\ /
V_______________________________________________________________ *)
-include "arithmetics/sigma_pi.ma".
include "arithmetics/primes.ma".
+include "arithmetics/sigma_pi.ma".
(* binomial coefficient *)
definition bc ≝ λn,k. n!/(k!*(n-k)!).
qed.
theorem bc_n_O: ∀n. bc n O = 1.
-#n >bceq <minus_n_O /2/
+#n >bceq <minus_n_O /2 by injective_plus_r/
qed.
theorem fact_minus: ∀n,k. k < n →
cut (m-(m-(S c)) = S c) [@plus_to_minus @plus_minus_m_m //] #eqSc
lapply (Hind c (le_S_S_to_le … lec)) #Hc
lapply (Hind (m - (S c)) ?) [@le_plus_to_minus //] >eqSc #Hmc
- >(plus_minus_m_m m c) in ⊢ (??(??(?%))) [|@le_S_S_to_le //]
+ >(plus_minus_m_m m c) in ⊢ (??(??(?%))); [|@le_S_S_to_le //]
>commutative_plus >(distributive_times_plus ? (S c))
@divides_plus
[>associative_times >(commutative_times (S c))
<associative_times @divides_times //
- |<(fact_minus …ltcm) in ⊢ (?(??%)?)
+ |<(fact_minus …ltcm) in ⊢ (?(??%)?);
<associative_times @divides_times //
>commutative_times @Hmc
]
theorem bc1: ∀n.∀k. k < n →
bc (S n) (S k) = (bc n k) + (bc n (S k)).
#n #k #ltkn > bceq >(bceq n) >(bceq n (S k))
->(div_times_times ?? (S k)) in ⊢ (???(?%?))
+>(div_times_times ?? (S k)) in ⊢ (???(?%?));
[|>(times_n_O 0) @lt_times // | //]
->associative_times in ⊢ (???(?(??%)?))
->commutative_times in ⊢ (???(?(??(??%))?))
-<associative_times in ⊢ (???(?(??%)?))
->(div_times_times ?? (n - k)) in ⊢ (???(??%))
+>associative_times in ⊢ (???(?(??%)?));
+>commutative_times in ⊢ (???(?(??(??%))?));
+<associative_times in ⊢ (???(?(??%)?));
+>(div_times_times ?? (n - k)) in ⊢ (???(??%)) ;
[|>(times_n_O 0) @lt_times //
- |@(le_plus_to_le_r k ??) <plus_minus_m_m /2/]
->associative_times in ⊢ (???(??(??%)))
+ |@(le_plus_to_le_r k ??) <plus_minus_m_m /2 by lt_to_le/]
+>associative_times in ⊢ (???(??(??%)));
>fact_minus // <plus_div
[<distributive_times_plus
- >commutative_plus in ⊢ (???%) <plus_n_Sm <plus_minus_m_m [|/2/] @refl
+ >commutative_plus in ⊢ (???%); <plus_n_Sm <plus_minus_m_m [|/2 by lt_to_le/] @refl
|<fact_minus // <associative_times @divides_times // @(bc2 n (S k)) //
|>associative_times >(commutative_times (S k))
- <associative_times @divides_times // @bc2 /2/
+ <associative_times @divides_times // @bc2 /2 by lt_to_le/
|>(times_n_O 0) @lt_times [@(le_1_fact (S k)) | //]
]
qed.
qed.
theorem binomial_law:∀a,b,n.
- (a+b)^n = Σ_{k < S n}((bc n k)*(a^(n-k))*(b^k)).
+ (a+b)^n = ∑_{k < S n}((bc n k)*(a^(n-k))*(b^k)).
#a #b #n (elim n) //
--n #n #Hind normalize in ⊢ (? ? % ?).
->bigop_Strue // >Hind >distributive_times_plus
-<(minus_n_n (S n)) <commutative_times <(commutative_times b)
-(* hint??? *)
->(bigop_distr ???? natDop ? a) >(bigop_distr ???? natDop ? b)
->bigop_Strue in ⊢ (??(??%)?) // <associative_plus
-<commutative_plus in ⊢ (??(? % ?) ?) >associative_plus @eq_f2
+-n #n #Hind normalize in ⊢ (??%?); >commutative_times
+>bigop_Strue // >Hind >distributive_times_plus_r
+<(minus_n_n (S n)) (* <commutative_times <(commutative_times b) *)
+(* da capire *)
+>(bigop_distr ??? 0 (mk_Dop ℕ O plusAC times (λn0:ℕ.sym_eq ℕ O (n0*O) (times_n_O n0))
+ distributive_times_plus) ? a)
+>(bigop_distr ???? (mk_Dop ℕ O plusAC times (λn0:ℕ.sym_eq ℕ O (n0*O) (times_n_O n0))
+ distributive_times_plus) ? b)
+>bigop_Strue in ⊢ (??(??%)?); // <associative_plus
+<commutative_plus in ⊢ (??(? % ?) ?); >associative_plus @eq_f2
[<minus_n_n >bc_n_n >bc_n_n normalize //
- |>bigop_0 >associative_plus >commutative_plus in ⊢ (??(??%)?)
