(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/chebishev".
-
include "nat/log.ma".
include "nat/pi_p.ma".
include "nat/factorization.ma".
include "nat/factorial2.ma".
+include "nat/o.ma".
definition prim: nat \to nat \def
-\lambda n. sigma_p (S n) primeb (\lambda p.(S O)).
+\lambda n. sigma_p (S n) primeb (\lambda p.1).
+
+theorem le_prim_n: \forall n. prim n \le n.
+intros.unfold prim. elim n
+ [apply le_n
+ |apply (bool_elim ? (primeb (S n1)));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > sym_plus.
+ rewrite < plus_n_Sm.
+ rewrite < plus_n_O.
+ apply le_S_S.assumption
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [apply le_S.assumption
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem not_prime_times_SSO: \forall n. 1 < n \to \lnot prime (2*n).
+intros.intro.elim H1.
+absurd (2 = 2*n)
+ [apply H3
+ [apply (witness ? ? n).reflexivity
+ |apply le_n
+ ]
+ |apply lt_to_not_eq.
+ rewrite > times_n_SO in ⊢ (? % ?).
+ apply lt_times_r.
+ assumption
+ ]
+qed.
+
+theorem eq_prim_prim_pred: \forall n. 1 < n \to
+(prim (2*n)) = (prim (pred (2*n))).
+intros.unfold prim.
+rewrite < S_pred
+ [rewrite > false_to_sigma_p_Sn
+ [reflexivity
+ |apply not_prime_to_primeb_false.
+ apply not_prime_times_SSO.
+ assumption
+ ]
+ |apply (trans_lt ? (2*1))
+ [simplify.apply lt_O_S
+ |apply lt_times_r.
+ assumption
+ ]
+ ]
+qed.
+
+theorem le_prim_n1: \forall n. 4 \le n \to prim (S(2*n)) \le n.
+intros.unfold prim. elim H
+ [simplify.apply le_n
+ |cut (sigma_p (2*S n1) primeb (λp:nat.1) = sigma_p (S (2*S n1)) primeb (λp:nat.1))
+ [apply (bool_elim ? (primeb (S(2*(S n1)))));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > sym_plus.
+ rewrite < plus_n_Sm.
+ rewrite < plus_n_O.
+ apply le_S_S.
+ rewrite < Hcut.
+ rewrite > times_SSO.
+ assumption
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [apply le_S.
+ rewrite < Hcut.
+ rewrite > times_SSO.
+ assumption
+ |assumption
+ ]
+ ]
+ |apply sym_eq.apply (eq_prim_prim_pred (S n1)).
+ apply le_S_S.apply (trans_le ? 4)
+ [apply leb_true_to_le.reflexivity
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+(* usefull to kill a successor in bertrand *)
+theorem le_prim_n2: \forall n. 7 \le n \to prim (S(2*n)) \le pred n.
+intros.unfold prim. elim H
+ [apply leb_true_to_le.reflexivity.
+ |cut (sigma_p (2*S n1) primeb (λp:nat.1) = sigma_p (S (2*S n1)) primeb (λp:nat.1))
+ [apply (bool_elim ? (primeb (S(2*(S n1)))));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > sym_plus.
+ rewrite < plus_n_Sm.
+ rewrite < plus_n_O.
+ simplify in ⊢ (? ? %).
+ rewrite > S_pred in ⊢ (? ? %)
+ [apply le_S_S.
+ rewrite < Hcut.
+ rewrite > times_SSO.
+ assumption
+ |apply (ltn_to_ltO ? ? H1)
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [simplify in ⊢ (? ? %).
+ apply (trans_le ? ? ? ? (le_pred_n n1)).
+ rewrite < Hcut.
+ rewrite > times_SSO.
+ assumption
+ |assumption
+ ]
+ ]
+ |apply sym_eq.apply (eq_prim_prim_pred (S n1)).
+ apply le_S_S.apply (trans_le ? 4)
+ [apply leb_true_to_le.reflexivity
+ |apply (trans_le ? ? ? ? H1).
+ apply leb_true_to_le.reflexivity
+ ]
+ ]
+ ]
+qed.
+
+(* da spostare *)
+theorem le_pred: \forall n,m. n \le m \to pred n \le pred m.
+apply nat_elim2;intros
+ [apply le_O_n
+ |apply False_ind.apply (le_to_not_lt ? ? H).
+ apply lt_O_S
+ |simplify.apply le_S_S_to_le.assumption
+ ]
+qed.
+
+(* la prova potrebbe essere migliorata *)
+theorem le_prim_n3: \forall n. 15 \le n \to
+prim n \le pred (n/2).
+intros.
+elim (or_eq_eq_S (pred n)).
+elim H1
+ [cut (n = S (2*a))
+ [rewrite > Hcut.
+ apply (trans_le ? (pred a))
+ [apply le_prim_n2.
+ apply (le_times_to_le 2)
+ [apply le_n_Sn
+ |apply le_S_S_to_le.
+ rewrite < Hcut.
+ assumption
+ ]
+ |apply le_pred.
