(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/iteration2.ma".
+set "baseuri" "cic:/matita/nat/iteration2".
include "nat/primes.ma".
include "nat/ord.ma".
+include "nat/generic_sigma_p.ma".
-let rec sigma_p n p (g:nat \to nat) \def
- match n with
- [ O \Rightarrow O
- | (S k) \Rightarrow
- match p k with
- [true \Rightarrow (g k)+(sigma_p k p g)
- |false \Rightarrow sigma_p k p g]
- ].
+
+(* 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. (sigma_p_gen n p nat g O plus).
+
+theorem symmetricIntPlus: symmetric nat plus.
+change with (\forall a,b:nat. (plus a b) = (plus b a)).
+intros.
+rewrite > sym_plus.
+reflexivity.
+qed.
+(*the following theorems on sigma_p in N are obtained by the more general ones
+ * in sigma_p_gen.ma
+ *)
theorem true_to_sigma_p_Sn:
\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
p n = true \to sigma_p (S n) p g =
(g n)+(sigma_p n p g).
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply true_to_sigma_p_Sn_gen.
+assumption.
qed.
theorem false_to_sigma_p_Sn:
\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
p n = false \to sigma_p (S n) p g = sigma_p n p g.
-intros.simplify.
-rewrite > H.reflexivity.
+intros.
+unfold sigma_p.
+apply false_to_sigma_p_Sn_gen.
+assumption.
+
qed.
theorem eq_sigma_p: \forall p1,p2:nat \to bool.
(\forall x. x < n \to p1 x = p2 x) \to
(\forall x. x < n \to g1 x = g2 x) \to
sigma_p n p1 g1 = sigma_p n p2 g2.
-intros 5.elim n
- [reflexivity
- |apply (bool_elim ? (p1 n1))
- [intro.
- rewrite > (true_to_sigma_p_Sn ? ? ? H3).
- rewrite > true_to_sigma_p_Sn
- [apply eq_f2
- [apply H2.apply le_n.
- |apply H
- [intros.apply H1.apply le_S.assumption
- |intros.apply H2.apply le_S.assumption
- ]
- ]
- |rewrite < H3.apply sym_eq.apply H1.apply le_n
- ]
- |intro.
- rewrite > (false_to_sigma_p_Sn ? ? ? H3).
- rewrite > false_to_sigma_p_Sn
- [apply H
- [intros.apply H1.apply le_S.assumption
- |intros.apply H2.apply le_S.assumption
- ]
- |rewrite < H3.apply sym_eq.apply H1.apply le_n
- ]
- ]
- ]
+intros.
+unfold sigma_p.
+apply eq_sigma_p_gen;
+ assumption.
+qed.
+
+theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.\forall n.
+(\forall x. x < n \to p1 x = p2 x) \to
+(\forall x. x < n \to p1 x = true \to g1 x = g2 x) \to
+sigma_p n p1 g1 = sigma_p n p2 g2.
+intros.
+unfold sigma_p.
+apply eq_sigma_p1_gen;
+ assumption.
qed.
theorem sigma_p_false:
\forall g: nat \to nat.\forall n.sigma_p n (\lambda x.false) g = O.
intros.
-elim n[reflexivity|simplify.assumption]
+unfold sigma_p.
+apply sigma_p_false_gen.
qed.
theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
sigma_p (k+n) p g
= sigma_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) + sigma_p n p g.
intros.
-elim k
- [reflexivity
- |apply (bool_elim ? (p (n1+n)))
- [intro.
- simplify in \vdash (? ? (? % ? ?) ?).
- rewrite > (true_to_sigma_p_Sn ? ? ? H1).
- rewrite > (true_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
- rewrite > assoc_plus.
- rewrite < H.reflexivity
- |intro.
- simplify in \vdash (? ? (? % ? ?) ?).
- rewrite > (false_to_sigma_p_Sn ? ? ? H1).
- rewrite > (false_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
- assumption.
- ]
- ]
+unfold sigma_p.
+apply (sigma_p_plusA_gen nat n k p g O plus)
+[ apply symmetricIntPlus.
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+| apply associative_plus
+]
qed.
theorem false_to_eq_sigma_p: \forall n,m:nat.n \le m \to
\forall p:nat \to bool.
\forall g: nat \to nat. (\forall i:nat. n \le i \to i < m \to
p i = false) \to sigma_p m p g = sigma_p n p g.
-intros 5.
-elim H
- [reflexivity
- |simplify.
- rewrite > H3
- [simplify.
- apply H2.
- intros.
- apply H3[apply H4|apply le_S.assumption]
- |assumption
- |apply le_n
- ]
- ]
+intros.
