |- context[Nat.eqb ?x ?y] ]
              => first [ is_var x; fail 1
                       | generalize x; intro ]
          end.
diff --git a/src/Parsers/Reflective/LogicalRelations.v b/src/Parsers/Reflective/LogicalRelations.v
index cce387d0e..bdd1605b3 100644
--- a/src/Parsers/Reflective/LogicalRelations.v
+++ b/src/Parsers/Reflective/LogicalRelations.v
@@ -307,7 +307,7 @@ Proof.
     apply list_rect_nodep_meaning_correct; simpl; eauto with nocore.
     simpler_meaning; meaning_tac. }
   { simpler_meaning; meaning_tac.
-    rewrite Plus.plus_comm; meaning_tac. }
+    rewrite Nat.add_comm; meaning_tac. }
 Qed.
 
 Local Hint Resolve push_var.
diff --git a/src/Parsers/Reflective/ParserSemantics.v b/src/Parsers/Reflective/ParserSemantics.v
index 1dac18bc9..24b8764e3 100644
--- a/src/Parsers/Reflective/ParserSemantics.v
+++ b/src/Parsers/Reflective/ParserSemantics.v
@@ -3,6 +3,7 @@ Require Import Fiat.Parsers.Reflective.Semantics.
 Require Import Fiat.Parsers.Splitters.RDPList.
 Require Import Fiat.Parsers.GenericRecognizer.
 Require Import Fiat.Common.Wf Fiat.Common.Wf2.
+Require Import Coq.Arith.PeanoNat.
 Set Implicit Arguments.
 
 Definition step_option_rec
@@ -26,7 +27,7 @@ Proof.
                                          (or_introl pf)
                                          up_to_G_length
                                          offset'
-                                         (len - len0_minus_len') (Minus.le_minus len len0_minus_len')))
+                                         (len - len0_minus_len') (Nat.le_sub_l len len0_minus_len')))
                        (fun _
                         => _)
                        (Compare_dec.lt_dec len len0)).
@@ -35,13 +36,13 @@ Proof.
             (fun is_valid
              => _)
             (fun _ => None)
-            (Sumbool.sumbool_of_bool (negb (EqNat.beq_nat valid_len0 0) && is_valid_nonterminal valids nt))%bool).
+            (Sumbool.sumbool_of_bool (negb (Nat.eqb valid_len0 0) && is_valid_nonterminal valids nt))%bool).
   refine (Some
             (fun offset' len0_minus_len' =>
                parse_nt len0 (pred valid_len0)
                         (or_intror (conj eq_refl (GenericRecognizer.pred_lt_beq _)))
                         (RDPList.filter_out_eq nt valids) offset'
-                        (len0 - len0_minus_len') (Minus.le_minus len0 len0_minus_len'))).
+                        (len0 - len0_minus_len') (Nat.le_sub_l len0 len0_minus_len'))).
   apply (proj1 (proj1 (Bool.andb_true_iff _ _) is_valid)).
 Defined.
 
diff --git a/src/Parsers/Reflective/PartialUnfold.v b/src/Parsers/Reflective/PartialUnfold.v
index a6262beff..d4b142fe8 100644
--- a/src/Parsers/Reflective/PartialUnfold.v
+++ b/src/Parsers/Reflective/PartialUnfold.v
@@ -135,7 +135,7 @@ Section normalization_by_evaluation.
        | Rbeq_nat
          => specific_meaning_apply2
               cnat cnat cbool (syntactify_bool _)
-              EqNat.beq_nat
+              Nat.eqb
        | Rmap A B
          => fun af
             => option_map (@syntactify_list _ _)
diff --git a/src/Parsers/Reflective/Reify.v b/src/Parsers/Reflective/Reify.v
index e25d1030b..07dd10765 100644
--- a/src/Parsers/Reflective/Reify.v
+++ b/src/Parsers/Reflective/Reify.v
@@ -185,7 +185,7 @@ Ltac reify_Term var term
                        | andbr => constr:(@Randbr)
                        | minusr => constr:(@Rminusr)
                        | Compare_dec.leb => constr:(@Rleb)
-                       | EqNat.beq_nat => constr:(@Rbeq_nat)
+                       | Nat.eqb => constr:(@Rbeq_nat)
                        | Equality.string_beq => constr:(@Rstring_beq)
                        | @Reflective.interp_RCharExpr _ _ => constr:(@Rinterp_RCharExpr_ascii)
                        end in
diff --git a/src/Parsers/Reflective/Semantics.v b/src/Parsers/Reflective/Semantics.v
index 197478e48..6ba082192 100644
--- a/src/Parsers/Reflective/Semantics.v
+++ b/src/Parsers/Reflective/Semantics.v
@@ -90,7 +90,7 @@ Definition interp_RLiteralTerm {T} (t : RLiteralTerm T) : interp_TypeCode T
           | Rpred => pred
           | Rplus => plus
           | Rleb => Compare_dec.leb
-          | Rbeq_nat => EqNat.beq_nat
+          | Rbeq_nat => Nat.eqb
           | Rstring_beq => Equality.string_beq
           | Rlength _ => @List.length _
           | Rbool_rect_nodep _ => @bool_rect_nodep _
diff --git a/src/Parsers/Reflective/SyntaxEquivalenceReflective.v b/src/Parsers/Reflective/SyntaxEquivalenceReflective.v
index 5738b7b4f..693c35103 100644
--- a/src/Parsers/Reflective/SyntaxEquivalenceReflective.v
+++ b/src/Parsers/Reflective/SyntaxEquivalenceReflective.v
@@ -103,7 +103,7 @@ Section Term_equiv.
     : option pointed_Prop
     := match e1, e2 return option _ with
        | RVar t0 x, RVar t1 y
-         => match beq_nat (fst x) (fst y), List.nth_error G (List.length G - S (fst x)) with
+         => match Nat.eqb (fst x) (fst y), List.nth_error G (List.length G - S (fst x)) with
             | true, Some v => eq_type_and_var v (existT _ (t0, t1) (snd x, snd y))
             | _, _ => None
             end
diff --git a/src/Parsers/SimpleRecognizerCorrect.v b/src/Parsers/SimpleRecognizerCorrect.v
index 029bae677..1c042a53d 100644
--- a/src/Parsers/SimpleRecognizerCorrect.v
+++ b/src/Parsers/SimpleRecognizerCorrect.v
@@ -1,5 +1,5 @@
 (** * Proof that SimpleRecognizer outputs correct parse trees *)
-Require Import Coq.Classes.Morphisms.
+Require Import Coq.Classes.Morphisms Coq.Arith.PeanoNat.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
 Require Import Fiat.Parsers.ContextFreeGrammar.Core.
@@ -56,7 +56,7 @@ Section convenience.
     | _ => progress simpl in *
     | _ => progress simpl_simple_parse_of_correct
     | [ |- exists p, Some ?x = Some p /\ _ ] => exists x
-    | [ H : ?x = 0 \/ _, H' : andb (EqNat.beq_nat ?x (S _)) _ = true |- _ ]
+    | [ H : ?x = 0 \/ _, H' : andb (Nat.eqb ?x (S _)) _ = true |- _ ]
       => destruct H
     | [ H : andb _ (char_at_matches _ _ _) = true |- _ ]
       => apply char_at_matches_is_char_no_ex in H; [ | assumption ]
@@ -104,7 +104,7 @@ Section convenience.
     | [ |- context[take _ (take _ _)] ]
       => setoid_rewrite take_take
     | [ H : context[substring (?offset + ?loc) _ _] |- context[substring (_ + ?offset) _ _] ]
-      => rewrite (Plus.plus_comm offset loc) in H
+      => rewrite (Nat.add_comm offset loc) in H
     | [ H : context[to_production (production_tl _)] |- _ ]
       => rewrite production_tl_correct in H
     | [ H : ?x = _, H' : context[List.tl ?x] |- _ ]
diff --git a/src/Parsers/SplitterFromParserADT.v b/src/Parsers/SplitterFromParserADT.v
index 0589a0273..81b2bde66 100644
--- a/src/Parsers/SplitterFromParserADT.v
+++ b/src/Parsers/SplitterFromParserADT.v
@@ -184,7 +184,7 @@ Section parser.
     prove_string_eq.
   Qed.
   Definition mis_char (ch : Ascii.ascii) (str : cRep (projT1 splitter_impl)) : bool
-    := (EqNat.beq_nat (mlength str) 1 && mchar_at_matches 0 (ascii_beq ch) str)%bool.
+    := (Nat.eqb (mlength str) 1 && mchar_at_matches 0 (ascii_beq ch) str)%bool.
   Global Arguments mis_char : simpl never.
   Definition mis_char_eq {arg st str} (H : AbsR (projT2 splitter_impl) str st)
   : mis_char arg st = is_char str arg.
@@ -197,8 +197,8 @@ Section parser.
       destruct H' as [H'0 H'1].
       rewrite H'0, H'1; simpl.
       rewrite ascii_lb; reflexivity. }
-    { destruct (EqNat.beq_nat (length str) 1) eqn:H'0; simpl; [ | reflexivity ].
-      apply EqNat.beq_nat_true in H'0.
+    { destruct (Nat.eqb (length str) 1) eqn:H'0; simpl; [ | reflexivity ].
+      apply (fun n m => proj1 (Nat.eqb_eq n m)) in H'0.
       destruct (get 0 str) as [ch|] eqn:H'1; simpl; [ | exfalso ].
       { destruct (ascii_beq arg ch) eqn:H''; [ | reflexivity ].
         apply ascii_bl in H''; subst.
diff --git a/src/Parsers/Splitters/BruteForce.v b/src/Parsers/Splitters/BruteForce.v
index b19120ff3..1f3afa1c5 100644
--- a/src/Parsers/Splitters/BruteForce.v
+++ b/src/Parsers/Splitters/BruteForce.v
@@ -44,15 +44,15 @@ Section brute_force_splitter.
         try solve [ reflexivity
                   | left; reflexivity
                   | rewrite !take_long; trivial; reflexivity
-                  | rewrite take_long at 1 by (rewrite Properties.substring_length; apply Min.min_case_strong; omega);
+                  | rewrite take_long at 1 by (rewrite Properties.substring_length; apply Nat.min_case_strong; omega);
                     symmetry;
-                    rewrite take_long at 1 by (rewrite Properties.substring_length; apply Min.min_case_strong; omega);
+                    rewrite take_long at 1 by (rewrite Properties.substring_length; apply Nat.min_case_strong; omega);
                     reflexivity ]. }
       { eapply expand_minimal_parse_of_production; [ .. | eassumption ];
         try solve [ reflexivity
                   | left; reflexivity
                   | apply bool_eq_empty; rewrite drop_length; omega
-                  | apply bool_eq_empty; rewrite drop_length, Properties.substring_length; apply Min.min_case_strong; omega ]. } }
+                  | apply bool_eq_empty; rewrite drop_length, Properties.substring_length; apply Nat.min_case_strong; omega ]. } }
     { exists n; simpl; repeat split; try assumption.
       right; apply List.in_map, in_up_to; assumption. }
   Qed.
diff --git a/src/Parsers/Splitters/RDPList.v b/src/Parsers/Splitters/RDPList.v
index b3b03c8ac..163604cd8 100644
--- a/src/Parsers/Splitters/RDPList.v
+++ b/src/Parsers/Splitters/RDPList.v
@@ -26,14 +26,14 @@ Section recursive_descent_parser_list.
   Notation rdp_list_production_carrierT := default_production_carrierT.
 