+ apply le_times_to_le_div
+ [apply lt_O_S
+ |apply le_n_Sn
+ ]
+ ]
+ |rewrite < H2.
+ apply S_pred.
+ apply (ltn_to_ltO ? ? H)
+ ]
+ |cut (n=2*(S a))
+ [rewrite > Hcut.
+ rewrite > eq_prim_prim_pred
+ [rewrite > times_SSO in ⊢ (? % ?).
+ change in ⊢ (? (? %) ?) with (S (2*a)).
+ rewrite > sym_times in ⊢ (? ? (? (? % ?))).
+ rewrite > lt_O_to_div_times
+ [apply (trans_le ? (pred a))
+ [apply le_prim_n2.
+ apply le_S_S_to_le.
+ apply (lt_times_to_lt 2)
+ [apply le_n_Sn
+ |apply le_S_S_to_le.
+ rewrite < Hcut.
+ apply le_S_S.
+ assumption
+ ]
+ |apply le_pred.
+ apply le_n_Sn
+ ]
+ |apply lt_O_S
+ ]
+ |apply le_S_S.
+ apply not_lt_to_le.intro.
+ apply (le_to_not_lt ? ? H).
+ rewrite > Hcut.
+ lapply (le_S_S_to_le ? ? H3) as H4.
+ apply (le_n_O_elim ? H4).
+ apply leb_true_to_le.reflexivity
+ ]
+ |rewrite > times_SSO.
+ rewrite > S_pred
+ [apply eq_f.assumption
+ |apply (ltn_to_ltO ? ? H)
+ ]
+ ]
+ ]
+qed.
(* This is chebishev psi function *)
definition A: nat \to nat \def
]
]
qed.
-
+
theorem lt_max_to_pi_p_primeb: \forall q,m. O < m \to max m (\lambda i.primeb i \land divides_b i m) < q \to
m = pi_p q (\lambda i.primeb i \land divides_b i m) (\lambda p.exp p (ord m p)).
intro.elim q
assumption.
qed.
-theorem le_ord_log: \forall n,p. O < n \to S O < p \to
+theorem le_ord_log: \forall n,p. O < n \to 1 < p \to
ord n p \le log p n.
intros.
rewrite > (exp_ord p) in ⊢ (? ? (? ? %))
theorem eq_ord_sigma_p:
\forall n,m,x. O < n \to prime x \to
exp x m \le n \to n < exp x (S m) \to
-ord n x=sigma_p m (λi:nat.divides_b (x\sup (S i)) n) (λx:nat.S O).
+ord n x=sigma_p m (λi:nat.divides_b (x\sup (S i)) n) (λx:nat.1).
intros.
lapply (prime_to_lt_SO ? H1).
rewrite > (exp_ord x n) in ⊢ (? ? ? (? ? (λi:?.? ? %) ?))
intros.
rewrite > eq_fact_pi_p.
apply (trans_eq ? ?
- (pi_p (S n) (λi:nat.leb (S O) i)
- (λn.pi_p (S n) primeb
- (\lambda p.(pi_p (log p n)
- (\lambda i.divides_b (exp p (S i)) n) (\lambda i.p))))))
+ (pi_p (S n) (λm:nat.leb (S O) m)
+ (λm.pi_p (S m) primeb
+ (\lambda p.(pi_p (log p m)
+ (\lambda i.divides_b (exp p (S i)) m) (\lambda i.p))))))
[apply eq_pi_p1;intros
[reflexivity
|apply pi_p_primeb1.
apply leb_true_to_le.assumption
]
|apply (trans_eq ? ?
- (pi_p (S n) (λi:nat.leb (S O) i)
- (λn:nat
- .pi_p (S n) (\lambda p.primeb p\land leb p n)
- (λp:nat.pi_p (log p n) (λi:nat.divides_b ((p)\sup(S i)) n) (λi:nat.p)))))
+ (pi_p (S n) (λm:nat.leb (S O) m)
+ (λm:nat
+ .pi_p (S m) (\lambda p.primeb p\land leb p m)
+ (λp:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p)))))
[apply eq_pi_p1
[intros.reflexivity
|intros.apply eq_pi_p1
]
]
|apply (trans_eq ? ?
- (pi_p (S n) (λi:nat.leb (S O) i)
+ (pi_p (S n) (λm:nat.leb (S O) m)
(λm:nat
.pi_p (S n) (λp:nat.primeb p∧leb p m)
(λp:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p)))))
|
*)
-theorem fact_pi_p2: \forall n. fact ((S(S O))*n) =
-pi_p (S ((S(S O))*n)) primeb
- (\lambda p.(pi_p (log p ((S(S O))*n))
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))*(exp p (mod ((S(S O))*n /(exp p (S i))) (S(S O)))))))).
+theorem fact_pi_p2: \forall n. fact (2*n) =
+pi_p (S (2*n)) primeb
+ (\lambda p.(pi_p (log p (2*n))
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))*(exp p (mod (2*n /(exp p (S i))) 2)))))).
intros.rewrite > fact_pi_p.
apply eq_pi_p1
[intros.reflexivity
]
qed.
