This commit is contained in:
Thomas Baruchel 2022-11-22 16:00:24 +01:00
parent c6142420cd
commit 9975cd0e5a

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@ -829,48 +829,60 @@ Proof.
reflexivity.
Qed.
Lemma tm_step_consecutive_pow2 :
forall (n k : nat) (l1 l2 : list bool) (b1 b2 b1' b2': bool),
tm_step n = l1 ++ b1 :: b2 :: l2
-> 2^k > S (length l1)
-> nth_error (tm_step n) ((length l1) + 2^k) = Some b1'
-> nth_error (tm_step n) (S (length l1) + 2^k) = Some b2'
-> (eqb b1 b2) = (eqb b1' b2').
Proof.
intros n k l1 l2 b1 b2 b1' b2'.
intros.
Admitted.
Require Import BinPosDef.
(* utiliser Pos.pred à partir de 2 *)
(* donc théorème sur S (S n) *)
Compute let b := Pos.of_nat 6 in Pos.lxor b (Pos.pred b).
Compute let b := Pos.of_nat 3 in Pos.pred b.
Compute let b := Pos.of_nat 7 in
Pos.size_nat (
match Pos.lxor b (Pos.pred b) with
| N0 => BinNums.xH (* arbitrary, never used *)
| Npos(p) => p
end).
Compute tm_step 4.
(*
= [false; true; true; false; true; false; false; true; true; false;
false; true; false; true; true; false]
*)
(* Autre construction de la suite, ici n est le nombre de termes *)
Fixpoint tm_bin (n: nat) : list bool :=
match n with
| 0 => nil
| S n' => let t := tm_bin n' in
let m := Pos.of_nat n' in
(xorb (hd false t)
(odd (Pos.size_nat
match Pos.lxor m (Pos.pred m) with
| N0 => BinNums.xH
| Npos(p) => p
end))) :: t
end.
(* on suppose que len est la longueur de la liste qui précède b0 et b1 *)
Compute let mylen := 5 in
let ind_b1 := Pos.of_nat (S mylen) in (* index of b1 *)
let ind_b0 := Pos.pred ind_b1 in (* index of b0 *)
even (Pos.size_nat (
match Pos.lxor ind_b0 ind_b1 with
| N0 => BinNums.xH (* arbitrary, never used *)
| Npos(p) => p
end)).
Theorem tm_step_consecutive : forall (n : nat) (l1 l2 : list bool) (b1 b2 : bool),
tm_step n = l1 ++ b1 :: b2 :: l2 ->
(eqb b1 b2) =
let ind_b2 := Pos.of_nat (S (length l1)) in (* index of b2 *)
let ind_b1 := Pos.pred ind_b2 in (* index of b1 *)
even (Pos.size_nat (
even (Pos.size_nat
match Pos.lxor ind_b1 ind_b2 with
| N0 => BinNums.xH (* arbitrary, never used *)
| N0 => BinNums.xH
| Npos(p) => p
end)).
end).
Proof.
intros n l1 l2 b1 b2.
destruct n.
- simpl. intros H. induction l1.
+ rewrite app_nil_l in H. discriminate.
+ destruct l1. rewrite app_nil_l in IHl1. discriminate. discriminate.
- rewrite tm_build.
Admitted.
(*
intros n l1 l2 b1 b2.
intros H.
induction l1.
@ -880,8 +892,8 @@ Proof.
assert (J: tl (tl (tm_morphism (tm_step n))) = l2).
{ replace (tm_morphism (tm_step n)) with (tm_step (S n)).
rewrite H. reflexivity. reflexivity. }
rewrite J. intros. rewrite <- H1. rewrite <- H2. reflexivity.
-
(* rewrite J *) intros. rewrite <- H1. rewrite <- H2. reflexivity.
- replace (S (length (a :: l1))) with (S (S (length l1))).
@ -924,3 +936,4 @@ Proof.
Lemma tm_step_head_2 : forall (n : nat),
tm_step (S n) = false :: true :: tl (tl (tm_step (S n))).
*)