\ /
V_______________________________________________________________ *)
-include "arithmetics/bigops.ma".
+include "arithmetics/sigma_pi.ma".
include "arithmetics/primes.ma".
(* binomial coefficient *)
qed.
theorem fact_minus: ∀n,k. k < n →
- (n-k)*(n - S k)! = (n - k)!.
+ (n - S k)!*(n-k) = (n - k)!.
#n #k (cases n)
[#ltO @False_ind /2/
- |#l #ltl >minus_Sn_m [>commutative_times //|@le_S_S_to_le //]
+ |#l #ltl >(minus_Sn_m k) [// |@le_S_S_to_le //]
]
qed.
∀k. k ≤n → k!*(n-k)! ∣ n!.
#n (elim n)
[#k #lek0 <(le_n_O_to_eq ? lek0) //
- |#m #Hind #k generalize in match H1;cases k
- [intro;simplify in ⊢ (? (? ? (? %)) ?);simplify in ⊢ (? (? % ?) ?);
- rewrite > sym_times;rewrite < times_n_SO;apply reflexive_divides
- |intro;elim (decidable_eq_nat n2 n1)
- [rewrite > H3;rewrite < minus_n_n;
- rewrite > times_n_SO in ⊢ (? ? %);apply reflexive_divides
- |lapply (H n2)
- [lapply (H (n1 - (S n2)))
- [change in ⊢ (? ? %) with ((S n1)*n1!);
- rewrite > (plus_minus_m_m n1 n2) in ⊢ (? ? (? (? %) ?))
- [rewrite > sym_plus;
- change in ⊢ (? ? (? % ?)) with ((S n2) + (n1 - n2));
- rewrite > sym_times in \vdash (? ? %);
- rewrite > distr_times_plus in ⊢ (? ? %);
- simplify in ⊢ (? (? ? (? %)) ?);
- apply divides_plus
- [rewrite > sym_times;
- change in ⊢ (? (? ? %) ?) with ((S n2)*n2!);
- rewrite > sym_times in ⊢ (? (? ? %) ?);
- rewrite < assoc_times;
- apply divides_times
- [rewrite > sym_times;assumption
- |apply reflexive_divides]
- |rewrite < fact_minus in ⊢ (? (? ? %) ?)
- [rewrite > sym_times in ⊢ (? (? ? %) ?);
- rewrite < assoc_times;
- apply divides_times
- [rewrite < eq_plus_minus_minus_minus in Hletin1;
- [rewrite > sym_times;rewrite < minus_n_n in Hletin1;
- rewrite < plus_n_O in Hletin1;assumption
- |lapply (le_S_S_to_le ? ? H2);
- elim (le_to_or_lt_eq ? ? Hletin2);
- [assumption
- |elim (H3 H4)]
- |constructor 1]
- |apply reflexive_divides]
- |lapply (le_S_S_to_le ? ? H2);
- elim (le_to_or_lt_eq ? ? Hletin2);
- [assumption
- |elim (H3 H4)]]]
- |apply le_S_S_to_le;assumption]
- |apply le_minus_m;apply le_S_S_to_le;assumption]
- |apply le_S_S_to_le;assumption]]]]
-qed.
-
-theorem bc1: \forall n.\forall k. k < n \to bc (S n) (S k) = (bc n k) + (bc n (S k)).
-intros.unfold bc.
-rewrite > (lt_to_lt_to_eq_div_div_times_times ? ? (S k)) in ⊢ (? ? ? (? % ?))
- [rewrite > sym_times in ⊢ (? ? ? (? (? ? %) ?)).
- rewrite < assoc_times in ⊢ (? ? ? (? (? ? %) ?)).
- rewrite > (lt_to_lt_to_eq_div_div_times_times ? ? (n - k)) in ⊢ (? ? ? (? ? %))
- [rewrite > assoc_times in ⊢ (? ? ? (? ? (? ? %))).
- rewrite > sym_times in ⊢ (? ? ? (? ? (? ? (? ? %)))).
- rewrite > fact_minus
- [rewrite < eq_div_plus
- [rewrite < distr_times_plus.
- simplify in ⊢ (? ? ? (? (? ? %) ?)).
- rewrite > sym_plus in ⊢ (? ? ? (? (? ? (? %)) ?)).
