Update

thue-morse.v

@@ -865,63 +865,6 @@ Qed.
 
 
 
-
-
-
-Lemma tm_morphism_app2 : forall (l hd tl : list bool),
-  tm_morphism l = hd ++ tl -> even (length hd) = true
-  -> exists l2, hd = tm_morphism l2.
-Proof.
-  intros l hd tl. intros H I.
-  exists (firstn (Nat.div2 (length hd)) l).
-  generalize I. generalize H.
-  induction l.
-
-  - assert (J: hd = nil). destruct hd. reflexivity.
-    simpl in H. inversion H.
-    rewrite J. reflexivity.
-  - 
-
-
-
-
-  induction tl.
-  rewrite <- app_nil_end in H.
-  exists l. symmetry. assumption.
-
-
-
-
-
-
-Lemma tm_step_even_prefix : forall (hd tl : list bool) (k : nat),
-  tm_step k = hd ++ tl -> even (length hd) = true
-  -> exists l, hd = tm_morphism l.
-Proof.
-  intros hd tl k. intros H I.
-  destruct k.
-  - assert (J: hd = nil). destruct hd. reflexivity.
-    simpl in H. inversion H.
-    symmetry in H2. apply app_eq_nil in H2.
-    destruct H2. rewrite H0 in I. simpl in I. inversion I.
-    rewrite J. exists []. reflexivity.
-  - rewrite <- tm_step_lemma in H.
-    exists (firstn (Nat.div2 (length hd)) (tm_step k)).
-    apply app_inv_tail with (l := tl).
-    rewrite <- H.
-
-
-
-
-
-  
-
-
-
-
-
-
-
 (*
   From a(0) to a(2n+1), there are n+1 terms equal to 0 and n+1 terms equal to 1 (see Hassan Tarfaoui link, Concours Général 1990). - Bernard Schott, Jan 21 2022 
    TODO Search "count_occ"
@@ -938,20 +881,10 @@ Proof.
     destruct H2. rewrite H0 in I. simpl in I. inversion I.
     rewrite J. reflexivity.
   - rewrite <- tm_step_lemma in H.
-
-
-
-
-
-    generalize I. generalize H. induction hd.
-    reflexivity.
-
-
-
-    rewrite <- length_zero_iff_nil. simpl.
-    
-
-PeanoNat.Nat.neq_succ_0: forall n : nat, S n <> 0
+    assert (J: hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step k))).
+    generalize I. generalize H. apply tm_morphism_app2.
+    rewrite J. apply tm_morphism_count_occ.
+Qed.