(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/iteration2".
-
include "nat/primes.ma".
include "nat/ord.ma".
include "nat/generic_iter_p.ma".
include "nat/count.ma".(*necessary just to use bool_to_nat and bool_to_nat_andb*)
-
(* sigma_p on nautral numbers is a specialization of sigma_p_gen *)
definition sigma_p: nat \to (nat \to bool) \to (nat \to nat) \to nat \def
\lambda n, p, g. (iter_p_gen n p nat g O plus).
assumption.
qed.
+theorem or_false_to_eq_sigma_p:
+\forall n,m:nat.\forall p:nat \to bool.
+\forall g: nat \to nat.
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = O)
+\to sigma_p m p g = sigma_p n p g.
+intros.
+unfold sigma_p.
+apply or_false_eq_baseA_to_eq_iter_p_gen
+ [intros.reflexivity
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem bool_to_nat_to_eq_sigma_p:
+\forall n:nat.\forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.
+(\forall i:nat.
+bool_to_nat (p1 i)*(g1 i) = bool_to_nat (p2 i)*(g2 i))
+\to sigma_p n p1 g1 = sigma_p n p2 g2.
+intros.elim n
+ [reflexivity
+ |generalize in match (H n1).
+ apply (bool_elim ? (p1 n1));intro
+ [apply (bool_elim ? (p2 n1));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [apply eq_f2
+ [simplify in H4.
+ rewrite > plus_n_O.
+ rewrite > plus_n_O in ⊢ (? ? ? %).
+ assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn
+ [change in ⊢ (? ? ? %) with (O + sigma_p n1 p2 g2).
+ apply eq_f2
+ [simplify in H4.
+ rewrite > plus_n_O.
+ assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |apply (bool_elim ? (p2 n1));intros
+ [rewrite > false_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [change in ⊢ (? ? % ?) with (O + sigma_p n1 p1 g1).
+ apply eq_f2
+ [simplify in H4.
+ rewrite < plus_n_O in H4.
+ assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn
+ [assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
theorem sigma_p2 :
\forall n,m:nat.
\forall p1,p2:nat \to bool.
]
qed.
+theorem eq_sigma_p_pred:
+\forall n,p,g. p O = true \to
+sigma_p (S n) (\lambda i.p (pred i)) (\lambda i.g(pred i)) =
+plus (sigma_p n p g) (g O).
+intros.
+unfold sigma_p.
+apply eq_iter_p_gen_pred
+ [assumption
+ |apply symmetricIntPlus
+ |apply associative_plus
+ |intros.apply sym_eq.apply plus_n_O
+ ]
+qed.
+
+(* monotonicity *)
+theorem le_sigma_p:
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to
+sigma_p n p g1 \le sigma_p n p g2.
+intros.
+elim n in H ⊢ %
+ [apply le_n.
+ |apply (bool_elim ? (p n1));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply le_plus
+ [apply H1[apply le_n|assumption]
+ |apply H.
+ intros.
+ apply H1[apply le_S.assumption|assumption]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H.
+ intros.
+ apply H1[apply le_S.assumption|assumption]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+(* a slightly more general result *)
+theorem le_sigma_p1:
+\forall n:nat. \forall p1,p2:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to
+bool_to_nat (p1 i)*(g1 i) \le bool_to_nat (p2 i)*g2 i) \to
+sigma_p n p1 g1 \le sigma_p n p2 g2.
+intros.
+elim n in H ⊢ %
+ [apply le_n.
+ |apply (bool_elim ? (p1 n1));intros
+ [apply (bool_elim ? (p2 n1));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply le_plus
+ [lapply (H1 n1) as H5
+ [rewrite > H2 in H5.
+ rewrite > H3 in H5.
+ simplify in H5.
+ rewrite < plus_n_O in H5.
+ rewrite < plus_n_O in H5.
+ assumption
+ |apply le_S_S.apply le_n
+ ]
+ |apply H.intros.
+ apply H1.apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [change in ⊢ (? ? %) with (O + sigma_p n1 p2 g2).
+ apply le_plus
+ [lapply (H1 n1) as H5
+ [rewrite > H2 in H5.
+ rewrite > H3 in H5.
+ simplify in H5.
+ rewrite < plus_n_O in H5.
+ assumption
+ |apply le_S_S.apply le_n
+ ]
+ |apply H.intros.
+ apply H1.apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |apply (bool_elim ? (p2 n1));intros
+ [rewrite > false_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [change in ⊢ (? % ?) with (O + sigma_p n1 p1 g1).
+ apply le_plus
+ [lapply (H1 n1) as H5
+ [rewrite > H2 in H5.
+ rewrite > H3 in H5.
+ simplify in H5.
+ rewrite < plus_n_O in H5.
+ assumption
+ |apply le_S_S.apply le_n
+ ]
+ |apply H.intros.
+ apply H1.apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H.intros.
+ apply H1.apply le_S.assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_sigma_p:
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to
+(\exists i. i < n \and (p i = true) \and (g1 i < g2 i)) \to
+sigma_p n p g1 < sigma_p n p g2.
+intros 4.
+elim n
+ [elim H1.clear H1.
+ elim H2.clear H2.
+ elim H1.clear H1.
+ apply False_ind.
+ apply (lt_to_not_le ? ? H2).
+ apply le_O_n
+ |apply (bool_elim ? (p n1));intros
+ [apply (bool_elim ? (leb (S (g1 n1)) (g2 n1)));intros
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [change with
+ (S (g1 n1)+sigma_p n1 p g1 \le g2 n1+sigma_p n1 p g2).