   Definition list_bin_eq
-    := Eval unfold list_bin in list_bin EqNat.beq_nat.
+    := Eval unfold list_bin in list_bin Nat.eqb.
 
   Definition rdp_list_is_valid_nonterminal : rdp_list_nonterminals_listT -> rdp_list_nonterminal_carrierT -> bool
     := fun ls nt => list_bin_eq nt ls.
 
   Local Ltac fix_list_bin_eq :=
     unfold rdp_list_is_valid_nonterminal;
-    change list_bin_eq with (list_bin EqNat.beq_nat) in *.
+    change list_bin_eq with (list_bin Nat.eqb) in *.
 
   Local Notation valid_nonterminals := (map fst (pregrammar_productions G)).
 
@@ -77,8 +77,8 @@ Section recursive_descent_parser_list.
       | _ => progress subst
       | [ |- nth _ _ _ = _ ] => apply nth_error_Some_nth
       | [ H : ?T |- _ <-> ?T ] => split; [ intro; assumption | intro ]
-      | [ H : is_true (list_bin _ _ _) |- _ ] => apply (list_in_bl (@beq_nat_true)) in H
-      | [ |- is_true (list_bin _ _ _) ] => apply list_in_lb; [ intros ???; subst; symmetry; apply beq_nat_refl | ]
+      | [ H : is_true (list_bin _ _ _) |- _ ] => apply (list_in_bl (fun n m => proj1 (Nat.eqb_eq n m))) in H
+      | [ |- is_true (list_bin _ _ _) ] => apply list_in_lb; [ intros ???; subst; symmetry; apply eq_sym, Nat.eqb_refl | ]
       | [ H : In _ (up_to _) |- _ ] => apply in_up_to_iff in H
       | [ |- In _ (up_to _) ] => apply in_up_to_iff
       | [ H : In ?nt _ |- _ ] => let H' := fresh in assert (H' : string_beq nt nt = false) by eauto; exfalso; clear -H'
@@ -165,7 +165,7 @@ Section recursive_descent_parser_list.
     simpl in *.
     unfold rdp_list_to_nonterminal, rdp_list_of_nonterminal, rdp_list_is_valid_nonterminal, rdp_list_initial_nonterminals_data, default_to_nonterminal, default_of_nonterminal.
     intros nt H.
-    apply (list_in_bl (@beq_nat_true)), in_up_to_iff in H.
+    apply (list_in_bl (fun n m => proj1 (Nat.eqb_eq n m))), in_up_to_iff in H.
     revert nt H.
     replace valid_nonterminals with (uniquize string_beq valid_nonterminals) by (rewrite HNoDup; reflexivity).
     induction valid_nonterminals as [|x xs IHxs].
@@ -186,7 +186,7 @@ Section recursive_descent_parser_list.
         rewrite (string_lb eq_refl); trivial. }
       { simpl Datatypes.length.
         intro H.
-        apply lt_S_n in H.
+        apply Nat.succ_lt_mono in H.
         etransitivity; [ | eapply (f_equal S), (IHxs _ H); clear IHxs ].
         rewrite first_index_default_S_cons; simpl.
         match goal with
@@ -252,12 +252,12 @@ Section recursive_descent_parser_list.
     induction (Lookup_string G nt) as [|p ps IHps]; simpl.
     { reflexivity. }
     { simpl.
-      rewrite minus_diag; simpl.
+      rewrite Nat.sub_diag; simpl.
       apply f_equal.
       etransitivity; [ | apply IHps ]; clear IHps.
       apply map_ext_in.
       intros a H; apply in_up_to_iff in H.
-      rewrite NPeano.Nat.sub_succ_r.
+      rewrite Nat.sub_succ_r.
       destruct (Datatypes.length ps - a) eqn:H'; try omega; [].
       reflexivity. }
   Qed.
@@ -292,7 +292,7 @@ Section recursive_descent_parser_list.
     intros nt Hnt.
     apply Forall_map, Forall_forall; unfold compose; simpl; intros ? H.
     fix_list_bin_eq.
-    apply list_in_bl in Hnt; [ | exact (EqNat.beq_nat_true) ].
+    apply list_in_bl in Hnt; [ | exact (fun n m => proj1 (Nat.eqb_eq n m)) ].
     apply in_up_to_iff in Hnt.
     unfold default_production_carrier_valid; simpl.
     repeat (apply Bool.andb_true_iff; split); apply leb_iff.
@@ -324,7 +324,7 @@ Section recursive_descent_parser_list.
     { apply rdp_list_initial_nonterminals_correct'.
       unfold rdp_list_is_valid_nonterminal, rdp_list_initial_nonterminals_data.
       fix_list_bin_eq.
-      apply list_in_lb; [ apply beq_nat_true_iff | ].
+      apply list_in_lb; [ apply Nat.eqb_eq | ].
       rewrite map_length.
       apply in_up_to; assumption. }
     { unfold rdp_list_to_nonterminal.
@@ -337,7 +337,7 @@ Section recursive_descent_parser_list.
   Qed.
 
   Definition filter_out_eq nt ls
-    := Eval unfold filter_out in filter_out (EqNat.beq_nat nt) ls.
+    := Eval unfold filter_out in filter_out (Nat.eqb nt) ls.
 
   Definition rdp_list_remove_nonterminal : rdp_list_nonterminals_listT -> rdp_list_nonterminal_carrierT -> rdp_list_nonterminals_listT
     := fun ls nt =>
@@ -345,7 +345,7 @@ Section recursive_descent_parser_list.
 
   Local Ltac fix_filter_out_eq :=
     unfold rdp_list_remove_nonterminal;
-    change filter_out_eq with (fun nt ls => filter_out (EqNat.beq_nat nt) ls); cbv beta;
+    change filter_out_eq with (fun nt ls => filter_out (Nat.eqb nt) ls); cbv beta;
     setoid_rewrite filter_out_filter.
 
   Local Ltac fix_eqs := try fix_filter_out_eq; try fix_list_bin_eq.
@@ -357,7 +357,7 @@ Section recursive_descent_parser_list.
     induction ls; trivial; simpl in *.
     repeat match goal with
              | [ |- context[if ?a then _ else _] ] => destruct a; simpl in *
-             | [ |- S _ <= S _ ] => solve [ apply Le.le_n_S; auto ]
+             | [ |- S _ <= S _ ] => solve [ apply (fun n m => proj1 (Nat.succ_le_mono n m)); auto ]
              | [ |- _ <= S _ ] => solve [ apply le_S; auto ]
            end.
   Qed.
@@ -368,7 +368,7 @@ Section recursive_descent_parser_list.
     intros ls prods H.
     unfold rdp_list_is_valid_nonterminal, rdp_list_nonterminals_listT_R, rdp_list_remove_nonterminal, ltof in *.
     fix_eqs.
-    apply list_in_bl in H; [ | solve [ intros; apply beq_nat_true; trivial ] ].
+    apply list_in_bl in H; [ | solve [ intros; apply Nat.eqb_eq; trivial ] ].
     match goal with
       | [ H : In ?prods ?ls |- context[filter ?f ?ls] ]
         => assert (~In prods (filter f ls))
@@ -376,7 +376,7 @@ Section recursive_descent_parser_list.
     { intro H'.
       apply filter_In in H'.
       destruct H' as [? H'].
-      rewrite <- beq_nat_refl in H'.
+      rewrite Nat.eqb_refl in H'.
       simpl in *; congruence. }
     { match goal with
         | [ |- context[filter ?f ?ls] ] => generalize dependent f; intros
@@ -389,8 +389,8 @@ Section recursive_descent_parser_list.
                | [ H : ~(_ \/ _) |- _ ] => apply Decidable.not_or in H
                | [ H : _ /\ _ |- _ ] => destruct H
                | [ H : ?x <> ?x |- _ ] => exfalso; apply (H eq_refl)
-               | _ => apply Lt.lt_n_S
-               | _ => apply Le.le_n_S
+               | _ => apply (fun n m => (proj1 (Nat.succ_lt_mono n m)))
+               | _ => apply (fun n m => (proj1 (Nat.succ_le_mono n m)))
                | _ => apply filter_list_dec
                | [ H : _ -> _ -> ?G |- ?G ] => apply H; auto
              end. }
@@ -418,10 +418,10 @@ Section recursive_descent_parser_list.
              | [ H : In _ (filter _ _) |- _ ] => apply filter_In in H
              | [ H : _ /\ _ |- _ ] => destruct H
              | [ H : ?T, H' : ~?T |- _ ] => destruct (H' H)
-             | [ H : list_bin _ _ _ = true |- _ ] => apply list_in_bl in H; [ | solve [ intros; apply beq_nat_true; trivial ] ]
+             | [ H : list_bin _ _ _ = true |- _ ] => apply list_in_bl in H; [ | solve [ intros; apply (fun n m => (proj1 (Nat.eqb_eq n m))); trivial ] ]
              | [ |- list_bin _ _ _ = true ] => apply list_in_lb
              | _ => progress subst
-             | [ |- context[beq_nat ?x ?x] ] => rewrite <- beq_nat_refl
+             | [ |- context[Nat.eqb ?x ?x] ] => rewrite Nat.eqb_refl
            end.
   Qed.
 
