Update

2023-01-14 21:37:37 +01:00 · 2023-01-14 21:37:37 +01:00 · ddedf950b6
commit ddedf950b6
parent 3c2f4121e1
1 changed files with 53 additions and 77 deletions
--- a/src/thue_morse2.v
+++ b/src/thue_morse2.v
@ -329,71 +329,6 @@ Qed.
   prefix being odd or even in front of a squared pattern.
 *)

-Lemma tm_step_odd_prefix_square_3 : forall (n : nat) (hd a tl : list bool),
-  tm_step n = hd ++ a ++ a ++ tl
-  -> length a = 3 -> even (length hd) = false.
-Proof.
-  intros n hd a tl. intros H I.
-  assert (J: Nat.even (length hd) || Nat.odd (length hd) = true).
-  apply Nat.orb_even_odd. apply orb_prop in J. destruct J.
-  assert (Z: a = [true;false;true] \/ a = [false;true;false]).
-  destruct a. inversion I. destruct a. inversion I.
-  destruct a. inversion I. destruct a.
-  - destruct b; destruct b0; destruct b1.
-    + replace ([true;true;true] ++ [true;true;true] ++ tl)
-      with ([true;true] ++ [true] ++ [true;true] ++ [true] ++ tl) in H.
-      apply tm_step_consecutive_identical' in H. inversion H.
-      reflexivity.
-    + replace ([true;true;false] ++ [true;true;false] ++ tl)
-      with ([true;true] ++ [false] ++ [true;true] ++ [false] ++ tl) in H.
-      apply tm_step_consecutive_identical' in H. inversion H.
-      reflexivity.
-    + left. reflexivity.
-    + replace (hd ++ [true;false;false] ++ [true;false;false] ++ tl)
-      with ((hd ++ [true]) ++ [false; false] ++ [true] ++ [false;false] ++ tl) in H.
-      apply tm_step_consecutive_identical' in H. inversion H.
-      rewrite <- app_assoc. reflexivity.
-    + replace (hd ++ [false;true;true] ++ [false;true;true] ++ tl)
-      with ((hd ++ [false]) ++ [true; true] ++ [false] ++ [true;true] ++ tl) in H.
-      apply tm_step_consecutive_identical' in H. inversion H.
-      rewrite <- app_assoc. reflexivity.
-    + right. reflexivity.
-    + replace ([false;false;true] ++ [false;false;true] ++ tl)
-      with ([false;false] ++ [true] ++ [false;false] ++ [true] ++ tl) in H.
-      apply tm_step_consecutive_identical' in H. inversion H.
-      reflexivity.
-    + replace ([false;false;false] ++ [false;false;false] ++ tl)
-      with ([false;false] ++ [false] ++ [false;false] ++ [false] ++ tl) in H.
-      apply tm_step_consecutive_identical' in H. inversion H.
-      reflexivity.
-  - inversion I.
-  - assert (K: count_occ Bool.bool_dec hd true
-                 = count_occ Bool.bool_dec hd false).
-    generalize H0. generalize H. apply tm_step_count_occ.
-    destruct Z; rewrite H1 in H.
-    replace (hd ++ [true;false;true] ++ [true;false;true] ++ tl)
-      with ((hd ++ [true;false;true;true]) ++ [false;true] ++ tl) in H.
-    assert (even (length (hd ++ [true;false;true;true])) = true).
-    rewrite app_length. rewrite Nat.even_add. rewrite H0. reflexivity.
-    assert (M: count_occ Bool.bool_dec (hd ++ [true;false;true;true]) true
-                  = count_occ Bool.bool_dec (hd ++ [true;false;true;true]) false).
-    generalize H2. generalize H. apply tm_step_count_occ.
-    rewrite count_occ_app in M. symmetry in M. rewrite count_occ_app in M.
-    rewrite K in M. rewrite Nat.add_cancel_l in M. inversion M.
-    rewrite <- app_assoc. reflexivity.
-    replace (hd ++ [false;true;false] ++ [false;true;false] ++ tl)
-      with ((hd ++ [false;true;false;false]) ++ [true;false] ++ tl) in H.
-    assert (even (length (hd ++ [false;true;false;false])) = true).
-    rewrite app_length. rewrite Nat.even_add. rewrite H0. reflexivity.
-    assert (M: count_occ Bool.bool_dec (hd ++ [false;true;false;false]) true
-                  = count_occ Bool.bool_dec (hd ++ [false;true;false;false]) false).
-    generalize H2. generalize H. apply tm_step_count_occ.
-    rewrite count_occ_app in M. symmetry in M. rewrite count_occ_app in M.
-    rewrite K in M. rewrite Nat.add_cancel_l in M. inversion M.