+unfold sigma_p.
+apply (false_to_eq_sigma_p_gen);
+ assumption.
qed.
theorem sigma_p2 :
sigma_p n p1
(\lambda x.sigma_p m p2 (g x)).
intros.
-elim n
- [simplify.reflexivity
- |apply (bool_elim ? (p1 n1))
- [intro.
- rewrite > (true_to_sigma_p_Sn ? ? ? H1).
- simplify in \vdash (? ? (? % ? ?) ?);
- rewrite > sigma_p_plus.
- rewrite < H.
- apply eq_f2
- [apply eq_sigma_p
- [intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H2).
- rewrite > (mod_plus_times ? ? ? H2).
- rewrite > H1.
- simplify.reflexivity
- |intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H2).
- rewrite > (mod_plus_times ? ? ? H2).
- rewrite > H1.
- simplify.reflexivity.
- ]
- |reflexivity
- ]
- |intro.
- rewrite > (false_to_sigma_p_Sn ? ? ? H1).
- simplify in \vdash (? ? (? % ? ?) ?);
- rewrite > sigma_p_plus.
- rewrite > H.
- apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m p2 (g x)))))
- [apply eq_f2
- [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
- [apply sigma_p_false
- |intros.
- rewrite > sym_plus.
- rewrite > (div_plus_times ? ? ? H2).
- rewrite > (mod_plus_times ? ? ? H2).
- rewrite > H1.
- simplify.reflexivity
- |intros.reflexivity.
- ]
- |reflexivity
- ]
- |reflexivity
- ]
- ]
- ]
+unfold sigma_p.
+apply (sigma_p2_gen n m p1 p2 nat g O plus)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+]
+qed.
+
+theorem sigma_p2' :
+\forall n,m:nat.
+\forall p1:nat \to bool.
+\forall p2:nat \to nat \to bool.
+\forall g: nat \to nat \to nat.
+sigma_p (n*m)
+ (\lambda x.andb (p1 (div x m)) (p2 (div x m) (mod x m)))
+ (\lambda x.g (div x m) (mod x m)) =
+sigma_p n p1
+ (\lambda x.sigma_p m (p2 x) (g x)).
+intros.
+unfold sigma_p.
+apply (sigma_p2_gen' n m p1 p2 nat g O plus)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+]
qed.
lemma sigma_p_gi: \forall g: nat \to nat.
\forall n,i.\forall p:nat \to bool.i < n \to p i = true \to
sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
-intros 2.
-elim n
- [apply False_ind.
- apply (not_le_Sn_O i).
- assumption
- |apply (bool_elim ? (p n1));intro
- [elim (le_to_or_lt_eq i n1)
- [rewrite > true_to_sigma_p_Sn
- [rewrite > true_to_sigma_p_Sn
- [rewrite < assoc_plus.
- rewrite < sym_plus in \vdash (? ? ? (? % ?)).
- rewrite > assoc_plus.
- apply eq_f2
- [reflexivity
- |apply H[assumption|assumption]
- ]
- |rewrite > H3.simplify.
- change with (notb (eqb n1 i) = notb false).
- apply eq_f.
- apply not_eq_to_eqb_false.
- unfold Not.intro.
- apply (lt_to_not_eq ? ? H4).
- apply sym_eq.assumption
- ]
- |assumption
- ]
- |rewrite > true_to_sigma_p_Sn
- [rewrite > H4.
- apply eq_f2
- [reflexivity
- |rewrite > false_to_sigma_p_Sn
- [apply eq_sigma_p
- [intros.
- elim (p x)
- [simplify.
- change with (notb false = notb (eqb x n1)).
- apply eq_f.
- apply sym_eq.
- apply not_eq_to_eqb_false.
- apply (lt_to_not_eq ? ? H5)
- |reflexivity
- ]
- |intros.reflexivity
- ]
- |rewrite > H3.
- rewrite > (eq_to_eqb_true ? ? (refl_eq ? n1)).
- reflexivity
- ]
- ]
- |assumption
- ]
- |apply le_S_S_to_le.assumption
- ]
- |rewrite > false_to_sigma_p_Sn
- [elim (le_to_or_lt_eq i n1)
- [rewrite > false_to_sigma_p_Sn
- [apply H[assumption|assumption]
- |rewrite > H3.reflexivity
- ]
- |apply False_ind.
- apply not_eq_true_false.
- rewrite < H2.
- rewrite > H4.
- assumption
- |apply le_S_S_to_le.assumption
- ]
- |assumption
- ]
- ]
- ]
+intros.