@@ -454,17 +454,17 @@ Section recursive_descent_parser_list.
              | [ H : string_beq _ _ = true |- _ ] => apply string_bl in H
              | [ H : context[string_beq ?x ?x] |- _ ] => rewrite (string_lb (eq_refl x)) in H
              | _ => progress simpl in *
-             | [ H : list_bin _ _ _ = true |- _ ] => apply list_in_bl in H; [ | solve [ intros; apply beq_nat_true; trivial ] ]
+             | [ H : list_bin _ _ _ = true |- _ ] => apply list_in_bl in H; [ | solve [ intros; apply (fun n m => (proj1 (Nat.eqb_eq n m))); trivial ] ]
              | [ |- list_bin _ _ _ = true ] => apply list_in_lb
              | [ |- context[list_bin ?x ?y ?z = false] ] => case_eq (list_bin x y z)
              | [ H : list_bin _ _ _ = false |- _ ] => apply Bool.not_true_iff_false in H
              | [ H : list_bin _ _ _ <> true |- _ ] => specialize (fun pf H' => H (list_in_lb pf H'))
              | [ H : ?A -> ?B |- _ ]
                => let H' := fresh in
-                  assert (H' : A) by (intros; subst; rewrite <- ?beq_nat_refl; reflexivity);
+                  assert (H' : A) by (intros; subst; rewrite ?Nat.eqb_refl; reflexivity);
                     specialize (H H'); clear H'
-             | [ H : beq_nat _ _ = true |- _ ] => apply beq_nat_true in H
-             | [ H : context[beq_nat ?x ?x] |- _ ] => rewrite <- beq_nat_refl in H
+             | [ H : Nat.eqb _ _ = true |- _ ] => apply (fun n m => (proj1 (Nat.eqb_eq n m))) in H
+             | [ H : context[Nat.eqb ?x ?x] |- _ ] => rewrite Nat.eqb_refl in H
            end.
   Qed.
 
@@ -475,7 +475,7 @@ Section recursive_descent_parser_list.
     fix_eqs.
     intro H.
     cut (Datatypes.length ls < Datatypes.length ls); try omega.
-    eapply Lt.lt_le_trans; [ eassumption | ].
+    eapply Nat.lt_le_trans; [ eassumption | ].
     apply filter_list_dec.
   Qed.
 
diff --git a/src/Parsers/StringLike/FirstChar.v b/src/Parsers/StringLike/FirstChar.v
index 670465cda..7e53e9123 100644
--- a/src/Parsers/StringLike/FirstChar.v
+++ b/src/Parsers/StringLike/FirstChar.v
@@ -2,7 +2,6 @@ Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Mapping predicates over [StringLike] things *)
 
 Require Import Coq.ZArith.ZArith.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
 Require Import Fiat.Parsers.StringLike.ForallChars.
@@ -111,7 +110,7 @@ Section for_first_char.
                | [ H : _ |- _ ] => rewrite drop_length in H
                | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
                | _ => erewrite singleton_unique; eassumption
-               | [ H : context[min] |- _ ] => revert H; apply Min.min_case_strong
+               | [ H : context[min] |- _ ] => revert H; apply Nat.min_case_strong
              end. }
     { match goal with
         | [ H : _ |- _ ] => apply H
@@ -138,7 +137,7 @@ Section for_first_char.
   Proof.
     rewrite (for_first_char__take 0).
     assert (H' : length (take 1 str) = 1)
-      by (rewrite take_length; apply Min.min_case_strong; omega).
+      by (rewrite take_length; apply Nat.min_case_strong; omega).
     destruct (singleton_exists _ H') as [ch H''].
     rewrite for_first_char_singleton_length by exact H'.
     split; intros.
@@ -361,7 +360,7 @@ Section for_first_char.
       destruct (get 0 (take 1 str)) as [ch|] eqn:Heq.
       { exists ch; split; auto.
         rewrite is_char_parts.
-        rewrite take_length; split; [ apply Min.min_case_strong; omega | assumption ]. }
+        rewrite take_length; split; [ apply Nat.min_case_strong; omega | assumption ]. }
       { apply no_first_char_empty in Heq.
         rewrite take_length, H in Heq.
         simpl in *; omega. } }
diff --git a/src/Parsers/StringLike/FirstCharSuchThat.v b/src/Parsers/StringLike/FirstCharSuchThat.v
index 29bfb322e..c21133931 100644
--- a/src/Parsers/StringLike/FirstCharSuchThat.v
+++ b/src/Parsers/StringLike/FirstCharSuchThat.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Mapping predicates over [StringLike] things *)
-
-Require Import Coq.Numbers.Natural.Peano.NPeano.
+Require Import Coq.Arith.PeanoNat.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
 Require Import Fiat.Parsers.StringLike.ForallChars.
@@ -55,7 +54,7 @@ Section is_first_char_such_that.
     try solve [ left; assumption
               | right; omega
               | apply for_first_char_nil; omega
-              | apply forall_chars_nil; rewrite ?take_length; try apply Min.min_case_strong; intros; omega
+              | apply forall_chars_nil; rewrite ?take_length; try apply Nat.min_case_strong; intros; omega
               | destruct (length str);
                 try solve [ left; repeat split; eauto; omega
                           | right; repeat split; eauto; omega ] ].
@@ -90,8 +89,8 @@ Section is_first_char_such_that.
              | _ => omega
              | [ H : ?A /\ ?B -> ?C |- _ ] => specialize (fun x y => H (conj x y))
              | [ H : _ -> ?A /\ ?B -> ?C |- _ ] => specialize (fun a x y => H a (conj x y))
-             | [ H : _ |- _ ] => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?drop_take, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Plus.plus_comm 2), <- ?Nat.lt_add_lt_sub_r in H
-             | _ => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?drop_take, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Plus.plus_comm 2), <- ?Nat.lt_add_lt_sub_r
+             | [ H : _ |- _ ] => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?drop_take, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Nat.add_comm 2), <- ?Nat.lt_add_lt_sub_r in H
+             | _ => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?drop_take, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Nat.add_comm 2), <- ?Nat.lt_add_lt_sub_r
              | _ => progress destruct_head or
              | [ |- _ /\ _ ] => split
              | [ |- _ <-> _ ] => split
diff --git a/src/Parsers/StringLike/ForallChars.v b/src/Parsers/StringLike/ForallChars.v
index cdb8395d3..eb2e7023c 100644
--- a/src/Parsers/StringLike/ForallChars.v
+++ b/src/Parsers/StringLike/ForallChars.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Mapping predicates over [StringLike] things *)
 
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Coq.ZArith.ZArith.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
@@ -153,7 +152,7 @@ Section forall_chars.
                | [ H : _ |- _ ] => rewrite drop_length in H
                | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
                | _ => erewrite singleton_unique; eassumption
-               | [ H : context[min] |- _ ] => revert H; apply Min.min_case_strong
+               | [ H : context[min] |- _ ] => revert H; apply Nat.min_case_strong
              end. }
     { match goal with
         | [ H : _ |- _ ] => apply (H 0)
@@ -185,7 +184,7 @@ Section forall_chars.
     intros n H'''.
     rewrite H''' in *.
     rewrite sub_plus in H'' by omega.
-    rewrite Minus.minus_diag in *.
+    rewrite Nat.sub_diag in *.
     specialize (H'' eq_refl).
     destruct H'' as [ch H''].
     exfalso; eapply H', H.
diff --git a/src/Parsers/StringLike/LastChar.v b/src/Parsers/StringLike/LastChar.v
index bb5a60467..d9b0a4d3c 100644
--- a/src/Parsers/StringLike/LastChar.v
+++ b/src/Parsers/StringLike/LastChar.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Mapping predicates over [StringLike] things *)
 Require Import Coq.ZArith.ZArith.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
 Require Import Fiat.Parsers.StringLike.ForallChars.
@@ -133,7 +132,7 @@ Section for_last_char.
                | [ H : _ |- _ ] => rewrite drop_length in H
                | [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
                | _ => erewrite singleton_unique; eassumption
-               | [ H : context[min] |- _ ] => revert H; apply Min.min_case_strong
+               | [ H : context[min] |- _ ] => revert H; apply Nat.min_case_strong
              end. }
     { match goal with
         | [ H : _ |- _ ] => apply H
@@ -390,7 +389,7 @@ Section for_last_char.
     destruct (for_last_char_combine HR H0 H1) as [H2|]; [ | assumption ].
     contradict H2.
     rewrite drop_length, take_length.
-    apply Min.min_case_strong; omega.
+    apply Nat.min_case_strong; omega.
   Qed.
 
