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Thomas Baruchel 2023-01-10 19:09:56 +01:00
parent 5a6a3dd60e
commit a7bb37b3a2
1 changed files with 56 additions and 9 deletions
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src/thue_morse2.v
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@ -17,27 +17,74 @@ Proof.
assert (I: even (length hd) = false).
assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
apply bool_dec. destruct J. assumption. 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.
rewrite app_assoc in H.
assert (K: count_occ Bool.bool_dec (hd ++ [b1;b1]) true
assert (L: count_occ Bool.bool_dec (hd ++ [b1;b1]) true
= count_occ Bool.bool_dec (hd ++ [b1;b1]) false).
assert (even (length (hd ++ [b1;b1])) = true).
rewrite app_length. rewrite Nat.even_add_even. assumption.
simpl. apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
generalize H0. 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 b1. simpl in L. inversion L. simpl in L. inversion L.
assert (J: {even (length (hd ++ [b1;b1] ++ a))
= false} + { ~ (even (length (hd ++ [b1;b1] ++ a))) = false}).
apply bool_dec. destruct J.
rewrite app_length in e. rewrite app_length in e.
rewrite Nat.add_assoc in e. rewrite Nat.add_shuffle0 in e.
rewrite Nat.even_add_even in e. rewrite Nat.even_add in e.
rewrite I in e. destruct (Nat.even (length a)). reflexivity.
simpl in e. inversion e. simpl.
apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
apply not_false_is_true in n0.
assert (K: even (length (hd ++ [b1;b1] ++ a ++ [b2;b2])) = true).
replace (hd ++ [b1;b1] ++ a ++ [b2;b2])
with ((hd ++ [b1;b1] ++ a) ++ [b2;b2]).
rewrite app_length. rewrite Nat.even_add.
rewrite n0. reflexivity.
rewrite <- app_assoc. apply app_inv_head_iff.
rewrite <- app_assoc. reflexivity.
assert (N: count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a ++ [b2;b2]) true
= count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a ++ [b2;b2]) false).
replace (hd ++ [b1;b1] ++ a ++ [b2;b2] ++ tl)
with ((hd ++ [b1;b1] ++ a ++ [b2;b2]) ++ tl) in H.
generalize K. generalize H. apply tm_step_count_occ.
rewrite <- app_assoc. apply app_inv_head_iff.
rewrite <- app_assoc. apply app_inv_head_iff.
rewrite <- app_assoc. reflexivity.
assert (M: count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a) true
= count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a) false).
replace (hd ++ [b1;b1] ++ a ++ [b2;b2] ++ tl)
with ((hd ++ [b1;b1] ++ a) ++ [b2;b2] ++ tl) in H.
generalize n0. generalize H. apply tm_step_count_occ.
rewrite <- app_assoc. apply app_inv_head_iff.
rewrite <- app_assoc. reflexivity.
replace (hd ++ [b1;b1] ++ a ++ [b2;b2])
with ((hd ++ [b1;b1] ++ a) ++ [b2;b2]) in N.
rewrite count_occ_app in N. symmetry in N.
rewrite count_occ_app in N. rewrite M in N.
rewrite Nat.add_cancel_l in N.
destruct b2. simpl in N. inversion N. simpl in N. inversion N.
rewrite <- app_assoc. apply app_inv_head_iff.
rewrite <- app_assoc. reflexivity.
Qed.
destruct n.
- simpl in H. destruct hd. simpl in H. assert (b1 = false).
inversion H. rewrite H0 in H. inversion H. inversion H.
symmetry in H2. apply app_eq_nil in H2. destruct H2. inversion H2.
- rewrite <- tm_step_lemma in H.
assert (I: even (length hd) = false).
assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
apply bool_dec. destruct J. assumption. apply not_false_is_true in n0.
Theorem tm_step_count_occ : forall (hd tl : list bool) (n : nat),
tm_step n = hd ++ tl -> even (length hd) = true
-> count_occ Bool.bool_dec hd true = count_occ Bool.bool_dec hd false.