+unfold sigma_p.
+apply (sigma_p_gi_gen)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+| assumption
+| assumption
+]
qed.
theorem eq_sigma_p_gh:
(\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to
(\forall j. j < n1 \to p2 j = true \to h1 j < n) \to
sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
-intros 4.
-elim n
- [generalize in match H5.
- elim n1
- [reflexivity
- |apply (bool_elim ? (p2 n2));intro
- [apply False_ind.
- apply (not_le_Sn_O (h1 n2)).
- apply H7
- [apply le_n|assumption]
- |rewrite > false_to_sigma_p_Sn
- [apply H6.
- intros.
- apply H7[apply le_S.apply H9|assumption]
- |assumption
- ]
- ]
- ]
- |apply (bool_elim ? (p1 n1));intro
- [rewrite > true_to_sigma_p_Sn
- [rewrite > (sigma_p_gi g n2 (h n1))
- [apply eq_f2
- [reflexivity
- |apply H
- [intros.
- rewrite > H1
- [simplify.
- change with ((\not eqb (h i) (h n1))= \not false).
- apply eq_f.
- apply not_eq_to_eqb_false.
- unfold Not.intro.
- apply (lt_to_not_eq ? ? H8).
- rewrite < H2
- [rewrite < (H2 n1)
- [apply eq_f.assumption|apply le_n|assumption]
- |apply le_S.assumption
- |assumption
- ]
- |apply le_S.assumption
- |assumption
- ]
- |intros.
- apply H2[apply le_S.assumption|assumption]
- |intros.
- apply H3[apply le_S.assumption|assumption]
- |intros.
- apply H4
- [assumption
- |generalize in match H9.
- elim (p2 j)
- [reflexivity|assumption]
- ]
- |intros.
- apply H5
- [assumption
- |generalize in match H9.
- elim (p2 j)
- [reflexivity|assumption]
- ]
- |intros.
- elim (le_to_or_lt_eq (h1 j) n1)
- [assumption
- |generalize in match H9.
- elim (p2 j)
- [simplify in H11.
- absurd (j = (h n1))
- [rewrite < H10.
- rewrite > H5
- [reflexivity|assumption|autobatch]
- |apply eqb_false_to_not_eq.
- generalize in match H11.
- elim (eqb j (h n1))
- [apply sym_eq.assumption|reflexivity]
- ]
- |simplify in H11.
- apply False_ind.
- apply not_eq_true_false.
- apply sym_eq.assumption
- ]
- |apply le_S_S_to_le.
- apply H6
- [assumption
- |generalize in match H9.
- elim (p2 j)
- [reflexivity|assumption]
- ]
- ]
- ]
- ]
- |apply H3[apply le_n|assumption]
- |apply H1[apply le_n|assumption]
- ]
- |assumption
- ]
- |rewrite > false_to_sigma_p_Sn
- [apply H
- [intros.apply H1[apply le_S.assumption|assumption]
- |intros.apply H2[apply le_S.assumption|assumption]
- |intros.apply H3[apply le_S.assumption|assumption]
- |intros.apply H4[assumption|assumption]
- |intros.apply H5[assumption|assumption]
- |intros.
- elim (le_to_or_lt_eq (h1 j) n1)
- [assumption
- |absurd (j = (h n1))
- [rewrite < H10.
- rewrite > H5
- [reflexivity|assumption|assumption]
- |unfold Not.intro.
- apply not_eq_true_false.
- rewrite < H7.
- rewrite < H10.
- rewrite > H4
- [reflexivity|assumption|assumption]
- ]
- |apply le_S_S_to_le.
- apply H6[assumption|assumption]
- ]
- ]
- |assumption
- ]
- ]
- ]
-qed.
-
-definition p_ord_times \def
-\lambda p,m,x.
- match p_ord x p with
- [pair q r \Rightarrow r*m+q].
-
-theorem eq_p_ord_times: \forall p,m,x.
-p_ord_times p m x = (ord_rem x p)*m+(ord x p).
-intros.unfold p_ord_times. unfold ord_rem.
-unfold ord.
-elim (p_ord x p).
-reflexivity.
+intros.
+unfold sigma_p.
+apply (eq_sigma_p_gh_gen nat O plus ? ? ? g h h1 n n1 p1 p2)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+]
qed.
-theorem div_p_ord_times:
-\forall p,m,x. ord x p < m \to p_ord_times p m x / m = ord_rem x p.
-intros.rewrite > eq_p_ord_times.
-apply div_plus_times.
-assumption.
-qed.
-
-theorem mod_p_ord_times:
-\forall p,m,x. ord x p < m \to p_ord_times p m x \mod m = ord x p.