-theorem fact_pi_p3: \forall n. fact ((S(S O))*n) =
-pi_p (S ((S(S O))*n)) primeb
- (\lambda p.(pi_p (log p ((S(S O))*n))
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))*
-pi_p (S ((S(S O))*n)) primeb
- (\lambda p.(pi_p (log p ((S(S O))*n))
- (\lambda i.true) (\lambda i.(exp p (mod ((S(S O))*n /(exp p (S i))) (S(S O))))))).
+theorem fact_pi_p3: \forall n. fact (2*n) =
+pi_p (S (2*n)) primeb
+ (\lambda p.(pi_p (log p (2*n))
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))*
+pi_p (S (2*n)) primeb
+ (\lambda p.(pi_p (log p (2*n))
+ (\lambda i.true) (\lambda i.(exp p (mod (2*n /(exp p (S i))) 2))))).
intros.
rewrite < times_pi_p.
rewrite > fact_pi_p2.
]
qed.
-theorem pi_p_primeb4: \forall n. S O < n \to
-pi_p (S ((S(S O))*n)) primeb
- (\lambda p.(pi_p (log p ((S(S O))*n))
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))
+theorem pi_p_primeb4: \forall n. 1 < n \to
+pi_p (S (2*n)) primeb
+ (\lambda p.(pi_p (log p (2*n))
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))
=
pi_p (S n) primeb
- (\lambda p.(pi_p (log p (S(S O)*n))
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))).
+ (\lambda p.(pi_p (log p (2*n))
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))).
intros.
apply or_false_eq_SO_to_eq_pi_p
[apply le_S_S.
]
qed.
-theorem pi_p_primeb5: \forall n. S O < n \to
-pi_p (S ((S(S O))*n)) primeb
+theorem pi_p_primeb5: \forall n. 1 < n \to
+pi_p (S (2*n)) primeb
(\lambda p.(pi_p (log p ((S(S O))*n))
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))
=
pi_p (S n) primeb
(\lambda p.(pi_p (log p n)
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))).
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))).
intros.
rewrite > (pi_p_primeb4 ? H).
apply eq_pi_p1;intros
qed.
theorem exp_fact_SSO: \forall n.
-exp (fact n) (S(S O))
+exp (fact n) 2
=
pi_p (S n) primeb
(\lambda p.(pi_p (log p n)
- (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))).
+ (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))).
intros.
rewrite > fact_pi_p.
rewrite < exp_pi_p.
(\lambda p.(pi_p (log p n)
(\lambda i.true) (\lambda i.(exp p (mod (n /(exp p (S i))) (S(S O))))))).
+theorem B_SSSO: B 3 = 6.
+reflexivity.
+qed.
+
+theorem B_SSSSO: B 4 = 6.
+reflexivity.
+qed.
+
+theorem B_SSSSSO: B 5 = 30.
+reflexivity.
+qed.
+
+theorem B_SSSSSSO: B 6 = 20.
+reflexivity.
+qed.
+
+theorem B_SSSSSSSO: B 7 = 140.
+reflexivity.
+qed.
+
+theorem B_SSSSSSSSO: B 8 = 70.
+reflexivity.
+qed.
+
theorem eq_fact_B:\forall n.S O < n \to
-fact ((S(S O))*n) = exp (fact n) (S(S O)) * B((S(S O))*n).
+fact (2*n) = exp (fact n) 2 * B(2*n).
intros. unfold B.
rewrite > fact_pi_p3.
apply eq_f2
]
]
qed.
-
+
+(* not usefull
theorem le_fact_A:\forall n.S O < n \to
-fact ((S(S O))*n) \le exp (fact n) (S(S O)) * A ((S(S O))*n).
+fact (2*n) \le exp (fact n) 2 * A (2*n).
intros.
rewrite > eq_fact_B
[apply le_times_r.
apply le_B_A
|assumption
]
-qed.
+qed. *)
theorem lt_SO_to_le_B_exp: \forall n.S O < n \to
-B ((S(S O))*n) \le exp (S(S O)) ((S(S O))*n).
+B (2*n) \le exp 2 (pred (2*n)).
intros.
apply (le_times_to_le (exp (fact n) (S(S O))))
[apply lt_O_exp.
qed.
theorem le_B_exp: \forall n.
-B ((S(S O))*n) \le exp (S(S O)) ((S(S O))*n).
+B (2*n) \le exp 2 (pred (2*n)).
intro.cases n
[apply le_n
|cases n1
- [simplify.apply le_S.apply le_S.apply le_n
+ [simplify.apply le_n
|apply lt_SO_to_le_B_exp.
apply le_S_S.apply lt_O_S.
]
]
qed.
-theorem lt_SO_to_le_exp_B: \forall n. S O < n \to
-exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n).
+theorem lt_SSSSO_to_le_B_exp: \forall n.4 < n \to
+B (2*n) \le exp 2 ((2*n)-2).
+intros.
+apply (le_times_to_le (exp (fact n) (S(S O))))
+ [apply lt_O_exp.
