From e4870cb63ef993dfd0d1ea9d230853f76d0c364c Mon Sep 17 00:00:00 2001
From: Thomas Baruchel <baruchel@gmx.com>
Date: Wed, 23 Nov 2022 21:44:45 +0100
Subject: [PATCH] Update

---
 thue-morse.v | 70 +++++++++++++++++++++++++++++++++++++++++-----------
 1 file changed, 55 insertions(+), 15 deletions(-)

diff --git a/thue-morse.v b/thue-morse.v
index da526ce..aea1aef 100644
--- a/thue-morse.v
+++ b/thue-morse.v
@@ -976,33 +976,75 @@ Lemma tm_step_add_range2_neighbor : forall (n m k : nat),
     = eqb
     (nth (k + 2^(n+m)) (tm_step (S n+m)) false) (nth (S k + 2^(n+m)) (tm_step (S n+m)) false).
 Proof.
-  intros n m k. intros H. induction m.
+  intros n m k. intros H.
+  induction m.
   - rewrite Nat.add_0_r. rewrite Nat.add_0_r. apply tm_step_next_range2_neighbor.
     apply H.
   - rewrite IHm. rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.
 
-    (*
-    assert (I: S k < 2^(S n + m)).
 
-      assert (2^n < 2^(S n + m)).
-      assert (n < S n + m). rewrite Nat.add_succ_comm.
+    assert (I : eqb (nth k (tm_step (S n + m)) false)
+                    (nth (S k) (tm_step (S n + m)) false)
+              = eqb (nth (k + 2 ^ S (n + m)) (tm_step (S (S n + m))) false)
+                    (nth (S k + 2 ^ S (n + m)) (tm_step (S (S n + m))) false)).
+    apply tm_step_next_range2_neighbor. induction m.
+    + rewrite Nat.add_0_r. simpl. generalize H. apply Nat.lt_lt_add_r.
+    + assert (2^n < 2^(S n + S m)).
+      assert (n < S n + S m). rewrite Nat.add_succ_comm.
       apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
       generalize H0. assert (1 < 2). apply Nat.lt_1_2. generalize H1.
       apply Nat.pow_lt_mono_r. generalize H0. generalize H.
       apply Nat.lt_trans.
-      generalize I.
+    + rewrite <- I. rewrite <- IHm.
 
-    apply tm_step_next_range2_neighbor. induction m.
+      assert (U: S k < 2^(S n+m)).
+      assert (J: k < 2^n). apply Nat.lt_succ_l. apply H.
+    assert (L: 2^n < 2^(S n + m)).
+    assert (M: n < S n + m). rewrite Nat.add_succ_comm.
+    apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
+    apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply M.
+    generalize L. generalize H. apply Nat.lt_trans.
+    assert (K := U). apply Nat.lt_succ_l in K.
 
+      assert (J: k < 2^n). apply Nat.lt_succ_l. apply H.
+
+      assert (nth_error (tm_step n) k = Some(nth k (tm_step n) false)).
+      generalize J. rewrite <- tm_size_power2. apply nth_error_nth'.
+      assert (nth_error (tm_step (S n + m)) k = Some(nth k (tm_step (S n + m)) false)).
+      generalize K. rewrite <- tm_size_power2. apply nth_error_nth'.
+
+      (*  assert (nth k (tm_step n) false = nth k (tm_step (S n + m)) false).  *)
+      assert (nth_error (tm_step n) k = nth_error (tm_step (S n + m)) k).
+      generalize K. generalize J. apply tm_step_stable.
+
+      rewrite <- H2 in H1. rewrite H0 in H1. inversion H1.
+      rewrite H4.
+
+      assert (nth_error (tm_step n) (S k) = Some(nth (S k) (tm_step n) false)).
+      generalize H. rewrite <- tm_size_power2. apply nth_error_nth'.
+      assert (nth_error (tm_step (S n + m)) (S k) = Some(nth (S k) (tm_step (S n + m)) false)).
+
+      generalize U. rewrite <- tm_size_power2. apply nth_error_nth'.
+      
+      assert (nth_error (tm_step n) (S k) = nth_error (tm_step (S n + m)) (S k)).
+
+      generalize U. generalize H. apply tm_step_stable.
+      rewrite H3 in H6. rewrite H5 in H6.
+      inversion H6. rewrite H8.
+      reflexivity.
+Qed.
+
+
+Lemma tm_step_add_range2_neighbor : forall (n m k : nat),
   S k < 2^n ->
     eqb (nth k (tm_step n) false) (nth (S k) (tm_step n) false)
     = eqb
-    (nth (k + 2^n) (tm_step (S n)) false) (nth (S k + 2^n) (tm_step (S n)) false).
-
-
-
-     *)
-
+    (nth (k + 2^(n+m)) (tm_step (S n+m)) false) (nth (S k + 2^(n+m)) (tm_step (S n+m)) false).
+Proof.
+  intros n m k. intros H. induction m.
+  - rewrite Nat.add_0_r. rewrite Nat.add_0_r. apply tm_step_next_range2_neighbor.
+    apply H.
+  - rewrite IHm. rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.
 
     assert (I : eqb (nth k (tm_step (S n + m)) false)
                     (nth (S k) (tm_step (S n + m)) false)
@@ -1042,7 +1084,6 @@ Proof.
       generalize H. rewrite <- tm_size_power2. apply nth_error_nth'.
       assert (nth_error (tm_step (S n + m)) (S k) = Some(nth (S k) (tm_step (S n + m)) false)).
 
-
       assert (2^n < 2^(S n + m)).
       assert (n < S n + m). rewrite Nat.add_succ_comm.
       apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
@@ -1068,7 +1109,6 @@ Proof.
       generalize H7. generalize H. apply tm_step_stable.
       rewrite H3 in H6. rewrite H5 in H6.
       inversion H6. rewrite H8.
-
       reflexivity.
 Qed.