-    rewrite <- app_assoc. reflexivity.
-  - rewrite <- Nat.negb_odd. rewrite H0. reflexivity.
-Qed.
-
 Lemma tm_step_odd_prefix_square : forall (n : nat) (hd a tl : list bool),
  tm_step n = hd ++ a ++ a ++ tl
  -> odd (length a) = true -> even (length hd) = false.
@ -401,18 +336,59 @@ Proof.
  intros n hd a tl. intros H I.
  assert (length a < 4). generalize I. generalize H. apply tm_step_odd_square.
  destruct a. inversion I. destruct a.
-  rewrite <- negb_true_iff. rewrite Nat.negb_even.
-  generalize H. apply tm_step_consecutive_identical.
-  destruct a. inversion I.
-  destruct a.
-  apply tm_step_odd_prefix_square_3 with (n := n) (a := [b;b0;b1]) (tl := tl).
-  assumption. reflexivity.
-  simpl in H0.
-  rewrite <- Nat.succ_lt_mono in H0.
-  rewrite <- Nat.succ_lt_mono in H0.
-  rewrite <- Nat.succ_lt_mono in H0.
-  rewrite <- Nat.succ_lt_mono in H0.
-  apply Nat.nlt_0_r in H0. contradiction H0.
+  - rewrite <- negb_true_iff. rewrite Nat.negb_even.
+    generalize H. apply tm_step_consecutive_identical.
+  - destruct a. inversion I. destruct a.
+
+    assert (J: {b0 = b1} + {b0 <> b1}). apply bool_dec. destruct J.
+    rewrite e in H.
+    replace (hd ++ [b;b1;b1] ++ [b;b1;b1] ++ tl)
+      with ((hd ++ [b]) ++ [b1;b1] ++ [b] ++ [b1;b1] ++ tl) in H.
+    assert (K: even (length [b]) = true). generalize H.
+    apply tm_step_consecutive_identical'. inversion K.
+    rewrite <- app_assoc. reflexivity.
+    assert (J: {b = b0} + {b <> b0}). apply bool_dec. destruct J.
+    rewrite e in H.
+    replace (hd ++ [b0;b0;b1] ++ [b0;b0;b1] ++ tl)
+      with (hd ++ [b0;b0] ++ [b1] ++ [b0;b0] ++ [b1] ++ tl) in H.
+    assert (K: even (length [b1]) = true). generalize H.
+    apply tm_step_consecutive_identical'. inversion K.
+    reflexivity.
+    assert (J: {b = b1} + {b <> b1}). apply bool_dec. destruct J.
+    rewrite e in H.
+
+    assert (J: Nat.even (length hd) || Nat.odd (length hd) = true).
+    apply Nat.orb_even_odd. apply orb_prop in J. destruct J.
+    assert (K: count_occ Bool.bool_dec hd true
+                 = count_occ Bool.bool_dec hd false).
+    generalize H1. generalize H. apply tm_step_count_occ.
+
+    assert (L: count_occ Bool.bool_dec (hd ++ [b1;b0;b1;b1]) true
+                  = count_occ Bool.bool_dec (hd ++ [b1;b0;b1;b1]) false).
+    assert (M: even (length (hd ++ [b1;b0;b1;b1])) = true).
+    rewrite app_length. rewrite Nat.even_add. rewrite H1. reflexivity.
+    replace (hd ++ [b1;b0;b1] ++ [b1;b0;b1] ++ tl)
+      with ((hd ++ [b1;b0;b1;b1]) ++ [b0;b1] ++ tl) in H.
+    generalize M. generalize H. apply tm_step_count_occ.
+    rewrite <- app_assoc. reflexivity.
+    rewrite count_occ_app in L. symmetry in L. rewrite count_occ_app in L.
+    rewrite K in L. rewrite Nat.add_cancel_l in L.
+
+    destruct b0; destruct b1; inversion L.
+    rewrite <- Nat.negb_odd. rewrite H1. reflexivity.
+
+    destruct b; destruct b0; destruct b1.
+    contradiction n0. contradiction n2. contradiction n1. contradiction n2.
+    contradiction n0. reflexivity.
+    contradiction n0. contradiction n2. reflexivity.
+    contradiction n1. reflexivity. contradiction n0. reflexivity.
+
+    simpl in H0.
+    rewrite <- Nat.succ_lt_mono in H0.
+    rewrite <- Nat.succ_lt_mono in H0.
+    rewrite <- Nat.succ_lt_mono in H0.
+    rewrite <- Nat.succ_lt_mono in H0.
+    apply Nat.nlt_0_r in H0. contradiction H0.
 Qed.