+ apply lt_O_fact
+ |rewrite < eq_fact_B
+ [rewrite < sym_times in ⊢ (? ? %).
+ rewrite > exp_SSO.
+ rewrite < assoc_times.
+ apply lt_SSSSO_to_fact.assumption
+ |apply lt_to_le.apply lt_to_le.
+ apply lt_to_le.assumption
+ ]
+ ]
+qed.
+
+theorem lt_SO_to_le_exp_B: \forall n. 1 < n \to
+exp 2 (2*n) \le 2 * n * B (2*n).
intros.
apply (le_times_to_le (exp (fact n) (S(S O))))
[apply lt_O_exp.
qed.
theorem le_exp_B: \forall n. O < n \to
-exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n).
+exp 2 (2*n) \le 2 * n * B (2*n).
intros.
elim H
[apply le_n
qed.
theorem eq_A_SSO_n: \forall n.O < n \to
-A((S(S O))*n) =
- pi_p (S ((S(S O))*n)) primeb
- (\lambda p.(pi_p (log p ((S(S O))*n) )
+A(2*n) =
+ pi_p (S (2*n)) primeb
+ (\lambda p.(pi_p (log p (2*n) )
(\lambda i.true) (\lambda i.(exp p (bool_to_nat (leb (S n) (exp p (S i))))))))
*A n.
intro.
qed.
theorem le_A_BA1: \forall n. O < n \to
-A((S(S O))*n) \le B((S(S O))*n)*A n.
+A(2*n) \le B(2*n)*A n.
intros.
rewrite > eq_A_SSO_n
[apply le_times_l.
qed.
theorem le_A_exp: \forall n.
-A((S(S O))*n) \le (exp (S(S O)) ((S(S O)*n)))*A n.
+A(2*n) \le (exp 2 (pred (2*n)))*A n.
intro.
-apply (trans_le ? (B((S(S O))*n)*A n))
+apply (trans_le ? (B(2*n)*A n))
[apply le_A_BA
|apply le_times_l.
apply le_B_exp
]
qed.
+theorem lt_SSSSO_to_le_A_exp: \forall n. 4 < n \to
+A(2*n) \le exp 2 ((2*n)-2)*A n.
+intros.
+apply (trans_le ? (B(2*n)*A n))
+ [apply le_A_BA
+ |apply le_times_l.
+ apply lt_SSSSO_to_le_B_exp.assumption
+ ]
+qed.
+
+theorem times_SSO_pred: \forall n. 2*(pred n) \le pred (2*n).
+intro.cases n
+ [apply le_n
+ |simplify.apply le_plus_r.
+ apply le_n_Sn
+ ]
+qed.
+
+theorem le_S_times_SSO: \forall n. O < n \to S n \le 2*n.
+intros.
+elim H
+ [apply le_n
+ |rewrite > times_SSO.
+ apply le_S_S.
+ apply (trans_le ? (2*n1))
+ [assumption
+ |apply le_n_Sn
+ ]
+ ]
+qed.
+
theorem le_A_exp1: \forall n.
-A(exp (S(S O)) n) \le (exp (S(S O)) ((S(S O))*(exp (S(S O)) n))).
+A(exp 2 n) \le (exp 2 ((2*(exp 2 n)-(S(S n))))).
intro.elim n
- [simplify.apply le_S_S.apply le_O_n
- |change with (A ((S(S O))*((S(S O)))\sup n1)≤ ((S(S O)))\sup((S(S O))*((S(S O))\sup(S n1)))).
- apply (trans_le ? ((exp (S(S O)) ((S(S O)*(exp (S(S O)) n1))))*A (exp (S(S O)) n1)))
+ [simplify.apply le_n
+ |change in ⊢ (? % ?) with (A(2*(exp 2 n1))).
+ apply (trans_le ? ((exp 2 (pred(2*(exp (S(S O)) n1))))*A (exp (S(S O)) n1)))
[apply le_A_exp
- |apply (trans_le ? ((S(S O))\sup((S(S O))*((S(S O)))\sup(n1))*(S(S O))\sup((S(S O))*((S(S O)))\sup(n1))))
+ |apply (trans_le ? ((2)\sup(pred (2*(2)\sup(n1)))*(2)\sup(2*(2)\sup(n1)-S (S n1))))
[apply le_times_r.
assumption
|rewrite < exp_plus_times.
- simplify.rewrite < plus_n_O in ⊢ (? ? (? ? (? ? %))).
- apply le_n
+ apply le_exp
+ [apply lt_O_S
+ |cut (S(S n1) \le 2*(exp 2 n1))
+ [apply le_S_S_to_le.
+ change in ⊢ (? % ?) with (S(pred (2*(2)\sup(n1)))+(2*(2)\sup(n1)-S (S n1))).
+ rewrite < S_pred
+ [rewrite > eq_minus_S_pred in ⊢ (? ? %).
+ rewrite < S_pred
+ [rewrite < eq_minus_plus_plus_minus
+ [rewrite > plus_n_O in ⊢ (? (? (? ? %) ?) ?).