- rewrite < plus_minus_m_m
- [rewrite > sym_times in ⊢ (? ? ? (? % ?)).
- reflexivity
- |apply lt_to_le.
- assumption
- ]
- |rewrite > (times_n_O O).
- apply lt_times;apply lt_O_fact
- |change in ⊢ (? (? % ?) ?) with ((S k)*k!);
- rewrite < sym_times in ⊢ (? ? %);
- rewrite > assoc_times;apply divides_times
- [apply reflexive_divides
- |apply bc2;apply le_S_S_to_le;constructor 2;assumption]
- |rewrite < fact_minus
- [rewrite > sym_times in ⊢ (? (? ? %) ?);rewrite < assoc_times;
- apply divides_times
- [apply bc2;assumption
- |apply reflexive_divides]
- |assumption]]
- |assumption]
- |apply lt_to_lt_O_minus;assumption
- |rewrite > (times_n_O O).
- apply lt_times;apply lt_O_fact]
-|apply lt_O_S
-|rewrite > (times_n_O O).
- apply lt_times;apply lt_O_fact]
-qed.
-
-theorem lt_O_bc: \forall n,m. m \le n \to O < bc n m.
-intro.elim n
- [apply (le_n_O_elim ? H).
- simplify.apply le_n
- |elim (le_to_or_lt_eq ? ? H1)
- [generalize in match H2.cases m;intro
- [rewrite > bc_n_O.apply le_n
- |rewrite > bc1
- [apply (trans_le ? (bc n1 n2))
- [apply H.apply le_S_S_to_le.apply lt_to_le.assumption
- |apply le_plus_n_r
- ]
- |apply le_S_S_to_le.assumption
- ]
+ |#m #Hind #k (cases k) normalize //
+ #c #lec cases (le_to_or_lt_eq … (le_S_S_to_le …lec))
+ [#ltcm
+ 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 //]
+ >commutative_plus >(distributive_times_plus ? (S c))
+ @divides_plus
+ [>associative_times >(commutative_times (S c))
+ <associative_times @divides_times //
+ |<(fact_minus …ltcm) in ⊢ (?(??%)?);
+ <associative_times @divides_times //
+ >commutative_times @Hmc
]
- |rewrite > H2.
- rewrite > bc_n_n.
- apply le_n
+ |#eqcm >eqcm <minus_n_n <times_n_1 //
]
]
qed.
-
-theorem exp_plus_sigma_p:\forall a,b,n.
-exp (a+b) n = sigma_p (S n) (\lambda k.true)
- (\lambda k.(bc n k)*(exp a (n-k))*(exp b k)).
-intros.elim n
- [simplify.reflexivity
- |simplify in ⊢ (? ? % ?).
- rewrite > true_to_sigma_p_Sn
- [rewrite > H;rewrite > sym_times in ⊢ (? ? % ?);
- rewrite > distr_times_plus in ⊢ (? ? % ?);
- rewrite < minus_n_n in ⊢ (? ? ? (? (? (? ? (? ? %)) ?) ?));
- rewrite > sym_times in ⊢ (? ? (? % ?) ?);
- rewrite > sym_times in ⊢ (? ? (? ? %) ?);
- rewrite > distributive_times_plus_sigma_p in ⊢ (? ? (? % ?) ?);
- rewrite > distributive_times_plus_sigma_p in ⊢ (? ? (? ? %) ?);
- rewrite > true_to_sigma_p_Sn in ⊢ (? ? (? ? %) ?)
- [rewrite < assoc_plus;rewrite < sym_plus in ⊢ (? ? (? % ?) ?);
- rewrite > assoc_plus;
- apply eq_f2
- [rewrite > sym_times;rewrite > assoc_times;autobatch paramodulation
- |rewrite > (sigma_p_gi ? ? O);
- [rewrite < (eq_sigma_p_gh ? S pred n1 (S n1) (λx:nat.true)) in ⊢ (? ? (? (? ? %) ?) ?)
