Update

src/thue_morse.v

@@ -898,6 +898,51 @@ Proof.
 Qed.
 
 
+Lemma tm_step_consecutive_identical :
+  forall (n : nat) (hd a tl : list bool) (b : bool),
+  tm_step n = hd ++ (b::b::nil) ++ tl
+  -> odd (length hd) = true.
+Proof.
+  intros n hd a tl b. intro H.
+  assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
+  apply bool_dec. destruct J.
+  - rewrite <- Nat.negb_even. rewrite e. reflexivity.
+  - apply not_false_is_true in n0.
+    assert (K: count_occ Bool.bool_dec hd true
+             = count_occ Bool.bool_dec hd false).
+    generalize n0. generalize H. apply tm_step_count_occ.
+    assert (L: count_occ Bool.bool_dec (hd ++ [b;b]) true
+             = count_occ Bool.bool_dec (hd ++ [b;b]) false).
+    assert (even (length (hd ++ [b;b])) = true).
+    rewrite app_length. rewrite Nat.even_add_even. assumption.
+    simpl. apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
+    generalize H0. rewrite app_assoc in H. generalize H.
+    apply tm_step_count_occ.
+    rewrite count_occ_app in L. rewrite count_occ_app in L.
+    rewrite K in L. rewrite Nat.add_cancel_l in L.
+    destruct b. simpl in L. inversion L. simpl in L. inversion L.
+Qed.
+
+
+Lemma tm_step_consecutive_identical' :
+  forall (n : nat) (hd a tl : list bool) (b1 b2 : bool),
+  tm_step n = hd ++ (b1::b1::nil) ++ a ++ (b2::b2::nil) ++ tl
+  -> even (length a) = true.
+Proof.
+  intros n hd a tl b1 b2. intros H.
+  assert (Nat.odd (length hd) = true).
+  generalize H. apply tm_step_consecutive_identical. apply hd.
+  rewrite app_assoc in H. rewrite app_assoc in H.
+  assert (Nat.odd (length ((hd ++ [b1;b1])++a)) = true).
+  generalize H. apply tm_step_consecutive_identical. apply hd.
+  rewrite app_length in H1. rewrite Nat.odd_add in H1.
+  rewrite app_length in H1. rewrite Nat.odd_add in H1. rewrite H0 in H1.
+  replace (Nat.odd (length a)) with (negb (Nat.even (length a))) in H1.
+  destruct (even (length a)). reflexivity. inversion H1.
+  reflexivity.
+Qed.
+
+
 Lemma tm_step_factor4_odd_prefix : forall (n : nat) (hd a tl : list bool),
   tm_step n = hd ++ a ++ tl
   -> length a = 4 -> odd (length hd) = true

src/thue_morse2.v

@@ -8,57 +8,6 @@ Require Import Bool.
 Import ListNotations.
 
 
-(**
-   The following lemma, while not of truly general use, is
-    used in several proofs and has therefore its own dedicated
-   section here.
-*)
-
-Lemma tm_step_consecutive_identical :
-  forall (n : nat) (hd a tl : list bool) (b : bool),
-  tm_step n = hd ++ (b::b::nil) ++ tl
-  -> odd (length hd) = true.
-Proof.
-  intros n hd a tl b. intro H.
-  assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
-  apply bool_dec. destruct J.
-  - rewrite <- Nat.negb_even. rewrite e. reflexivity.
-  - apply not_false_is_true in n0.
-    assert (K: count_occ Bool.bool_dec hd true
-             = count_occ Bool.bool_dec hd false).
-    generalize n0. generalize H. apply tm_step_count_occ.
-    assert (L: count_occ Bool.bool_dec (hd ++ [b;b]) true
-             = count_occ Bool.bool_dec (hd ++ [b;b]) false).
-    assert (even (length (hd ++ [b;b])) = true).
-    rewrite app_length. rewrite Nat.even_add_even. assumption.
-    simpl. apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
-    generalize H0. rewrite app_assoc in H. generalize H.
-    apply tm_step_count_occ.
-    rewrite count_occ_app in L. rewrite count_occ_app in L.
-    rewrite K in L. rewrite Nat.add_cancel_l in L.
-    destruct b. simpl in L. inversion L. simpl in L. inversion L.
-Qed.
-
-
-Lemma tm_step_consecutive_identical' :
-  forall (n : nat) (hd a tl : list bool) (b1 b2 : bool),
-  tm_step n = hd ++ (b1::b1::nil) ++ a ++ (b2::b2::nil) ++ tl
-  -> even (length a) = true.
-Proof.
-  intros n hd a tl b1 b2. intros H.
-  assert (Nat.odd (length hd) = true).
-  generalize H. apply tm_step_consecutive_identical. apply hd.
-  rewrite app_assoc in H. rewrite app_assoc in H.
-  assert (Nat.odd (length ((hd ++ [b1;b1])++a)) = true).
-  generalize H. apply tm_step_consecutive_identical. apply hd.
-  rewrite app_length in H1. rewrite Nat.odd_add in H1.
-  rewrite app_length in H1. rewrite Nat.odd_add in H1. rewrite H0 in H1.
-  replace (Nat.odd (length a)) with (negb (Nat.even (length a))) in H1.
-  destruct (even (length a)). reflexivity. inversion H1.
-  reflexivity.
-Qed.
-
-
 (**
    The following lemmas and theorems are all related to
    squares of odd length in the Thue-Morse sequence.