+ apply le_n
+ |assumption
+ ]
+ |apply lt_to_lt_O_minus.
+ apply (lt_to_le_to_lt ? (2*(S(S n1))))
+ [rewrite > times_n_SO in ⊢ (? % ?).
+ rewrite > sym_times.
+ apply lt_times_l1
+ [apply lt_O_S
+ |apply le_n
+ ]
+ |apply le_times_r.
+ assumption
+ ]
+ ]
+ |unfold.rewrite > times_n_SO in ⊢ (? % ?).
+ apply le_times
+ [apply le_n_Sn
+ |apply lt_O_exp.
+ apply lt_O_S
+ ]
+ ]
+ |elim n1
+ [apply le_n
+ |apply (trans_le ? (2*(S(S n2))))
+ [apply le_S_times_SSO.
+ apply lt_O_S
+ |apply le_times_r.
+ assumption
+ ]
+ ]
+ ]
+ ]
]
]
]
-qed.
+qed.
theorem monotonic_A: monotonic nat le A.
unfold.intros.
]
]
qed.
-
+
+(*
theorem le_A_exp2: \forall n. O < n \to
A(n) \le (exp (S(S O)) ((S(S O)) * (S(S O)) * n)).
intros.
]
]
qed.
+*)
(* example *)
theorem A_SO: A (S O) = S O.
reflexivity.
qed.
-(* da spostare *)
-theorem or_eq_eq_S: \forall n.\exists m.
-n = (S(S O))*m \lor n = S ((S(S O))*m).
-intro.elim n
- [apply (ex_intro ? ? O).
- left.reflexivity
- |elim H.elim H1
- [apply (ex_intro ? ? a).
- right.apply eq_f.assumption
- |apply (ex_intro ? ? (S a)).
- left.rewrite > H2.
- apply sym_eq.
- apply times_SSO
- ]
- ]
-qed.
-
+(*
(* a better result *)
theorem le_A_exp3: \forall n. S O < n \to
-A(n) \le exp (pred n) (S(S O))*(exp (S(S O)) ((S(S O)) * n)).
+A(n) \le exp (pred n) (2*(exp 2 (2 * n)).
intro.
apply (nat_elim1 n).
intros.
apply False_ind.
apply (lt_to_not_le ? ? H1).
rewrite > H3.
- apply le_n
+ apply le_
]
]
qed.
+*)
theorem le_A_exp4: \forall n. S O < n \to
-A(n) \le exp (pred n) (S(S O))*(exp (S(S O)) ((S(S O)) * (pred n))).
+A(n) \le (pred n)*exp 2 ((2 * n) -3).
intro.
apply (nat_elim1 n).
intros.
elim H2
[elim (le_to_or_lt_eq (S O) a)
[rewrite > H3 in ⊢ (? % ?).
- apply (trans_le ? ((exp (S(S O)) ((S(S O)*a)))*A a))
+ apply (trans_le ? (exp 2 (pred(2*a))*A a))
[apply le_A_exp
- |apply (trans_le ? (((S(S O)))\sup((S(S O))*a)*
- ((pred a)\sup((S(S O)))*((S(S O)))\sup((S(S O))*(pred a)))))
+ |apply (trans_le ? (2\sup(pred(2*a))*((pred a)*2\sup((2*a)-3))))
[apply le_times_r.
apply H
[rewrite > H3.
]
|assumption
]
- |rewrite > sym_times.
+ |rewrite < H3.
+ rewrite < assoc_times.
+ rewrite > sym_times in ⊢ (? (? % ?) ?).
rewrite > assoc_times.
- rewrite < exp_plus_times.
- apply (trans_le ?
- (pred a\sup((S(S O)))*(S(S O))\sup(S(S O))*(S(S O))\sup((S(S O))*(pred m))))
- [rewrite > assoc_times.
- apply le_times_r.
- rewrite < exp_plus_times.
+ apply le_times
+ [rewrite > H3.
+ elim a[apply le_n|simplify.apply le_plus_n_r]
+ |rewrite < exp_plus_times.
apply le_exp
[apply lt_O_S
- |rewrite < H3.
- simplify.
- rewrite < plus_n_O.
- apply le_S.apply le_S.
- rewrite < plus_n_O;
- apply le_S_S_to_le;
- rewrite > plus_n_Sm in \vdash (? ? %);
- rewrite < S_pred;
- [change in \vdash (? % ?) with ((S (pred a + pred a)) + m);
- apply le_plus_l;
- apply le_S_S_to_le;
- rewrite < S_pred;
- [rewrite > plus_n_Sm;rewrite > sym_plus;
- rewrite > plus_n_Sm;
- rewrite > H3;simplify in \vdash (? ? %);
- rewrite < plus_n_O;rewrite < S_pred;
- [apply le_n
- |apply lt_to_le;assumption]
- |apply lt_to_le;assumption]
- |apply lt_to_le;assumption]]
- |apply le_times_l.
- rewrite > times_exp.
- apply monotonic_exp1.
- rewrite > H3.
- rewrite > sym_times.