- <associative_plus >bigop_0 // @eq_f2
- [>(bigop_op n ??? natACop) @same_bigop //
+ |>(bigop_0 ??? plusA) >associative_plus >commutative_plus in ⊢ (??(??%)?);
+ <associative_plus >(bigop_0 n ?? plusA) @eq_f2
+ [cut (∀a,b. plus a b = plusAC a b) [//] #Hplus >Hplus
+ >(bigop_op n ??? plusAC) @same_bigop //
#i #ltin #_ <associative_times >(commutative_times b)
- >(associative_times ?? b) <(distributive_times_plus_r (b^(S i)))
+ >(associative_times ?? b) <Hplus <(distributive_times_plus_r (b^(S i)))
@eq_f2 // <associative_times >(commutative_times a)
>associative_times cut(∀n.a*a^n = a^(S n)) [#n @commutative_times] #H
>H <minus_Sn_m // <(distributive_times_plus_r (a^(n-i)))
@eq_f2 // @sym_eq >commutative_plus @bc1 //
- |>bc_n_O >bc_n_O normalize //
+ |>bc_n_O >bc_n_O normalize //
]
]
qed.
theorem exp_S_sigma_p:∀a,n.
-(S a)^n = Σ_{k < S n}((bc n k)*a^(n-k)).
+ (S a)^n = ∑_{k < S n}((bc n k)*a^(n-k)).
#a #n cut (S a = a + 1) // #H >H
>binomial_law @same_bigop //
qed.
-(*
-theorem exp_Sn_SSO: \forall n. exp (S n) 2 = S((exp n 2) + 2*n).
-intros.simplify.
-rewrite < times_n_SO.
-rewrite < plus_n_O.
-rewrite < sym_times.simplify.
-rewrite < assoc_plus.
-rewrite < sym_plus.
-reflexivity.
-qed. *)
+(************ mid value *************)
+lemma eqb_sym: ∀a,b. eqb a b = eqb b a.
+#a #b cases (true_or_false (eqb a b)) #Hab
+ [>(eqb_true_to_eq … Hab) //
+ |>Hab @sym_eq @not_eq_to_eqb_false
+ @(not_to_not … (eqb_false_to_not_eq … Hab)) //
+ ]
+qed-.
+
+definition M ≝ λm.bc (S(2*m)) m.
+
+lemma Mdef : ∀m. M m = bc (S(2*m)) m.
+// qed.
+theorem lt_M: ∀m. O < m → M m < exp 2 (2*m).
+#m #posm @(lt_times_n_to_lt_l 2)
+cut (∀a,b. a^(S b) = a^b*a) [//] #expS <expS
+cut (2 = 1+1) [//] #H2 >H2 in ⊢ (??(?%?));
+>binomial_law
+@(le_to_lt_to_lt ?
+ (∑_{k < S (S (2*m)) | orb (eqb k m) (eqb k (S m))}
+ (bc (S (2*m)) k*1^(S (2*m)-k)*1^k)))
+ [>(bigop_diff ??? plusAC … m)
+ [>(bigop_diff ??? plusAC … (S m))
+ [<(pad_bigop1 … (S(S(2*m))) 0)
+ [cut (∀a,b. plus a b = plusAC a b) [//] #Hplus <Hplus <Hplus
+ whd in ⊢ (? ? (? ? (? ? %)));
+ cut (∀a. 2*a = a + a) [//] #H2a >commutative_times >H2a
+ <exp_1_n <exp_1_n <exp_1_n <exp_1_n
+ <times_n_1 <times_n_1 <times_n_1 <times_n_1
+ @le_plus
+ [@le_n
+ |>Mdef <plus_n_O >bceq >bceq
+ cut (∀a,b.S a - (S b) = a -b) [//] #Hminus >Hminus
+ normalize in ⊢ (??(??(??(?(?%?))))); <plus_n_O <minus_plus_m_m
+ <commutative_times in ⊢ (??(??%));
+ cut (S (2*m)-m = S m)
+ [>H2a >plus_n_Sm >commutative_plus <minus_plus_m_m //]
+ #Hcut >Hcut //
+ ]
+ |#i #_ #_ >(eqb_sym i m) >(eqb_sym i (S m))
+ cases (eqb m i) cases (eqb (S m) i) //
+ |@le_O_n
+ ]
+ |>(eqb_sym (S m) m) >(eq_to_eqb_true ? ? (refl ? (S m)))
+ >(not_eq_to_eqb_false m (S m))
+ [// |@(not_to_not … (not_eq_n_Sn m)) //]
+ |@le_S_S @le_S_S //
+ ]
+ |>(eq_to_eqb_true ?? (refl ? m)) //
+ |@le_S @le_S_S //
+ ]
+ |@lt_sigma_p
+ [//
+ |#i #lti #Hi @le_n
+ |%{0} %
+ [@lt_O_S
+ |%2 %
+ [% // >(not_eq_to_eqb_false 0 (S m)) //
+ >(not_eq_to_eqb_false 0 m) // @lt_to_not_eq //
+ |>bc_n_O <exp_1_n <exp_1_n @le_n
+ ]
+ ]
+ ]
+ ]
+qed.
+
projects / helm.git / blobdiff