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thue-morse.v
13
thue-morse.v
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@ -23,15 +23,6 @@ Fixpoint tm_step (n: nat) : list bool :=
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end.
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(* ad hoc more or less general lemmas *)
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Lemma negb_map_explode : forall (l1 l2 : list bool),
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map negb (l1 ++ l2) = map negb l1 ++ map negb l2.
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Proof.
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intros l1 l2.
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induction l1.
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- reflexivity.
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- simpl. rewrite IHl1. reflexivity.
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Qed.
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Lemma negb_double_map : forall (l : list bool),
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map negb (map negb l) = l.
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Proof.
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@ -117,7 +108,7 @@ Lemma tm_morphism_rev : forall (l : list bool),
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Proof.
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intro l. induction l.
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- reflexivity.
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- simpl. rewrite negb_map_explode.
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- simpl. rewrite map_app.
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rewrite IHl. rewrite tm_morphism_concat.
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rewrite <- app_assoc. simpl.
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rewrite negb_involutive. reflexivity.
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@ -306,7 +297,7 @@ Proof.
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- reflexivity.
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- simpl. rewrite tm_step_lemma. rewrite IHn. rewrite tm_morphism_concat.
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simpl in IHn. rewrite IHn. rewrite tm_build_negb. rewrite IHn.
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rewrite negb_map_explode. rewrite negb_double_map.
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rewrite map_app. rewrite negb_double_map.
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reflexivity.
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Qed.
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