- cases a
- [apply le_n
- |simplify.
- rewrite < plus_n_Sm.
- apply le_S.
- apply le_n
+ |apply (trans_le ? (m+(m-3)))
+ [apply le_plus_l.
+ cases m[apply le_n|apply le_n_Sn]
+ |simplify.rewrite < plus_n_O.
+ rewrite > eq_minus_plus_plus_minus
+ [apply le_n
+ |rewrite > H3.
+ apply (trans_le ? (2*2))
+ [simplify.apply (le_n_Sn 3)
+ |apply le_times_r.assumption
+ ]
+ ]
+ ]
]
]
]
]
- |rewrite < H4 in H3.
- simplify in H3.
- rewrite > H3.
- simplify.
- apply le_S_S.apply le_S_S.apply le_O_n
+ |rewrite > H3.rewrite < H4.simplify.
+ apply le_S_S.apply lt_O_S
|apply not_lt_to_le.intro.
apply (lt_to_not_le ? ? H1).
rewrite > H3.
rewrite > H3.
rewrite > times_SSO.
apply le_n_Sn
- |apply (trans_le ? ((exp (S(S O)) ((S(S O)*(S a))))*A (S a)))
+ |apply (trans_le ? ((exp 2 (pred(2*(S a))))*A (S a)))
[apply le_A_exp
- |apply (trans_le ? (((S(S O)))\sup((S(S O))*S a)*
- (a\sup((S(S O)))*((S(S O)))\sup((S(S O))*a))))
+ |apply (trans_le ? ((2\sup(pred (2*S a)))*(a*(exp 2 ((2*(S a))-3)))))
[apply le_times_r.
- apply (trans_le ? ? ? (H (S a) ? ?));
+ apply H
[rewrite > H3.
apply le_S_S.
- simplify.
- rewrite > plus_n_SO.
- apply le_plus_r.
- rewrite < plus_n_O.
+ apply le_S_times_SSO.
assumption
|apply le_S_S.assumption
- |simplify in ⊢ (? (? (? % ?) ?) ?);
- apply le_times_r;apply le_exp;
- [apply le_S;apply le_n
- |apply le_times_r;simplify;apply le_n]
]
- |rewrite > sym_times.
- rewrite > assoc_times.
- rewrite < exp_plus_times.
- rewrite > H3;
- change in ⊢ (? ? (? (? % ?) ?)) with ((S (S O))*a);
- rewrite < times_exp;
- rewrite < sym_times in \vdash (? ? (? % ?));
- rewrite > assoc_times;
- apply le_times_r;
- rewrite < times_n_Sm in ⊢ (? (? ? (? ? %)) ?);
- rewrite > sym_plus in \vdash (? (? ? %) ?);
- rewrite > assoc_plus in \vdash (? (? ? %) ?);
- rewrite > exp_plus_times;apply le_times_r;
- rewrite < distr_times_plus in ⊢ (? (? ? %) ?);
- simplify in ⊢ (? ? (? ? (? ? %)));rewrite < plus_n_O;
- apply le_n
+ |rewrite > H3.
+ change in ⊢ (? ? (? % ?)) with (2*a).
+ rewrite > times_SSO.
+ change in ⊢ (? (? (? ? %) ?) ?) with (S(2*a)).
+ rewrite > minus_Sn_m
+ [change in ⊢ (? (? ? (? ? %)) ?) with (2*(exp 2 (S(2*a)-3))).
+ rewrite < assoc_times in ⊢ (? (? ? %) ?).
+ rewrite < assoc_times.
+ rewrite > sym_times in ⊢ (? (? % ?) ?).
+ rewrite > sym_times in ⊢ (? (? (? % ?) ?) ?).
+ rewrite > assoc_times.
+ apply le_times_r.
+ rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_O_S
+ |rewrite < eq_minus_plus_plus_minus
+ [rewrite > plus_n_O in ⊢ (? (? (? ? %) ?) ?).
+ apply le_n
+ |apply le_S_S.
+ apply O_lt_const_to_le_times_const.
+ assumption
+ ]
+ ]
+ |apply le_S_S.
+ apply O_lt_const_to_le_times_const.
+ assumption
+ ]
]
]
]
]
qed.
+theorem le_n_SSSSSSSSO_to_le_A_exp:
+\forall n. n \le 8 \to A(n) \le exp 2 ((2 * n) -3).
+intro.cases n
+ [intro.apply le_n
+ |cases n1
+ [intro.apply le_n
+ |cases n2
+ [intro.apply le_n
+ |cases n3
+ [intro.apply leb_true_to_le.reflexivity
+ |cases n4
+ [intro.apply leb_true_to_le.reflexivity
+ |cases n5
+ [intro.apply leb_true_to_le.reflexivity
+ |cases n6
+ [intro.apply leb_true_to_le.reflexivity
+ |cases n7
+ [intro.apply leb_true_to_le.reflexivity
+ |cases n8
+ [intro.apply leb_true_to_le.reflexivity
+ |intro.apply False_ind.
+ apply (lt_to_not_le ? ? H).