+ apply le_plus
+ [apply leb_true_to_le.assumption
+ |apply le_sigma_p.intros.
+ apply H1
+ [apply lt_to_le.apply le_S_S.assumption
+ |assumption
+ ]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+ [unfold lt.
+ rewrite > plus_n_Sm.
+ apply le_plus
+ [apply H1
+ [apply le_n
+ |assumption
+ ]
+ |apply H
+ [intros.apply H1
+ [apply lt_to_le.apply le_S_S.assumption
+ |assumption
+ ]
+ |elim H2.clear H2.
+ elim H5.clear H5.
+ elim H2.clear H2.
+ apply (ex_intro ? ? a).
+ split
+ [split
+ [elim (le_to_or_lt_eq a n1)
+ [assumption
+ |absurd (g1 a < g2 a)
+ [assumption
+ |apply leb_false_to_not_le.
+ rewrite > H2.
+ assumption
+ ]
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+ [apply H
+ [intros.apply H1
+ [apply lt_to_le.apply le_S_S.assumption
+ |assumption
+ ]
+ |elim H2.clear H2.
+ elim H4.clear H4.
+ elim H2.clear H2.
+ apply (ex_intro ? ? a).
+ split
+ [split
+ [elim (le_to_or_lt_eq a n1)
+ [assumption
+ |apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H6.
+ rewrite < H3.
+ rewrite < H2.
+ reflexivity
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
theorem sigma_p_divides:
\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to
\forall g: nat \to nat.
qed.
theorem sigma_p_plus_1: \forall n:nat. \forall f,g:nat \to nat.
-sigma_p n (\lambda b:nat. true) (\lambda a:nat.(f a) + (g a)) =
-sigma_p n (\lambda b:nat. true) f + sigma_p n (\lambda b:nat. true) g.
+\forall p.
+sigma_p n p (\lambda a:nat.(f a) + (g a)) =
+sigma_p n p f + sigma_p n p g.
intros.
elim n
[ simplify.
reflexivity
-| rewrite > true_to_sigma_p_Sn
- [ rewrite > (true_to_sigma_p_Sn n1 (\lambda c:nat.true) f)
- [ rewrite > (true_to_sigma_p_Sn n1 (\lambda c:nat.true) g)
- [ rewrite > assoc_plus in \vdash (? ? ? %).
- rewrite < assoc_plus in \vdash (? ? ? (? ? %)).
- rewrite < sym_plus in \vdash (? ? ? (? ? (? % ?))).
- rewrite > assoc_plus in \vdash (? ? ? (? ? %)).
- rewrite < assoc_plus in \vdash (? ? ? %).
- apply eq_f.
- assumption
- | reflexivity
- ]
- | reflexivity
- ]
- | reflexivity
- ]
-]
+| apply (bool_elim ? (p n1)); intro;
+ [ rewrite > true_to_sigma_p_Sn
+ [ rewrite > (true_to_sigma_p_Sn n1 p f)
+ [ rewrite > (true_to_sigma_p_Sn n1 p g)
+ [ rewrite > assoc_plus in \vdash (? ? ? %).
+ rewrite < assoc_plus in \vdash (? ? ? (? ? %)).
+ rewrite < sym_plus in \vdash (? ? ? (? ? (? % ?))).
+ rewrite > assoc_plus in \vdash (? ? ? (? ? %)).
+ rewrite < assoc_plus in \vdash (? ? ? %).
+ apply eq_f.
+ assumption]]]
+ assumption
+ | rewrite > false_to_sigma_p_Sn
+ [ rewrite > (false_to_sigma_p_Sn n1 p f)
+ [ rewrite > (false_to_sigma_p_Sn n1 p g)
+ [assumption]]]
+ assumption
+]]
qed.
-
theorem eq_sigma_p_sigma_p_times1 : \forall n,m:nat.\forall f:nat \to nat.
sigma_p (n*m) (\lambda x:nat.true) f =
sigma_p m (\lambda x:nat.true)
apply eq_sigma_p_sigma_p_times1.
qed.
-
theorem sigma_p_times:\forall n,m:nat.
\forall f,f1,f2:nat \to bool.
\forall g:nat \to nat \to nat.
|assumption
]
qed.
-
\ No newline at end of file
+
+
+theorem sigma_p2_eq:
+\forall g: nat \to nat \to nat.
+\forall h11,h12,h21,h22: nat \to nat \to nat.
+\forall n1,m1,n2,m2.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to
+p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true
+\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j
+\land h11 i j < n1 \land h12 i j < m1) \to
+(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to
+p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true
+\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j
+\land (h21 i j) < n2 \land (h22 i j) < m2) \to
+sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) =
+sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?).
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+
+apply(iter_p_gen_2_eq nat O plus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intro.
+ rewrite < (plus_n_O).
+ reflexivity
+| assumption
+| assumption
+]
+qed.
+
+theorem sigma_p_sigma_p:
+\forall g: nat \to nat \to nat.
+\forall n,m.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall x,y. x < n \to y < m \to
+ (p11 x \land p12 x y) = (p21 y \land p22 y x)) \to
+sigma_p n p11 (\lambda x:nat.sigma_p m (p12 x) (\lambda y. g x y)) =
+sigma_p m p21 (\lambda y:nat.sigma_p n (p22 y) (\lambda x. g x y)).
+intros.
+unfold sigma_p.unfold sigma_p.
+apply (iter_p_gen_iter_p_gen ? ? ? sym_plus assoc_plus)
+ [intros.apply sym_eq.apply plus_n_O.
+ |assumption
+ ]
+qed.
\ No newline at end of file
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