   Lemma for_last_char_exists_get (str : String) P (H : length str >= 1)
diff --git a/src/Parsers/StringLike/LastCharSuchThat.v b/src/Parsers/StringLike/LastCharSuchThat.v
index 33b515e3d..da0e69828 100644
--- a/src/Parsers/StringLike/LastCharSuchThat.v
+++ b/src/Parsers/StringLike/LastCharSuchThat.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Mapping predicates over [StringLike] things *)
 Require Import Coq.ZArith.ZArith.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Fiat.Parsers.StringLike.Core.
 Require Import Fiat.Parsers.StringLike.Properties.
 Require Import Fiat.Parsers.StringLike.ForallChars.
@@ -44,7 +43,7 @@ Section is_after_last_char_such_that.
     unfold is_after_last_char_such_that.
     split; [ apply forall_chars_nil | apply for_last_char_nil ];
     rewrite ?take_length, ?drop_length;
-    try apply Min.min_case_strong; omega.
+    try apply Nat.min_case_strong; omega.
   Qed.
 
   Lemma after_last_char_such_that_0
@@ -79,11 +78,11 @@ Section is_after_last_char_such_that.
     destruct_head_hnf and;
     destruct_head_hnf or;
     destruct_head_hnf and; repeat split; trivial;
-      try rewrite Min.min_idempotent in *;
+      try rewrite Nat.min_idempotent in *;
       try solve [ left; assumption
                 | right; omega
                 | apply for_last_char_nil; omega
-                | apply forall_chars_nil; rewrite ?drop_length, ?take_length; try apply Min.min_case_strong; intros; omega
+                | apply forall_chars_nil; rewrite ?drop_length, ?take_length; try apply Nat.min_case_strong; intros; omega
                 | destruct (length str);
                   try solve [ left; repeat split; eauto; omega
                             | right; repeat split; eauto; omega ] ].
@@ -126,8 +125,8 @@ Section is_after_last_char_such_that.
              | _ => omega
              | [ H : ?A /\ ?B -> ?C |- _ ] => specialize (fun x y => H (conj x y))
              | [ H : _ -> ?A /\ ?B -> ?C |- _ ] => specialize (fun a x y => H a (conj x y))
-             | [ H : _ |- _ ] => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?take_drop, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Plus.plus_comm 2), <- ?Nat.lt_add_lt_sub_r in H
-             | _ => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?take_drop, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Plus.plus_comm 2), <- ?Nat.lt_add_lt_sub_r
+             | [ H : _ |- _ ] => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?take_drop, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Nat.add_comm 2), <- ?Nat.lt_add_lt_sub_r in H
+             | _ => progress rewrite ?drop_0, ?drop_drop, ?take_take, ?drop_length, ?take_length, ?take_drop, ?Nat.add_1_r, ?Nat.le_sub_le_add_r, <- ?(Nat.add_comm 2), <- ?Nat.lt_add_lt_sub_r
              | _ => progress destruct_head or
              | [ |- _ /\ _ ] => split
              | [ |- _ <-> _ ] => split
@@ -138,9 +137,9 @@ Section is_after_last_char_such_that.
                => eapply for_last_char__add_drop; eexact H
 
              | [ H : for_last_char (drop _ _) _ |- _ ]
-               => rewrite <- for_last_char__drop in H by (rewrite ?take_length; apply Min.min_case_strong; omega)
+               => rewrite <- for_last_char__drop in H by (rewrite ?take_length; apply Nat.min_case_strong; omega)
              | [ |- forall_chars (drop _ (take _ _)) _ ]
-               => rewrite drop_take, <- minus_Sn_m, minus_diag by omega; simpl;
+               => rewrite drop_take, Nat.sub_succ_l, Nat.sub_diag by omega; simpl;
                     apply (proj1 (forall_chars_take _ _))
              | [ H : forall_chars (drop ?n ?str) ?P |- forall_chars (drop (S ?n) ?str) ?P ]
                => rewrite <- (drop_drop str 1 n); apply (proj1 (forall_chars_drop _ _)); assumption
@@ -222,7 +221,7 @@ Section is_after_last_char_such_that.
                  => destruct (fun pf => for_last_char_combine (T := T') pf H H'); [ cbv beta; tauto | | ];
                     clear H H'
                | [ H : length (take _ _) = 0 |- _ ] => rewrite take_length in H
-               | [ H : min ?x ?y = 0 |- _ ] => revert H; apply (Min.min_case_strong x y)
+               | [ H : min ?x ?y = 0 |- _ ] => revert H; apply (Nat.min_case_strong x y)
                | _ => intro
                end..
       | let H := fresh in
diff --git a/src/Parsers/StringLike/OcamlString.v b/src/Parsers/StringLike/OcamlString.v
index 66129921a..83e6a13f3 100644
--- a/src/Parsers/StringLike/OcamlString.v
+++ b/src/Parsers/StringLike/OcamlString.v
@@ -1,5 +1,4 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Coq.ZArith.ZArith.
 Require Import Coq.Strings.Ascii.
 Require Import Coq.Strings.String.
@@ -38,19 +37,19 @@ Local Ltac t' :=
     | _ => exfalso; congruence
     | _ => rewrite substring_length
     | _ => rewrite <- plus_n_O
-    | _ => rewrite <- Minus.minus_n_O
-    | _ => rewrite Min.min_l by omega
-    | _ => rewrite Min.min_r by omega
+    | _ => rewrite Nat.sub_0_r
+    | _ => rewrite Nat.min_l by omega
+    | _ => rewrite Nat.min_r by omega
     | _ => rewrite substring_correct3 by assumption
     | _ => rewrite substring_substring
     | _ => rewrite substring_correct3'
     | _ => rewrite substring_correct0
     | _ => rewrite Nat.sub_min_distr_l
     | _ => rewrite Nat.add_sub
-    | _ => rewrite Min.min_0_r
-    | _ => rewrite Min.min_0_l
+    | _ => rewrite Nat.min_0_r
+    | _ => rewrite Nat.min_0_l
     | _ => rewrite Nat.add_1_r
-    | _ => rewrite <- Min.min_assoc
+    | _ => rewrite <- Nat.min_assoc
     | [ |- String.get _ _ = _ ] => apply StringProperties.get_correct
     | _ => progress rewrite ?string_beq_correct, ?ascii_beq_correct, ?string_copy_length
     | [ H : _ |- _ ] => progress rewrite ?string_beq_correct, ?ascii_beq_correct in H
@@ -59,11 +58,11 @@ Local Ltac t' :=
         by (rewrite <- Nat.sub_min_distr_l; apply f_equal2; omega)
     | [ |- context[min (?m + ?n) ?m] ]
       => replace (min (m + n) m) with (m + min n 0)
-        by (rewrite <- Min.plus_min_distr_l; apply f_equal2; omega)
-    | _ => rewrite Min.min_comm; reflexivity
+        by (rewrite <- Nat.add_min_distr_l; apply f_equal2; omega)
+    | _ => rewrite Nat.min_comm; reflexivity
     | [ |- context[string_eq_dec ?x ?y] ] => destruct (string_eq_dec x y)
     | [ H : _ <> _ |- False ] => apply H; clear H
-    | _ => apply Max.max_case_strong; intro; apply substring_correct4; omega
+    | _ => apply Nat.max_case_strong; intro; apply substring_correct4; omega
     | [ H : Coq.Strings.String.length ?s = 1 |- _ ] => is_var s; destruct s
     | [ H : S (Coq.Strings.String.length ?s) = 1 |- _ ] => is_var s; destruct s
     | _ => eexists; rewrite (ascii_lb eq_refl); reflexivity
@@ -77,9 +76,9 @@ Local Ltac t' :=
         | rewrite substring_correct2 by omega ]
     | _ => rewrite <- substring_correct3'; apply substring_correct2; omega
     | [ H : forall n, Coq.Strings.String.get n _ = Coq.Strings.String.get n _ |- _ ] => apply get_correct in H
-    | [ H : EqNat.beq_nat _ _ = true |- _ ] => apply EqNat.beq_nat_true in H
-    | [ H : EqNat.beq_nat _ _ = false |- _ ] => apply EqNat.beq_nat_false in H
-    | [ H : context[EqNat.beq_nat ?x ?y] |- _ ] => destruct (EqNat.beq_nat x y) eqn:?
+    | [ H : Nat.eqb _ _ = true |- _ ] => apply (proj1 (Nat.eqb_eq _ _)) in H
+    | [ H : Nat.eqb _ _ = false |- _ ] => apply (proj1 (Nat.eqb_neq _ _)) in H
+    | [ H : context[Nat.eqb ?x ?y] |- _ ] => destruct (Nat.eqb x y) eqn:?
     | [ H : context[option_beq ascii_beq _ _] |- _ ] => rewrite (option_beq_correct (@ascii_bl) (@ascii_lb)) in H
     | [ H : context[option_eq_dec ?beq ?bl ?lb ?x ?y] |- _ ] => destruct (option_eq_dec beq bl lb x y)
     | _ => progress change (andb true) with (fun x : bool => x) in *
@@ -107,7 +106,7 @@ Local Ltac t' :=
     | [ |- String.substring ?n ?m ?s = String.substring ?n ?m' ?s ]
       => not constr_eq m m'; replace m with m' by omega
     | [ |- String.substring ?n ?m ?s = String.substring ?n ?m' ?s ]
-      => not constr_eq m m'; replace m with m' by (repeat apply Min.min_case_strong; intros; omega)
+      => not constr_eq m m'; replace m with m' by (repeat apply Nat.min_case_strong; intros; omega)
   end.
 
