\big[op,nil]_{i∈[0,n[ |p i}(f i) = \big[op,nil]_{i < n|p i}(f i).
#n #p #B #nil #op #f <minus_n_O @same_bigop //
qed.
-
+
+theorem bigop_I_gen: ∀a,b,p,B.∀nil.∀op:Aop B nil.∀f:nat→B. a ≤b →
+\big[op,nil]_{i∈[a,b[ |p i}(f i) = \big[op,nil]_{i < b|leb a i ∧ p i}(f i).
+#a #b elim b // -b #b #Hind #p #B #nil #op #f #lea
+cut (∀a,b. a ≤ b → S b - a = S (b -a))
+ [#a #b cases a // #a1 #lta1 normalize >eq_minus_S_pred >S_pred
+ /2 by lt_plus_to_minus_r/] #Hcut
+cases (le_to_or_lt_eq … lea) #Ha
+ [cases (true_or_false (p b)) #Hcase
+ [>bigop_Strue [2: >Hcase >(le_to_leb_true a b) // @le_S_S_to_le @Ha]
+ >(Hcut … (le_S_S_to_le … Ha))
+ >bigop_Strue
+ [@eq_f2
+ [@eq_f <plus_minus_m_m [//|@le_S_S_to_le //] @Hind
+ |@Hind @le_S_S_to_le //
+ ]
+ |<plus_minus_m_m // @le_S_S_to_le //
+ ]
+ |>bigop_Sfalse [2: >Hcase cases (leb a b)//]
+ >(Hcut … (le_S_S_to_le … Ha)) >bigop_Sfalse
+ [@Hind @le_S_S_to_le // | <plus_minus_m_m // @le_S_S_to_le //]
+ ]
+ |<Ha <minus_n_n whd in ⊢ (??%?); <(bigop_false a B nil op f) in ⊢ (??%?);
+ @same_bigop // #i #ltia >(not_le_to_leb_false a i) // @lt_to_not_le //
+ ]
+qed.
+
theorem bigop_sumI: ∀a,b,c,p,B.∀nil.∀op:Aop B nil.∀f:nat→B.
a ≤ b → b ≤ c →
\big[op,nil]_{i∈[a,c[ |p i}(f i) =
>div_plus_times /2 by monotonic_lt_minus_l/
>Hp1 >(mod_plus_times …) /2 by refl, monotonic_lt_minus_l, eq_f/
|>bigop_Sfalse // >Hind >(pad_bigop (S n*k2)) // @same_bigop
- #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2/
- #eqi >eqi in ⊢ (???%); >div_plus_times /2/
+ #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2 by plus_minus/
+ #eqi >eqi in ⊢ (???%); >div_plus_times /2 by refl, monotonic_lt_minus_l, trans_eq/
]
qed.
]
qed.
-(* lemma div_mod_exchange: ∀i,n,m. i < n*m → i\n = i mod m. *)
-
(* commutation *)
theorem bigop_commute: ∀n,m,p11,p12,p21,p22,B.∀nil.∀op:ACop B nil.∀f.
0 < n → 0 < m →
V_______________________________________________________________ *)
include "arithmetics/primes.ma".
-include "arithmetics/bigops.ma".
+include "arithmetics/sigma_pi.ma".
(* binomial coefficient *)
definition bc ≝ λn,k. n!/(k!*(n-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 ⊢ (??%?); >commutative_times
->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
+>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 >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.
+
+(************ 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)
- |change in ⊢ (? ? %) with (exp 2 (S(2*m))).
- change in ⊢ (? ? (? % ?)) with (1+1).
- rewrite > exp_plus_sigma_p.
- apply (le_to_lt_to_lt ? (sigma_p (S (S (2*m))) (λk:nat.orb (eqb k m) (eqb k (S m)))
- (λk:nat.bc (S (2*m)) k*(1)\sup(S (2*m)-k)*(1)\sup(k))))
- [rewrite > (sigma_p_gi ? ? m)
- [rewrite > (sigma_p_gi ? ? (S m))
- [rewrite > (false_to_eq_sigma_p O (S(S(2*m))))
- [simplify in ⊢ (? ? (? ? (? ? %))).
- simplify in ⊢ (? % ?).
- rewrite < exp_SO_n.rewrite < exp_SO_n.
- rewrite < exp_SO_n.rewrite < exp_SO_n.
- rewrite < times_n_SO.rewrite < times_n_SO.
- rewrite < times_n_SO.rewrite < times_n_SO.
- apply le_plus
- [unfold M.apply le_n
- |apply le_plus_l.unfold M.
- change in \vdash (? ? %) with (fact (S(2*m))/(fact (S m)*(fact ((2*m)-m)))).
- simplify in \vdash (? ? (? ? (? ? (? (? % ?))))).
- rewrite < plus_n_O.rewrite < minus_plus_m_m.
- rewrite < sym_times in \vdash (? ? (? ? %)).
- change in \vdash (? % ?) with (fact (S(2*m))/(fact m*(fact (S(2*m)-m)))).
- simplify in \vdash (? (? ? (? ? (? (? (? %) ?)))) ?).
- rewrite < plus_n_O.change in \vdash (? (? ? (? ? (? (? % ?)))) ?) with (S m + m).
- rewrite < minus_plus_m_m.
- apply le_n
- ]
- |apply le_O_n
- |intros.
- elim (eqb i m);elim (eqb i (S m));reflexivity
- ]
- |apply le_S_S.apply le_S_S.
- apply le_times_n.
- apply le_n_Sn
- |rewrite > (eq_to_eqb_true ? ? (refl_eq ? (S m))).
- rewrite > (not_eq_to_eqb_false (S m) m)
- [reflexivity
- |intro.apply (not_eq_n_Sn m).