+ apply leb_true_to_le.reflexivity
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem le_A_exp5: \forall n. A(n) \le exp 2 ((2 * n) -3).
+intro.
+apply (nat_elim1 n).
+intros.
+elim (decidable_le 9 m)
+ [elim (or_eq_eq_S m).
+ elim H2
+ [rewrite > H3 in ⊢ (? % ?).
+ apply (trans_le ? (exp 2 (pred(2*a))*A a))
+ [apply le_A_exp
+ |apply (trans_le ? (2\sup(pred(2*a))*(2\sup((2*a)-3))))
+ [apply le_times_r.
+ apply H.
+ rewrite > H3.
+ apply lt_m_nm
+ [apply (trans_lt ? 4)
+ [apply lt_O_S
+ |apply (lt_times_to_lt 2)
+ [apply lt_O_S
+ |rewrite < H3.assumption
+ ]
+ ]
+ |apply le_n
+ ]
+ |rewrite < H3.
+ rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_O_S
+ |simplify.rewrite < plus_n_O.
+ rewrite > eq_minus_plus_plus_minus
+ [apply le_plus_l.
+ apply le_pred_n
+ |apply (trans_le ? 9)
+ [apply leb_true_to_le. reflexivity
+ |assumption
+ ]
+ ]
+ ]
+ ]
+ ]
+ |apply (trans_le ? (A (2*(S a))))
+ [apply monotonic_A.
+ rewrite > H3.
+ rewrite > times_SSO.
+ apply le_n_Sn
+ |apply (trans_le ? ((exp 2 ((2*(S a))-2))*A (S a)))
+ [apply lt_SSSSO_to_le_A_exp.
+ apply le_S_S.
+ apply (le_times_to_le 2)
+ [apply le_n_Sn.
+ |apply le_S_S_to_le.rewrite < H3.assumption
+ ]
+ |apply (trans_le ? ((2\sup((2*S a)-2))*(exp 2 ((2*(S a))-3))))
+ [apply le_times_r.
+ apply H.
+ rewrite > H3.
+ apply le_S_S.
+ apply lt_m_nm
+ [apply (lt_to_le_to_lt ? 4)
+ [apply lt_O_S
+ |apply (le_times_to_le 2)
+ [apply lt_O_S
+ |apply le_S_S_to_le.
+ rewrite < H3.assumption
+ ]
+ ]
+ |apply le_n
+ ]
+ |rewrite > times_SSO.
+ rewrite < H3.
+ rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_O_S
+ |cases m
+ [apply le_n
+ |cases n1
+ [apply le_n
+ |simplify.
+ rewrite < minus_n_O.
+ rewrite < plus_n_O.
+ rewrite < plus_n_Sm.
+ simplify.rewrite < minus_n_O.
+ rewrite < plus_n_Sm.
+ apply le_n
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ |apply le_n_SSSSSSSSO_to_le_A_exp.
+ apply le_S_S_to_le.
+ apply not_le_to_lt.
+ assumption
+ ]
+qed.
+
+theorem le_exp_Al:\forall n. O < n \to exp 2 n \le A (2 * n).
+intros.
+apply (trans_le ? ((exp 2 (2*n))/(2*n)))
+ [apply le_times_to_le_div
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_S
+ |assumption
+ ]
+ |simplify in ⊢ (? ? (? ? %)).
+ rewrite < plus_n_O.
+ rewrite > exp_plus_times.
+ apply le_times_l.
+ apply le_times_SSO_n_exp_SSO_n
+ ]
+ |apply le_times_to_le_div2
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_S
+ |assumption
+ ]
+ |apply (trans_le ? ((B (2*n)*(2*n))))
+ [rewrite < sym_times in ⊢ (? ? %).
+ apply le_exp_B.
+ assumption
+ |apply le_times_l.
+ apply le_B_A
+ ]
+ ]
+ ]
+qed.
+
+theorem le_exp_A2:\forall n. 1 < n \to exp 2 (n / 2) \le A n.
+intros.
+apply (trans_le ? (A(n/2*2)))
+ [rewrite > sym_times.
+ apply le_exp_Al.
+ elim (le_to_or_lt_eq ? ? (le_O_n (n/2)))
+ [assumption
+ |apply False_ind.
+ apply (lt_to_not_le ? ? H).
+ rewrite > (div_mod n 2)
+ [rewrite < H1.
+ change in ⊢ (? % ?) with (n\mod 2).
+ apply le_S_S_to_le.
+ apply lt_mod_m_m.
+ apply lt_O_S
+ |apply lt_O_S
+ ]
+ ]
+ |apply monotonic_A.
+ rewrite > (div_mod n 2) in ⊢ (? ? %).
+ apply le_plus_n_r.
+ apply lt_O_S
+ ]
+qed.
+
theorem eq_sigma_pi_SO_n: \forall n.
sigma_p n (\lambda i.true) (\lambda i.S O) = n.
intro.elim n
(* the inequalities *)
theorem le_exp_priml: \forall n. O < n \to
-exp (S(S O)) ((S(S O))*n) \le exp ((S(S O))*n) (S(prim ((S(S O))*n))).