 Local Ltac t := repeat t'.
@@ -123,7 +122,7 @@ Module Export Ocaml.
     := { get n s := String.safe_get s n;
          take n s := String.sub s 0 n;
          drop n s := String.sub s n (String.length s - n);
-         is_char s ch := ((EqNat.beq_nat (String.length s) 1)
+         is_char s ch := ((Nat.eqb (String.length s) 1)
                             && (option_beq ascii_beq (String.safe_get s 0) (Some ch)))%bool;
          bool_eq s1 s2 := Zbool.Zeq_bool (z_of_int (Ocaml.String.compare s1 s2)) 0%Z }.
 
diff --git a/src/Parsers/StringLike/Properties.v b/src/Parsers/StringLike/Properties.v
index ff95f9929..56bd38370 100644
--- a/src/Parsers/StringLike/Properties.v
+++ b/src/Parsers/StringLike/Properties.v
@@ -1,7 +1,6 @@
 Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 (** * Theorems about string-like types *)
 
-Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Coq.ZArith.ZArith.
 Require Import Fiat.Common.
 Require Import Fiat.Common.List.Operations.
@@ -147,10 +146,10 @@ Section String.
   Proof.
     destruct (le_ge_dec (length str) n).
     { rewrite take_long by assumption.
-      rewrite Min.min_r by assumption.
+      rewrite Nat.min_r by assumption.
       reflexivity. }
     { rewrite take_short_length by assumption.
-      rewrite Min.min_l by assumption.
+      rewrite Nat.min_l by assumption.
       reflexivity. }
   Qed.
 
@@ -188,14 +187,14 @@ Section String.
       destruct H0' as [H''|H'']; try clear H0'; [ apply f_equal | exfalso ].
       { apply le_proof_irrelevance. }
       { setoid_subst s1.
-        eapply lt_irrefl; eassumption. }
+        eapply Nat.lt_irrefl; eassumption. }
     Qed.
 
     Definition strle_right (H' : s1 =s s2)
     : H0' = or_intror H'.
     Proof.
       destruct H0' as [H''|H'']; try clear H0'; [ exfalso | apply f_equal ].
-      { setoid_subst s1; eapply lt_irrefl; eassumption. }
+      { setoid_subst s1; eapply Nat.lt_irrefl; eassumption. }
       { apply dec_eq_uip.
         decide equality. }
     Qed.
@@ -208,7 +207,7 @@ Section String.
   : False.
   Proof.
     rewrite H' in H0'.
-    eapply lt_irrefl; eassumption.
+    eapply Nat.lt_irrefl; eassumption.
   Qed.
 
   Lemma singleton_exists_unique : forall s, length s = 1 -> exists !ch, s ~= [ ch ].
@@ -236,7 +235,7 @@ Section String.
   Lemma take_empty {str n} (H' : length str = 0) : take n str =s str.
   Proof.
     apply bool_eq_empty; rewrite ?drop_length, ?take_length; trivial.
-    apply Min.min_case_strong; omega.
+    apply Nat.min_case_strong; omega.
   Qed.
 
   Definition get_first_char_nonempty' str (H' : length str <> 0) : Char.
@@ -380,7 +379,7 @@ Section String.
     { intros.
       rewrite !(@get_drop (length _ - _)).
       rewrite drop_length, drop_drop.
-      rewrite <- Nat.sub_add_distr, (plus_comm m), Nat.sub_add_distr, Nat.sub_add by omega.
+      rewrite <- Nat.sub_add_distr, (Nat.add_comm m), Nat.sub_add_distr, Nat.sub_add by omega.
       match goal with
         | [ |- context[match ?e with _ => _ end] ] => destruct e eqn:?
       end.
@@ -413,9 +412,9 @@ Section String.
         congruence. }
       { intros ? H''.
         rewrite fold'_cons.
-        rewrite !H'', minus_diag, H'.
+        rewrite !H'', Nat.sub_diag, H'.
         rewrite drop_length, H''; simpl.
-        rewrite <- minus_n_O.
+        rewrite Nat.sub_0_r.
         apply f_equal.
         rewrite <- fold'_drop by omega; reflexivity. } }
     { intro H'.
@@ -475,12 +474,12 @@ Section String.
     { intros.
       rewrite !(@get_drop (length _ - _)).
       rewrite drop_length, drop_drop.
-      rewrite <- Nat.sub_add_distr, (plus_comm m), Nat.sub_add_distr, Nat.sub_add by omega.
+      rewrite <- Nat.sub_add_distr, (Nat.add_comm m), Nat.sub_add_distr, Nat.sub_add by omega.
       match goal with
         | [ |- context[match ?e with _ => _ end] ] => destruct e eqn:?
       end.
       { rewrite drop_drop;
-        rewrite <- Nat.sub_add_distr, (plus_comm m), Nat.sub_add_distr, Nat.sub_add by omega;
+        rewrite <- Nat.sub_add_distr, (Nat.add_comm m), Nat.sub_add_distr, Nat.sub_add by omega;
         rewrite <- IHlen by omega; reflexivity. }
       { reflexivity. } }
   Qed.
@@ -510,10 +509,10 @@ Section String.
         congruence. }
       { intros n H''.
         rewrite fold_lookahead'_cons.
-        rewrite !H'', minus_diag, H'.
+        rewrite !H'', Nat.sub_diag, H'.
         replace (S n - n) with 1 by omega.
         rewrite drop_length, H''; simpl.
-        rewrite <- minus_n_O.
+        rewrite Nat.sub_0_r.
         apply f_equal.
         rewrite <- fold_lookahead'_drop by omega; reflexivity. } }
     { intro H'.
@@ -565,14 +564,14 @@ Section String.
              | [ H : length ?str = 0, H' : is_true (take _ ?str ~= [ _ ]) |- _ ]
                => apply length_singleton in H'
              | [ H : ?x = 0, H' : context[?x] |- _ ] => rewrite H in H'
-             | [ H : context[min _ _] |- _ ] => revert H; apply Min.min_case_strong; intros
+             | [ H : context[min _ _] |- _ ] => revert H; apply Nat.min_case_strong; intros
            end.
   Qed.
 
   Lemma take_min_length (str : String) n
   : take n str =s take (min (length str) n) str.
   Proof.
-    apply Min.min_case_strong; [ | reflexivity ].
+    apply Nat.min_case_strong; [ | reflexivity ].
     intro H.
     rewrite !take_long by (assumption || reflexivity).
     reflexivity.
@@ -581,7 +580,7 @@ Section String.
   Lemma drop_min_length (str : String) n
   : drop n str =s drop (min (length str) n) str.
   Proof.
-    apply Min.min_case_strong; [ | reflexivity ].
+    apply Nat.min_case_strong; [ | reflexivity ].
     intro H.
     apply bool_eq_empty; rewrite drop_length; omega.
   Qed.
@@ -626,7 +625,7 @@ Section String.
       | specialize (IHlen (drop 1 str));
         rewrite drop_length in IHlen;
         specialize_by omega;
-        try rewrite <- !H, !minus_diag;
+        try rewrite <- !H, !Nat.sub_diag;
         try rewrite <- fold'_drop in IHlen by omega ].
 
   Lemma to_string_length str
@@ -658,7 +657,7 @@ Section String.
              | [ |- context[length (take _ _)] ] => rewrite take_length
              | [ |- context[_ - 0] ] => rewrite Nat.sub_0_r
              | [ H : context[_ - 0] |- _ ] => rewrite Nat.sub_0_r in H
-             | [ |- context[fold' _ _ (drop _ _) _] ] => rewrite <- fold'_drop by (rewrite ?take_length; try apply Min.min_case_strong; omega)
+             | [ |- context[fold' _ _ (drop _ _) _] ] => rewrite <- fold'_drop by (rewrite ?take_length; try apply Nat.min_case_strong; omega)
              | [ H : context[fold' _ _ (drop _ _) _] |- _ ] => rewrite <- fold'_drop in H by omega
              | [ H : ?x = ?y |- context[?x - ?y] ] => replace (x - y) with 0 by omega
              | [ H : ?y = ?x |- context[?x - ?y] ] => replace (x - y) with 0 by omega
@@ -672,12 +671,12 @@ Section String.
              | [ |- context[?x + 1] ] => rewrite Nat.add_1_r
              | [ H : S _ = ?x |- context[match ?x with _ => _ end] ] => rewrite <- H
              | [ H : S _ = ?x, H' : context[match ?x with _ => _ end] |- _ ] => rewrite <- H in H'
-             | [ H : context[?x - ?x] |- _ ] => rewrite minus_diag in H
-             | [ |- context[?x - ?x] ] => rewrite minus_diag
+             | [ H : context[?x - ?x] |- _ ] => rewrite Nat.sub_diag in H
+             | [ |- context[?x - ?x] ] => rewrite Nat.sub_diag
              | [ H : context[get _ (take _ _)] |- _ ] => rewrite get_take_lt in H by omega
-             | [ H : context[min (pred ?x) ?x] |- _ ] => rewrite (Min.min_l (pred x) x) in H by omega
+             | [ H : context[min (pred ?x) ?x] |- _ ] => rewrite (Nat.min_l (pred x) x) in H by omega
              | [ |- context[take _ (drop _ _)] ] => rewrite take_drop
-             | [ |- context[min ?x ?y] ] => rewrite (Min.min_l x y) by omega
+             | [ |- context[min ?x ?y] ] => rewrite (Nat.min_l x y) by omega
              | [ |- context[get _ (drop 0 ?s)] ] => rewrite (drop_0 s)
              | [ |- context[get _ (drop _ (drop _ _))] ] => rewrite drop_drop
              | [ H : S ?x = ?y, H' : S ?z = ?y |- _ ]
@@ -773,9 +772,9 @@ Section String.
         | [ |- context[min ?x ?y] ]
           => match goal with
                | [ |- context[min y x] ]
-                 => rewrite (Min.min_comm x y)
+                 => rewrite (Nat.min_comm x y)
              end
-        | _ => repeat apply Min.min_case_strong; omega
+        | _ => repeat apply Nat.min_case_strong; omega
       end.
 
     Local Ltac substring_t := repeat substring_t'.
@@ -824,14 +823,14 @@ Section String.
     Proof.
       pose proof (fun y x => @substring_substring str 0 z x y) as H'; simpl in *.
       setoid_rewrite Nat.sub_0_r in H'.
-      setoid_rewrite Min.min_comm in H'.
+      setoid_rewrite Nat.min_comm in H'.
       rewrite <- !H', H; reflexivity.
     Qed.
 