-intros.rewrite > eq_p_ord_times.
-apply mod_plus_times.
-assumption.
-qed.
theorem sigma_p_divides:
\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to
sigma_p (S n) (\lambda x.divides_b x n)
(\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
intros.
-cut (O < p)
- [rewrite < sigma_p2.
- apply (trans_eq ? ?
- (sigma_p (S n*S m) (\lambda x:nat.divides_b (x/S m) n)
- (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m)))))
- [apply sym_eq.
- apply (eq_sigma_p_gh g ? (p_ord_times p (S m)))
- [intros.
- lapply (divides_b_true_to_lt_O ? ? H H4).
- apply divides_to_divides_b_true
- [rewrite > (times_n_O O).
- apply lt_times
- [assumption
- |apply lt_O_exp.assumption
- ]
- |apply divides_times
- [apply divides_b_true_to_divides.assumption
- |apply (witness ? ? (p \sup (m-i \mod (S m)))).
- rewrite < exp_plus_times.
- apply eq_f.
- rewrite > sym_plus.
- apply plus_minus_m_m.
- autobatch
- ]
- ]
- |intros.
- lapply (divides_b_true_to_lt_O ? ? H H4).
- unfold p_ord_times.
- rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
- [change with ((i/S m)*S m+i \mod S m=i).
- apply sym_eq.
- apply div_mod.
- apply lt_O_S
- |assumption
- |unfold Not.intro.
- apply H2.
- apply (trans_divides ? (i/ S m))
- [assumption|
- apply divides_b_true_to_divides;assumption]
- |apply sym_times.
- ]
- |intros.
- apply le_S_S.
- apply le_times
- [apply le_S_S_to_le.
- change with ((i/S m) < S n).
- apply (lt_times_to_lt_l m).
- apply (le_to_lt_to_lt ? i)
- [autobatch|assumption]
- |apply le_exp
- [assumption
- |apply le_S_S_to_le.
- apply lt_mod_m_m.
- apply lt_O_S
- ]
- ]
- |intros.
- cut (ord j p < S m)
- [rewrite > div_p_ord_times
- [apply divides_to_divides_b_true
- [apply lt_O_ord_rem
- [elim H1.assumption
- |apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |cut (n = ord_rem (n*(exp p m)) p)
- [rewrite > Hcut2.
- apply divides_to_divides_ord_rem
- [apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord_rem.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |assumption
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |intros.
- cut (ord j p < S m)
- [rewrite > div_p_ord_times
- [rewrite > mod_p_ord_times
- [rewrite > sym_times.
- apply sym_eq.
- apply exp_ord
- [elim H1.assumption
- |apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut2.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |assumption
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |intros.
- rewrite > eq_p_ord_times.
- rewrite > sym_plus.
- apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
- [apply lt_plus_l.
- apply le_S_S.
- cut (m = ord (n*(p \sup m)) p)
- [rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > sym_times.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |reflexivity
- ]
- ]
- |change with (S (ord_rem j p)*S m \le S n*S m).
- apply le_times_l.
- apply le_S_S.
- cut (n = ord_rem (n*(p \sup m)) p)
- [rewrite > Hcut1.
- apply divides_to_le
- [apply lt_O_ord_rem
- [elim H1.assumption
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |apply divides_to_divides_ord_rem
- [apply (divides_b_true_to_lt_O ? ? ? H4).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- ]
- |unfold ord_rem.
- rewrite > sym_times.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |reflexivity
- ]
- ]
- ]
- ]
- |apply eq_sigma_p
- [intros.
- elim (divides_b (x/S m) n);reflexivity
- |intros.reflexivity
- ]
- ]
-|elim H1.apply lt_to_le.assumption
+unfold sigma_p.
+apply (sigma_p_divides_gen nat O plus n m p ? ? ? g)
+[ assumption
+| assumption
+| assumption
+| apply symmetricIntPlus
+| apply associative_plus
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+]
+qed.
+
+theorem distributive_times_plus_sigma_p: \forall n,k:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+k*(sigma_p n p g) = sigma_p n p (\lambda i:nat.k * (g i)).
+intros.
+apply (distributive_times_plus_sigma_p_generic nat plus O times n k p g)
+[ apply symmetricIntPlus
+| apply associative_plus
+| intros.
+ apply sym_eq.
+ apply plus_n_O
+| apply symmetric_times
+| apply distributive_times_plus
+| intros.
+ rewrite < (times_n_O a).
+ reflexivity
]
qed.
-
projects / helm.git / blobdiff