- apply sym_eq.assumption
+#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
]
- |apply le_S.apply le_S_S.
- apply le_times_n.
- apply le_n_Sn
- |rewrite > (eq_to_eqb_true ? ? (refl_eq ? (S m))).
- reflexivity
+ |>(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 //
]
- |rewrite > (bool_to_nat_to_eq_sigma_p (S(S(2*m))) ? (\lambda k.true) ?
- (\lambda k.bool_to_nat (eqb k m\lor eqb k (S m))*(bc (S (2*m)) k*(1)\sup(S (2*m)-k)*(1)\sup(k))))
- in \vdash (? % ?)
- [apply lt_sigma_p
- [intros.elim (eqb i m\lor eqb i (S m))
- [rewrite > sym_times.rewrite < times_n_SO.apply le_n
- |apply le_O_n
- ]
- |apply (ex_intro ? ? O).
- split
- [split[apply lt_O_S|reflexivity]
- |rewrite > (not_eq_to_eqb_false ? ? (not_eq_O_S m)).
- rewrite > (not_eq_to_eqb_false ? ? (lt_to_not_eq ? ? H)).
- simplify in \vdash (? % ?).
- rewrite < exp_SO_n.rewrite < exp_SO_n.
- rewrite > bc_n_O.simplify.
- apply le_n
- ]
+ |>(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
]
- |intros.rewrite > sym_times in \vdash (? ? ? %).
- rewrite < times_n_SO.
- reflexivity
]
]
]
qed.
-
(*
theorem exp_Sn_SSO: \forall n. exp (S n) 2 = S((exp n 2) + 2*n).
intros.simplify.
rewrite < sym_plus.
reflexivity.
qed. *)
-*)
\ No newline at end of file
(* monotonicity; these roperty should be expressed at a more
genral level *)
+theorem le_sigma:
+∀n:nat. ∀p1,p2:nat → bool. ∀g1,g2:nat → nat.
+(∀i. i < n → p1 i = true → p2 i = true ) →
+(∀i. i < n → p1 i = true → g1 i ≤ g2 i ) →
+ ∑_{i < n | p1 i} (g1 i) ≤ ∑_{i < n | p2 i} (g2 i).
+#n #p1 #p2 #g1 #g2 elim n
+ [#_ #_ @le_n
+ |#n1 #Hind #H1 #H2
+ lapply (Hind ??)
+ [#j #ltin1 #Hgj @(H2 … Hgj) @le_S //
+ |#j #ltin1 #Hp1j @(H1 … Hp1j) @le_S //
+ ] -Hind #Hind
+ cases (true_or_false (p2 n1)) #Hp2
+ [>bigop_Strue in ⊢ (??%); [2:@Hp2]
+ cases (true_or_false (p1 n1)) #Hp1
+ [>bigop_Strue [2:@Hp1] @(le_plus … Hind) @H2 //
+ |>bigop_Sfalse [2:@Hp1] @le_plus_a //
+ ]
+ |cut (p1 n1 = false)
+ [cases (true_or_false (p1 n1)) #Hp1
+ [>(H1 … (le_n ?) Hp1) in Hp2; #H destruct (H) | //]
+ ] #Hp1
+ >bigop_Sfalse [2:@Hp1] >bigop_Sfalse [2:@Hp2] //
+ ]
+ ]
+qed.
+
+theorem lt_sigma_p:
+∀n:nat. ∀p1,p2:nat → bool. ∀g1,g2:nat → nat.
+(∀i. i < n → p1 i = true → p2 i = true) →
+(∀i. i < n → p1 i = true → g1 i ≤ g2 i ) →
+(∃i. i < n ∧ ((p1 i = true) ∧ (g1 i < g2 i)
+ ∨ (p1 i = false ∧ (p2 i = true) ∧ (0 < g2 i)))) →
+ ∑_{i < n | p1 i} (g1 i) < ∑_{i < n | p2 i} (g2 i).
+#n #p1 #p2 #g1 #g2 #H1 #H2 * #k * #ltk *
+ [* #p1k #gk
+ lapply (H1 k ltk p1k) #p2k
+ >(bigop_diff p1 ?? plusAC … ltk p1k)
+ >(bigop_diff p2 ?? plusAC … ltk p2k) whd
+ cut (∀a,b. S a + b = S(a +b)) [//] #Hplus <Hplus @le_plus
+ [@gk
+ |@le_sigma
+ [#i #ltin #H @true_to_andb_true
+ [@(andb_true_l … H) | @(H1 i ltin) @(andb_true_r … H)]
+ |#i #ltin #H @(H2 i ltin) @(andb_true_r … H)
+ ]
+ ]
+ |* * #p1k #p2k #posg2
+ >(bigop_diff p2 ?? plusAC … ltk p2k) whd
+ cut (∀a. S 0 + a = S a) [//] #H0 <(H0 (bigop n ?????)) @le_plus
+ [@posg2
+ |@le_sigma
+ [#i #ltin #H @true_to_andb_true
+ [cases (true_or_false (eqb k i)) #Hc >Hc
+ [normalize <H <p1k >(eqb_true_to_eq … Hc) //|//]
+ |@(H1 i ltin) @H]
+ |#i #ltin #H @(H2 i ltin) @H
+ ]
+ ]
+qed.
+
theorem le_pi:
∀n.∀p:nat → bool.∀g1,g2:nat → nat.
(∀i.i<n → p i = true → g1 i ≤ g2 i ) →
]
]
qed.
-
+
theorem exp_sigma: ∀n,a,p.
∏_{i < n | p i} a = exp a (∑_{i < n | p i} 1).
#n #a #p elim n // #n1 cases (true_or_false (p n1)) #Hcase
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