     Lemma substring_min_length str x y
     : substring x (min y (length str)) str =s substring x y str.
     Proof.
-      apply Min.min_case_strong; try reflexivity.
+      apply Nat.min_case_strong; try reflexivity.
       intro H.
       apply substring_correct4; omega.
     Qed.
@@ -840,16 +839,16 @@ Section String.
     : length (substring offset len s) = len.
     Proof.
       rewrite substring_length; destruct Hshort as [Hshort|Hshort];
-      try rewrite Min.min_r by (clear -Hshort; omega);
+      try rewrite Nat.min_r by (clear -Hshort; omega);
       subst; simpl;
-      rewrite <- ?NPeano.Nat.sub_min_distr_r, ?NPeano.Nat.add_sub, ?minus_diag, ?Min.min_0_r;
+      rewrite <- ?Nat.sub_min_distr_r, ?Nat.add_sub, ?Nat.sub_diag, ?Nat.min_0_r;
       reflexivity.
     Qed.
   End substring.
 
   Lemma char_at_matches_is_char_no_ex offset len str P
         (Hlen : offset + len <= length str)
-  : (EqNat.beq_nat len 1 && char_at_matches offset str P)%bool = true
+  : (Nat.eqb len 1 && char_at_matches offset str P)%bool = true
     <-> match get offset str with
           | Some ch => (P ch /\ substring offset len str ~= [ch])%string_like
           | None => False
@@ -859,7 +858,7 @@ Section String.
     { split; intro H.
       { apply Bool.andb_true_iff in H.
         destruct H as [H0 H1].
-        apply EqNat.beq_nat_true in H0; subst.
+        apply Nat.eqb_eq in H0; subst.
         erewrite char_at_matches_correct in H1 by eassumption.
         split; try assumption.
         rewrite get_drop in Heq.
@@ -871,22 +870,22 @@ Section String.
         apply Bool.andb_true_iff; split; [ | eassumption ].
         apply length_singleton in H1.
         rewrite substring_length, <- Nat.sub_min_distr_r, Nat.add_sub in H1.
-        apply EqNat.beq_nat_true_iff.
+        apply Nat.eqb_eq.
         revert H1.
-        apply Min.min_case_strong; intros; omega. } }
+        apply Nat.min_case_strong; intros; omega. } }
     { split; [ intro H | intros [] ].
       rewrite get_drop in Heq.
       apply no_first_char_empty in Heq.
       rewrite drop_length in Heq.
       apply Bool.andb_true_iff in H.
       destruct H as [H0 H1].
-      apply EqNat.beq_nat_true in H0; subst.
+      apply Nat.eqb_eq in H0; subst.
       omega. }
   Qed.
 
   Lemma char_at_matches_is_char offset len str P
         (Hlen : offset + len <= length str)
-  : (EqNat.beq_nat len 1 && char_at_matches offset str P)%bool = true
+  : (Nat.eqb len 1 && char_at_matches offset str P)%bool = true
     <-> (exists ch, P ch /\ substring offset len str ~= [ch])%string_like.
   Proof.
     rewrite char_at_matches_is_char_no_ex by assumption.
@@ -907,7 +906,7 @@ Section String.
       rewrite drop_length in Heq.
       apply length_singleton in H1.
       assert (len = 0) by omega; subst.
-      rewrite substring_length, <- Nat.sub_min_distr_r, Nat.add_sub, Min.min_0_r in H1.
+      rewrite substring_length, <- Nat.sub_min_distr_r, Nat.add_sub, Nat.min_0_r in H1.
       omega. }
   Qed.
 
@@ -970,7 +969,7 @@ Section Iso.
                end. }
       { intro n'; rewrite !get_S.
         rewrite drop_take, !drop_of_string; simpl.
-        rewrite NPeano.Nat.sub_0_r.
+        rewrite Nat.sub_0_r.
         destruct str; simpl;
         rewrite !IHn; simpl; trivial.
         destruct n; reflexivity. } }
@@ -1098,8 +1097,8 @@ Ltac simpl_string_like_no_setoid_step :=
   | [ H : @length ?Char ?HSLM ?str = @length ?Char ?HSLM ?str' |- _ ]
     => first [ generalize dependent (@length Char HSLM str'); intros; subst; clear str'
              | generalize dependent (@length Char HSLM str); intros; subst; clear str ]
-  | [ H : _ |- _ ] => progress rewrite ?take_length, ?drop_length, ?Min.min_0_r, ?Min.min_0_l, ?Nat.sub_0_l in H
-  | _ => progress rewrite ?take_length, ?drop_length, ?Min.min_0_r, ?Min.min_0_l, ?Nat.sub_0_l
+  | [ H : _ |- _ ] => progress rewrite ?take_length, ?drop_length, ?Nat.min_0_r, ?Nat.min_0_l, ?Nat.sub_0_l in H
+  | _ => progress rewrite ?take_length, ?drop_length, ?Nat.min_0_r, ?Nat.min_0_l, ?Nat.sub_0_l
   end.
 Ltac simpl_string_like_step :=
   first [ progress simpl_string_like_no_setoid_step
diff --git a/src/Parsers/StringLike/String.v b/src/Parsers/StringLike/String.v
index 83cfb16bc..f4cf71e53 100644
--- a/src/Parsers/StringLike/String.v
+++ b/src/Parsers/StringLike/String.v
@@ -3,7 +3,7 @@ Require Export Fiat.Common.Coq__8_4__8_5__Compat.
 Require Import Coq.Strings.Ascii.
 Require Import Coq.Strings.String.
 Require Import Coq.ZArith.ZArith.
-Require Import Coq.Numbers.Natural.Peano.NPeano.
+Require Import Coq.Arith.PeanoNat.
 Require Import Fiat.Common Fiat.Common.Equality.
 Require Import Fiat.Common.StringOperations Fiat.Common.StringFacts.
 Require Import Fiat.Parsers.StringLike.Core.
@@ -61,19 +61,19 @@ Local Ltac t :=
            | _ => exfalso; congruence
            | _ => rewrite substring_length
            | _ => rewrite <- plus_n_O
-           | _ => rewrite <- Minus.minus_n_O
-           | _ => rewrite Min.min_l by omega
-           | _ => rewrite Min.min_r by omega
+           | _ => rewrite Nat.sub_0_r
+           | _ => rewrite Nat.min_l by omega
+           | _ => rewrite Nat.min_r by omega
            | _ => rewrite substring_correct3 by assumption
            | _ => rewrite substring_substring
            | _ => rewrite substring_correct3'
            | _ => rewrite substring_correct0
            | _ => rewrite Nat.sub_min_distr_l
            | _ => rewrite Nat.add_sub
-           | _ => rewrite Min.min_0_r
-           | _ => rewrite Min.min_0_l
+           | _ => rewrite Nat.min_0_r
+           | _ => rewrite Nat.min_0_l
            | _ => rewrite Nat.add_1_r
-           | _ => rewrite <- Min.min_assoc
+           | _ => rewrite <- Nat.min_assoc
            | [ H : ?x = Some _ |- context[match ?x with _ => _ end] ] => rewrite H
            | _ => progress rewrite ?string_beq_correct, ?ascii_beq_correct, ?string_copy_length
            | [ H : _ |- _ ] => progress rewrite ?string_beq_correct, ?ascii_beq_correct in H
@@ -82,11 +82,11 @@ Local Ltac t :=
                by (rewrite <- Nat.sub_min_distr_l; apply f_equal2; omega)
            | [ |- context[min (?m + ?n) ?m] ]
              => replace (min (m + n) m) with (m + min n 0)
-               by (rewrite <- Min.plus_min_distr_l; apply f_equal2; omega)
-           | _ => rewrite Min.min_comm; reflexivity
+               by (rewrite <- Nat.add_min_distr_l; apply f_equal2; omega)
+           | _ => rewrite Nat.min_comm; reflexivity
            | [ |- context[string_eq_dec ?x ?y] ] => destruct (string_eq_dec x y)
            | [ H : _ <> _ |- False ] => apply H; clear H
-           | _ => apply Max.max_case_strong; intro; apply substring_correct4; omega
+           | _ => apply Nat.max_case_strong; intro; apply substring_correct4; omega
            | [ H : String.length ?s = 1 |- _ ] => is_var s; destruct s
            | [ H : S (String.length ?s) = 1 |- _ ] => is_var s; destruct s
            | _ => eexists; rewrite (ascii_lb eq_refl); reflexivity
@@ -124,7 +124,7 @@ Lemma substring_take_drop (str : String) n m
 Proof.
   simpl.
   rewrite substring_substring; simpl.
-  apply Min.min_case_strong; simpl; trivial; [].
+  apply Nat.min_case_strong; simpl; trivial; [].
   intro H.
   apply substring_correct4; omega.
 Qed.
diff --git a/src/QueryStructure/Implementation/Constraints/ConstraintChecksRefinements.v b/src/QueryStructure/Implementation/Constraints/ConstraintChecksRefinements.v
index 94f6f4572..3a41df4ff 100644
--- a/src/QueryStructure/Implementation/Constraints/ConstraintChecksRefinements.v
+++ b/src/QueryStructure/Implementation/Constraints/ConstraintChecksRefinements.v
@@ -819,7 +819,7 @@ Lemma refine_constraint_check_into_QueryResultComp :
                               (R tup2 /\ P' tup2))))
          (Bind
             (Count (For (QueryResultComp R (fun tup => Where (P tup) Return tup))))
-            (fun count => ret (negb (beq_nat count 0)))).
+            (fun count => ret (negb (Nat.eqb count 0)))).
 Proof.
   Local Transparent Count.
   unfold refine, Count, UnConstrQuery_In;
@@ -853,7 +853,7 @@ Lemma refine_constraint_check_into_query' :
                               (GetUnConstrRelation c tbl tup2 /\ P' tup2))))
          (Bind
             (Count (For (UnConstrQuery_In c tbl (fun tup => Where (P tup) Return tup))))
-            (fun count => ret (negb (beq_nat count 0)))).
+            (fun count => ret (negb (Nat.eqb count 0)))).
 Proof.
   intros; rewrite refine_constraint_check_into_QueryResultComp; eauto.
   reflexivity.
@@ -869,7 +869,7 @@ Corollary refine_constraint_check_into_query :
                            (GetUnConstrRelation c tbl tup2 /\ P (indexedRawTuple tup2)))))
       (Bind
          (Count (For (UnConstrQuery_In c tbl (fun tup => Where (P tup) Return tup))))
-         (fun count => ret (negb (beq_nat count 0)))).
+         (fun count => ret (negb (Nat.eqb count 0)))).
 Proof.
   intros.
   setoid_rewrite refine_constraint_check_into_query'; eauto.
@@ -888,7 +888,7 @@ Lemma refine_constraint_check_into_query'' :
                               (R tup2 /\ P' tup2))))
          (Bind
             (Count (For (QueryResultComp R (fun tup => Where (P tup) Return tup))))
-            (fun count => ret (negb (beq_nat count 0)))).
+            (fun count => ret (negb (Nat.eqb count 0)))).
 Proof.
   Local Transparent Count.
   unfold refine, Count, UnConstrQuery_In;
@@ -942,7 +942,7 @@ Lemma refine_functional_dependency_check_into_query :
                                         Where (tupleAgree_computational ref tup args1 /\
                                                ~ tupleAgree_computational ref tup args2)
                                               Return tup)))
-            (fun count => ret (beq_nat count 0))).
+            (fun count => ret (Nat.eqb count 0))).
 Proof.
   intros * is_dec ** .
 