- [rewrite > (sigma_p_gi ? ? O) in ⊢ (? ? ? %);
- [rewrite > assoc_plus;apply eq_f2
- [autobatch paramodulation;
- |rewrite < sigma_p_plus_1;
- rewrite < (eq_sigma_p_gh ? S pred n1 (S n1) (λx:nat.true)) in \vdash (? ? ? %);
- [apply eq_sigma_p
- [intros;reflexivity
- |intros;rewrite > sym_times;rewrite > assoc_times;
- rewrite > sym_times in ⊢ (? ? (? (? ? %) ?) ?);
- rewrite > assoc_times;rewrite < assoc_times in ⊢ (? ? (? (? ? %) ?) ?);
- rewrite > sym_times in ⊢ (? ? (? (? ? (? % ?)) ?) ?);
- change in ⊢ (? ? (? (? ? (? % ?)) ?) ?) with (exp a (S (n1 - S x)));
- rewrite < (minus_Sn_m ? ? H1);rewrite > minus_S_S;
- rewrite > sym_times in \vdash (? ? (? ? %) ?);
- rewrite > assoc_times;
- rewrite > sym_times in ⊢ (? ? (? ? (? ? %)) ?);
- change in \vdash (? ? (? ? (? ? %)) ?) with (exp b (S x));
- rewrite > assoc_times in \vdash (? ? (? ? %) ?);
- rewrite > sym_times in \vdash (? ? (? % ?) ?);
- rewrite > sym_times in \vdash (? ? (? ? %) ?);
- rewrite > assoc_times in \vdash (? ? ? %);
- rewrite > sym_times in \vdash (? ? ? %);
- rewrite < distr_times_plus;
- rewrite > sym_plus;rewrite < bc1;
- [reflexivity|assumption]]
- |intros;simplify;reflexivity
- |intros;simplify;reflexivity
- |intros;apply le_S_S;assumption
- |intros;reflexivity
- |intros 2;elim j
- [simplify in H2;destruct H2
- |simplify;reflexivity]
- |intro;elim j
- [simplify in H2;destruct H2
- |simplify;apply le_S_S_to_le;assumption]]]
- |apply le_S_S;apply le_O_n
- |reflexivity]
- |intros;simplify;reflexivity
- |intros;simplify;reflexivity
- |intros;apply le_S_S;assumption
- |intros;reflexivity
- |intros 2;elim j
- [simplify in H2;destruct H2
- |simplify;reflexivity]
- |intro;elim j
- [simplify in H2;destruct H2
- |simplify;apply le_S_S_to_le;assumption]]
- |apply le_S_S;apply le_O_n
- |reflexivity]]
- |reflexivity]
- |reflexivity]]
+
+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 ⊢ (???(?%?));
+ [|>(times_n_O 0) @lt_times // | //]
+>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 ⊢ (???(??(??%)));
+>fact_minus // <plus_div
+ [<distributive_times_plus
+ >commutative_plus in ⊢ (???%); <plus_n_Sm <plus_minus_m_m [|/2/] @refl
+ |<fact_minus // <associative_times @divides_times // @(bc2 n (S k)) //
+ |>associative_times >(commutative_times (S k))
+ <associative_times @divides_times // @bc2 /2/
+ |>(times_n_O 0) @lt_times [@(le_1_fact (S k)) | //]
+ ]
qed.
-theorem exp_S_sigma_p:\forall a,n.
-exp (S a) n = sigma_p (S n) (\lambda k.true) (\lambda k.(bc n k)*(exp a (n-k))).
-intros.
-rewrite > plus_n_SO.
-rewrite > exp_plus_sigma_p.
-apply eq_sigma_p;intros
- [reflexivity
- |rewrite < exp_SO_n.
- rewrite < times_n_SO.
- reflexivity
+theorem lt_O_bc: ∀n,m. m ≤ n → O < bc n m.
+#n (elim n)
+ [#m #lemO @(le_n_O_elim ? lemO) //
+ |-n #n #Hind #m (cases m) //
+ #m #lemn cases(le_to_or_lt_eq … (le_S_S_to_le …lemn)) //
+ #ltmn >bc1 // >(plus_O_n 0) @lt_plus @Hind /2/
]
+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 (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
+ [<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 //
+ #i #ltin #_ <associative_times >(commutative_times b)
+ >(associative_times ?? b) <(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 //
+ ]
+ ]
+qed.
+
+theorem exp_S_sigma_p:∀a,n.
+(S a)^n = Σ_{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 < assoc_plus.
rewrite < sym_plus.
reflexivity.
-qed.
-
+qed. *)
+*)
\ No newline at end of file
projects / helm.git / blobdiff