+exp 2 (2*n) \le exp (2*n) (S(prim (2*n))).
intros.
-apply (trans_le ? ((((S(S O))*n*(B ((S(S O))*n))))))
+apply (trans_le ? (((2*n*(B (2*n))))))
[apply le_exp_B.assumption
- |change in ⊢ (? ? %) with ((((S(S O))*n))*(((S(S O))*n))\sup (prim ((S(S O))*n))).
+ |change in ⊢ (? ? %) with (((2*n))*((2*n))\sup (prim (2*n))).
apply le_times_r.
- apply (trans_le ? (A ((S(S O))*n)))
+ apply (trans_le ? (A (2*n)))
[apply le_B_A
|apply le_Al
]
]
qed.
-alias num (instance 0) = "natural number".
-
theorem le_exp_prim4l: \forall n. O < n \to
exp 2 (S(4*n)) \le exp (4*n) (S(prim (4*n))).
intros.
qed.
theorem le_priml: \forall n. O < n \to
-(S(S O))*n \le (S (log (S(S O)) ((S(S O))*n)))*S(prim ((S(S O))*n)).
+2*n \le (S (log 2 (2*n)))*S(prim (2*n)).
intros.
rewrite < (eq_log_exp (S(S O))) in ⊢ (? % ?)
[apply (trans_le ? ((log (S(S O)) (exp ((S(S O))*n) (S(prim ((S(S O))*n)))))))
]
qed.
-theorem le_exp_primr: \forall n. S O < n \to
-exp n (prim n) \le exp (pred n) ((S(S O))*(S(S O)))*(exp (S(S O)) ((S(S O))*(S(S O)) * (pred n))).
+theorem le_exp_primr: \forall n.
+exp n (prim n) \le exp 2 (2*(2*n-3)).
intros.
-apply (trans_le ? (exp (A n) (S(S O))))
+apply (trans_le ? (exp (A n) 2))
[change in ⊢ (? ? %) with ((A n)*((A n)*(S O))).
rewrite < times_n_SO.
apply leA_r2
- |apply (trans_le ? (exp (exp (pred n) (S(S O))*(exp (S(S O)) ((S(S O)) * (pred n)))) (S(S O))))
- [apply monotonic_exp1.
- apply le_A_exp4.
+ |rewrite > sym_times.
+ rewrite < exp_exp_times.
+ apply monotonic_exp1.
+ apply le_A_exp5
+ ]
+qed.
+
+(* bounds *)
+theorem le_primr: \forall n. 1 < n \to prim n \le 2*(2*n-3)/log 2 n.
+intros.
+apply le_times_to_le_div
+ [apply lt_O_log
+ [apply lt_to_le.assumption
+ |assumption
+ ]
+ |apply (trans_le ? (log 2 (exp n (prim n))))
+ [rewrite > sym_times.
+ apply log_exp2
+ [apply le_n
+ |apply lt_to_le.assumption
+ ]
+ |rewrite < (eq_log_exp 2) in ⊢ (? ? %)
+ [apply le_log
+ [apply le_n
+ |apply le_exp_primr
+ ]
+ |apply le_n
+ ]
+ ]
+ ]
+qed.
+
+theorem le_priml1: \forall n. O < n \to
+2*n/((log 2 n)+2) - 1 \le prim (2*n).
+intros.
+apply le_plus_to_minus.
+apply le_times_to_le_div2
+ [rewrite > sym_plus.
+ simplify.apply lt_O_S
+ |rewrite > sym_times in ⊢ (? ? %).
+ rewrite < plus_n_Sm.
+ rewrite < plus_n_Sm in ⊢ (? ? (? ? %)).
+ rewrite < plus_n_O.
+ rewrite < sym_plus.
+ rewrite < log_exp
+ [simplify in ⊢ (? ? (? (? (? ? (? % ?))) ?)).
+ apply le_priml.
assumption
- |rewrite < times_exp.
- rewrite > exp_exp_times.
- rewrite > exp_exp_times.
- rewrite > sym_times in ⊢ (? (? ? (? ? %)) ?).
- rewrite < assoc_times in ⊢ (? (? ? (? ? %)) ?).
- apply le_n
+ |apply le_n
+ |assumption
]
]
qed.
+(*
+theorem prim_SSSSSSO: \forall n.30\le n \to O < prim (8*n) - prim n.
+intros.
+apply lt_to_lt_O_minus.
+change in ⊢ (? ? (? (? % ?))) with (2*4).
+rewrite > assoc_times.
+apply (le_to_lt_to_lt ? (2*(2*n-3)/log 2 n))
+ [apply le_primr.apply (trans_lt ? ? ? ? H).
+ apply leb_true_to_le.reflexivity
+ |apply (lt_to_le_to_lt ? (2*(4*n)/((log 2 (4*n))+2) - 1))
+ [elim H
+ [
+normalize in ⊢ (%);simplify.
+ |apply le_priml1.
+*)
+
+
+
projects / helm.git / blobdiff