@@ -994,7 +994,7 @@ Lemma refine_functional_dependency_check_into_query' :
                                         Where (tupleAgree_computational tup ref args1 /\
                                                ~ tupleAgree_computational tup ref args2)
                                               Return tup)))
-            (fun count => ret (beq_nat count 0))).
+            (fun count => ret (Nat.eqb count 0))).
 Proof.
   intros * is_dec ** .
 
@@ -1122,7 +1122,7 @@ Lemma refine_uniqueness_check_into_query' :
                                         Where (GetAttributeRaw tup attr
                                                = GetAttributeRaw tup' attr)
                                               Return tup')));
-            (ret (beq_nat c 0))).
+            (ret (Nat.eqb c 0))).
 Proof.
   intros.
   setoid_replace (forall tup', GetUnConstrRelation c idx tup' ->
diff --git a/src/QueryStructure/Implementation/DataStructures/Bags/BagsProperties.v b/src/QueryStructure/Implementation/DataStructures/Bags/BagsProperties.v
index fa07743a0..909a28a83 100644
--- a/src/QueryStructure/Implementation/DataStructures/Bags/BagsProperties.v
+++ b/src/QueryStructure/Implementation/DataStructures/Bags/BagsProperties.v
@@ -65,7 +65,7 @@ Section BagsProperties.
   Proof.
     unfold _BagInsertCount, _bcount; intros;
     rewrite binsert_enumerate; simpl; destruct (bfind_matcher search_term item); simpl; eauto.
-    rewrite plus_comm; reflexivity.
+    rewrite Nat.add_comm; reflexivity.
   Qed.
 
 End BagsProperties.
diff --git a/src/QueryStructure/Implementation/DataStructures/Bags/CountingListBags.v b/src/QueryStructure/Implementation/DataStructures/Bags/CountingListBags.v
index 0d9bdb835..2f65d8eed 100644
--- a/src/QueryStructure/Implementation/DataStructures/Bags/CountingListBags.v
+++ b/src/QueryStructure/Implementation/DataStructures/Bags/CountingListBags.v
@@ -213,7 +213,7 @@ Section CountingListBags.
     + trivial.
     + simpl; destruct (search_term a);
       simpl; rewrite <- IHcontainer; simpl; eauto.
-      rewrite (plus_comm 1); eauto using plus_assoc.
+      rewrite (Nat.add_comm 1); eauto using Nat.add_assoc.
   Qed.
 
   Lemma CountingList_BagCountCorrect :
@@ -222,7 +222,7 @@ Section CountingListBags.
     unfold BagCountCorrect, CountingListAsBag_bcount, CountingListAsBag_bfind,
     CountingList_RepInv; intros;
     pose proof (CountingList_BagCountCorrect_aux (ccontents container) search_term 0) as temp;
-    rewrite plus_0_r in temp; simpl in temp.
+    rewrite Nat.add_0_r in temp; simpl in temp.
     simpl; [ apply temp ].
   Qed.
 
diff --git a/src/QueryStructure/Implementation/DataStructures/Bags/ListBags.v b/src/QueryStructure/Implementation/DataStructures/Bags/ListBags.v
index 2e8b9ca80..44654cd65 100644
--- a/src/QueryStructure/Implementation/DataStructures/Bags/ListBags.v
+++ b/src/QueryStructure/Implementation/DataStructures/Bags/ListBags.v
@@ -84,7 +84,7 @@ Section ListBags.
   Proof.
     unfold BagCountCorrect, ListAsBag_bcount, ListAsBag_bfind; intros;
     pose proof (List_BagCountCorrect_aux container search_term 0) as temp;
-    rewrite plus_0_r in temp; simpl in temp; exact temp.
+    rewrite Nat.add_0_r in temp; simpl in temp; exact temp.
   Qed.
 
   Lemma List_BagDeleteCorrect :
diff --git a/src/QueryStructure/Implementation/DataStructures/Bags/RangeTreeBags.v b/src/QueryStructure/Implementation/DataStructures/Bags/RangeTreeBags.v
index fbd43c797..4b0feba8f 100644
--- a/src/QueryStructure/Implementation/DataStructures/Bags/RangeTreeBags.v
+++ b/src/QueryStructure/Implementation/DataStructures/Bags/RangeTreeBags.v
@@ -2,6 +2,7 @@ Require Export Fiat.QueryStructure.Implementation.DataStructures.Bags.BagsInterf
         Fiat.QueryStructure.Implementation.DataStructures.Bags.BagsProperties.
 
 Require Import
+        Coq.Arith.PeanoNat
         Coq.FSets.FMapInterface
         Coq.FSets.FMapFacts
         Coq.FSets.FMapAVL
@@ -434,13 +435,13 @@ Module RangeTreeBag (X : OrderedType).
         eapply binsert_RepInv; eapply containerCorrect; eauto.
         eapply binsert_RepInv; eapply bempty_RepInv.
       - rewrite <- is_eq.
-        rewrite binsert_enumerate_weak with (RepInv0 := RepInv) (ValidUpdate0 := ValidUpdate) in H;
+        rewrite binsert_enumerate_weak with (RepInv := RepInv) (ValidUpdate := ValidUpdate) in H;
           intuition; subst.
         destruct (FindWithDefault_dec (projection item') bempty container)
           as [ [bag' (mapsto & equality)] | (not_found & equality) ];
         rewrite equality in *; clear equality.
         eapply containerCorrect; eauto.
-        rewrite benumerate_empty_eq_nil with (RepInv0 := RepInv) (ValidUpdate0 := ValidUpdate) in H0;
+        rewrite benumerate_empty_eq_nil with (RepInv := RepInv) (ValidUpdate := ValidUpdate) in H0;
           simpl in H0; intuition.
         reflexivity.
         destruct (FindWithDefault_dec (projection item') bempty container)
@@ -565,7 +566,7 @@ Module RangeTreeBag (X : OrderedType).
       rewrite values_add_not_in by assumption.
       simpl.
       rewrite binsert_enumerate.
-      rewrite benumerate_empty_eq_nil with (RepInv0 := RepInv) (ValidUpdate0 := ValidUpdate); eauto.
+      rewrite benumerate_empty_eq_nil with (RepInv := RepInv) (ValidUpdate := ValidUpdate); eauto.
       apply bempty_RepInv.
     Qed.
 
@@ -578,7 +579,7 @@ Module RangeTreeBag (X : OrderedType).
       pose proof (RangeTreeBag_btraverse_closed container (minkey, maxkey)).
       rewrite <- Heqbs in H; clear Heqbs.
       rewrite length_flatten.
-      setoid_rewrite Plus.plus_comm at 1.
+      setoid_rewrite Nat.add_comm at 1.
       replace (fun (y : TKey * BagType) (x : nat) => bcount (snd y) searchterm + x)
       with (compose plus (fun (y : TKey * BagType) => bcount (snd y) searchterm)) by reflexivity.
       rewrite !foldright_compose.
diff --git a/src/QueryStructure/Implementation/DataStructures/Bags/TreeBags.v b/src/QueryStructure/Implementation/DataStructures/Bags/TreeBags.v
index 3684daa8e..bdb8de851 100644
--- a/src/QueryStructure/Implementation/DataStructures/Bags/TreeBags.v
+++ b/src/QueryStructure/Implementation/DataStructures/Bags/TreeBags.v
@@ -4,6 +4,7 @@ Require Export Fiat.QueryStructure.Implementation.DataStructures.Bags.BagsInterf
         Fiat.QueryStructure.Implementation.DataStructures.Bags.BagsProperties.
 Require Import
         Coq.FSets.FMapInterface Coq.FSets.FMapFacts
+        Coq.Arith.PeanoNat
         Fiat.Common
         Fiat.Common.List.ListFacts
         Fiat.Common.List.FlattenList
@@ -203,13 +204,13 @@ Module TreeBag (Import M: WS).
         eapply binsert_RepInv; eapply containerCorrect; eauto.
         eapply binsert_RepInv; eapply bempty_RepInv.
       - rewrite <- is_eq.
-        rewrite binsert_enumerate_weak with (RepInv0 := RepInv) (ValidUpdate0 := ValidUpdate) in H; intuition; subst.
+        rewrite binsert_enumerate_weak with (RepInv := RepInv) (ValidUpdate := ValidUpdate) in H; intuition; subst.
 
         destruct (FindWithDefault_dec (projection item') bempty container)
           as [ [bag' (mapsto & equality)] | (not_found & equality) ];
         rewrite equality in *; clear equality.
         eapply containerCorrect; eauto.
-        rewrite benumerate_empty_eq_nil with (RepInv0 := RepInv) (ValidUpdate0 := ValidUpdate) in H0;
+        rewrite benumerate_empty_eq_nil with (RepInv := RepInv) (ValidUpdate := ValidUpdate) in H0;
           simpl in H0; intuition.
         reflexivity.
         destruct (FindWithDefault_dec (projection item') bempty container)
@@ -368,7 +369,7 @@ Module TreeBag (Import M: WS).
       simpl.
 
       rewrite binsert_enumerate.
-      rewrite benumerate_empty_eq_nil with (RepInv0 := RepInv) (ValidUpdate0 := ValidUpdate); eauto.
+      rewrite benumerate_empty_eq_nil with (RepInv := RepInv) (ValidUpdate := ValidUpdate); eauto.
       apply bempty_RepInv.
     Qed.
 
@@ -396,7 +397,7 @@ Module TreeBag (Import M: WS).
       + rewrite length_flatten.
         rewrite (fold_over_Values _ _ (fun acc bag => acc + bcount bag search_term)) by eauto.
         rewrite <- fold_left_rev_right.
-        setoid_rewrite Plus.plus_comm at 1.
+        setoid_rewrite Nat.add_comm at 1.
         replace (fun (y : BagType) (x : nat) => bcount y search_term + x)
         with (compose plus (fun bag => bcount bag search_term)) by reflexivity.
         rewrite !foldright_compose.
diff --git a/src/QueryStructure/Implementation/DataStructures/Bags/TrieBags.v b/src/QueryStructure/Implementation/DataStructures/Bags/TrieBags.v
index f36a69e4c..696f21eaf 100644
--- a/src/QueryStructure/Implementation/DataStructures/Bags/TrieBags.v
+++ b/src/QueryStructure/Implementation/DataStructures/Bags/TrieBags.v
@@ -1078,7 +1078,7 @@ Module TrieBag (X:OrderedType).
         replaceXMapfold.
         unfold XMap.fold at 2.
 
-        rewrite Permutation_KeyBasedPartition with (key0 := key0)
+        rewrite Permutation_KeyBasedPartition with (key := key0)
                                                      (bst_m := SubTrieMapBST H1).
 
         pose proof (@partition_after_KeyBasedPartition_and_add
@@ -1980,9 +1980,9 @@ Module TrieBag (X:OrderedType).
             rewrite filter_Prefix; eauto.
           + rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
             match goal with
-            | [ H0 : TrieOK _ _ |- _ ] =>
-                rewrite Permutation_KeyBasedPartition with (key0 := key0)
-                                                           (bst_m := SubTrieMapBST H0)
+            | [ H : TrieOK _ _ |- _ ] =>
+                rewrite Permutation_KeyBasedPartition with (key := key0)
+                                                           (bst_m := SubTrieMapBST H)
             end.
             simpl.
             apply find_2 in e0.
@@ -2012,9 +2012,9 @@ Module TrieBag (X:OrderedType).
           + rewrite filter_Prefix; eauto.
           + rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
             match goal with
-            | [ H : TrieOK _ _ |- _ ] =>
-                rewrite Permutation_KeyBasedPartition with (key0 := key0)
-                                                           (bst_m := SubTrieMapBST' H)
+            | [ H0 : TrieOK _ _ |- _ ] =>
+                rewrite Permutation_KeyBasedPartition with (key := key0)
+                                                           (bst_m := SubTrieMapBST' H0)
             end.
             simpl in *.
             match goal with
@@ -2209,7 +2209,7 @@ Module TrieBag (X:OrderedType).
               eapply filter_negb_Prefix; eauto; reflexivity.
             + simpl.
               rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-              rewrite Permutation_KeyBasedPartition with (key0 := key0)
+              rewrite Permutation_KeyBasedPartition with (key := key0)
                                                            (bst_m := SubTrieMapBST containerCorrect).
               simpl in *.
               apply find_2 in e1.
@@ -2264,7 +2264,7 @@ Module TrieBag (X:OrderedType).
               eapply filter_negb_Prefix; eauto; reflexivity.
             + simpl.
               rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-              rewrite Permutation_KeyBasedPartition with (key0 := key0)
+              rewrite Permutation_KeyBasedPartition with (key := key0)
                                                            (bst_m := SubTrieMapBST containerCorrect).
               simpl.
               rewrite <- (@not_find_in_iff _ (XMap.Bst (SubTrieMapBST' containerCorrect)) key0) in e1.
@@ -2338,7 +2338,7 @@ Module TrieBag (X:OrderedType).
             + rewrite (H (l ++ [key0])); simpl; eauto.
               rewrite partition_filter_eq.
               rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-              rewrite Permutation_KeyBasedPartition with (key0 := key0)
+              rewrite Permutation_KeyBasedPartition with (key := key0)
                                                            (bst_m := SubTrieMapBST containerCorrect).
               simpl.
               apply find_2 in e1.
@@ -2373,7 +2373,7 @@ Module TrieBag (X:OrderedType).
               eapply filter_Prefix; eauto.
             + simpl.
               rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-              rewrite Permutation_KeyBasedPartition with (key0 := key0)
+              rewrite Permutation_KeyBasedPartition with (key := key0)
                                                            (bst_m := SubTrieMapBST containerCorrect).
               rewrite <- (@not_find_in_iff _ (XMap.Bst (SubTrieMapBST' containerCorrect)) key0) in e1.
               pose proof (KeyBasedPartition_fst_singleton_None key0 (XMap.Bst (SubTrieMapBST containerCorrect)) e1) as singleton.
@@ -2559,7 +2559,7 @@ Module TrieBag (X:OrderedType).
               f_equiv.
               * symmetry; f_equiv; eapply filter_Prefix; eauto.
               * rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-                rewrite Permutation_KeyBasedPartition with (key0 := key0)
+                rewrite Permutation_KeyBasedPartition with (key := key0)
                                                              (bst_m := SubTrieMapBST containerCorrect).
                 rewrite (Permutation_benumerate_add key0 bag'' (XMap.Bst (SubTrieMapBST containerCorrect))).
                 rewrite map_app, flatten_app.
@@ -2636,7 +2636,7 @@ Module TrieBag (X:OrderedType).
               * rewrite filter_Prefix; eauto; reflexivity.
               * simpl.
                 rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-                rewrite Permutation_KeyBasedPartition with (key0 := key0)
+                rewrite Permutation_KeyBasedPartition with (key := key0)
                                                              (bst_m := SubTrieMapBST containerCorrect).
                 simpl.
                 rewrite <- (@not_find_in_iff _ (XMap.Bst (SubTrieMapBST' containerCorrect)) key0) in e1.
@@ -2700,7 +2700,7 @@ Module TrieBag (X:OrderedType).
             + rewrite (H (l ++ [key0])); simpl; eauto.
               rewrite partition_filter_eq.
               rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-              rewrite Permutation_KeyBasedPartition with (key0 := key0)
+              rewrite Permutation_KeyBasedPartition with (key := key0)
                                                            (bst_m := SubTrieMapBST containerCorrect).
               simpl.
               apply find_2 in e1.
@@ -2735,7 +2735,7 @@ Module TrieBag (X:OrderedType).
             + rewrite filter_Prefix; eauto.
             + simpl.
               rewrite <- (fun H => @fold_1 _ m H (list BagType) [ ] (fun k trie a => Trie_enumerate trie ++ a)) by eauto.
-              rewrite Permutation_KeyBasedPartition with (key0 := key0)
+              rewrite Permutation_KeyBasedPartition with (key := key0)
                                                            (bst_m := SubTrieMapBST containerCorrect).
               rewrite <- (@not_find_in_iff _ (XMap.Bst (SubTrieMapBST' containerCorrect)) key0) in e1.
               pose proof (KeyBasedPartition_fst_singleton_None key0 (XMap.Bst (SubTrieMapBST containerCorrect)) e1) as singleton.
diff --git a/src/QueryStructure/Implementation/Operations/General/InsertRefinements.v b/src/QueryStructure/Implementation/Operations/General/InsertRefinements.v
index 8a5bcae96..eade4097e 100644
--- a/src/QueryStructure/Implementation/Operations/General/InsertRefinements.v
+++ b/src/QueryStructure/Implementation/Operations/General/InsertRefinements.v
@@ -660,7 +660,6 @@ Section InsertRefinements.
           as rewriteSat
             by (apply functional_extensionality; intros; rewrite GetRelDropConstraints;
                 reflexivity).
-      rewrite GetRelDropConstraints in H', H'',  H''''.
       setoid_rewrite rewriteSat in H''''; clear rewriteSat.
       (* Resume not-terribleness *)
       generalize (Iterate_Decide_Comp_BoundedIndex _ _ H''') as H3';
diff --git a/src/QueryStructure/Specification/Representation/QueryStructure.v b/src/QueryStructure/Specification/Representation/QueryStructure.v
index 9c5c5aad8..d40f88542 100644
--- a/src/QueryStructure/Specification/Representation/QueryStructure.v
+++ b/src/QueryStructure/Specification/Representation/QueryStructure.v
@@ -50,7 +50,7 @@ Section BuildQueryStructureConstraints.
     (* No need to check if the indices are in the list of Relations, because
      they are Bounded Strings. *)
     unfold crossRelationR, GetNRelSchemaHeading, GetNRelSchema; simpl.
-    rewrite H, H0;  exact (Some c).
+    rewrite e, e0;  exact (Some c).
   Defined.
 
   Fixpoint BuildQueryStructureConstraints'