(* *)
(**************************************************************************)
-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".
+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. (sigma_p_gen n p nat g O plus).
+\lambda n, p, g. (iter_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)).
(g n)+(sigma_p n p g).
intros.
unfold sigma_p.
-apply true_to_sigma_p_Sn_gen.
+apply true_to_iter_p_gen_Sn.
assumption.
qed.
p n = false \to sigma_p (S n) p g = sigma_p n p g.
intros.
unfold sigma_p.
-apply false_to_sigma_p_Sn_gen.
+apply false_to_iter_p_gen_Sn.
assumption.
-
-qed.
+qed.
theorem eq_sigma_p: \forall p1,p2:nat \to bool.
\forall g1,g2: nat \to nat.\forall n.
sigma_p n p1 g1 = sigma_p n p2 g2.
intros.
unfold sigma_p.
-apply eq_sigma_p_gen;
+apply eq_iter_p_gen;
assumption.
qed.
sigma_p n p1 g1 = sigma_p n p2 g2.
intros.
unfold sigma_p.
-apply eq_sigma_p1_gen;
+apply eq_iter_p_gen1;
assumption.
qed.
\forall g: nat \to nat.\forall n.sigma_p n (\lambda x.false) g = O.
intros.
unfold sigma_p.
-apply sigma_p_false_gen.
+apply iter_p_gen_false.
qed.
theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
= sigma_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) + sigma_p n p g.
intros.
unfold sigma_p.
-apply (sigma_p_plusA_gen nat n k p g O plus)
+apply (iter_p_gen_plusA nat n k p g O plus)
[ apply symmetricIntPlus.
| intros.
apply sym_eq.
p i = false) \to sigma_p m p g = sigma_p n p g.
intros.
unfold sigma_p.
-apply (false_to_eq_sigma_p_gen);
+apply (false_to_eq_iter_p_gen);
assumption.
qed.
(\lambda x.sigma_p m p2 (g x)).
intros.
unfold sigma_p.
-apply (sigma_p2_gen n m p1 p2 nat g O plus)
+apply (iter_p_gen2 n m p1 p2 nat g O plus)
[ apply symmetricIntPlus
| apply associative_plus
| intros.
\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.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 (iter_p_gen2' n m p1 p2 nat g O plus)
[ apply symmetricIntPlus
| apply associative_plus
| intros.
sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
intros.
unfold sigma_p.
-apply (sigma_p_gi_gen)
+apply (iter_p_gen_gi)
[ apply symmetricIntPlus
| apply associative_plus
| intros.
(\forall j. j < n1 \to p2 j = true \to p1 (h1 j) = true) \to
(\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.
+sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 p2 g.
intros.
unfold sigma_p.
-apply (eq_sigma_p_gh_gen nat O plus ? ? ? g h h1 n n1 p1 p2)
+apply (eq_iter_p_gen_gh nat O plus ? ? ? g h h1 n n1 p1 p2)
[ apply symmetricIntPlus
| apply associative_plus
| intros.
(\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
intros.
unfold sigma_p.
-apply (sigma_p_divides_gen nat O plus n m p ? ? ? g)
+apply (iter_p_gen_divides nat O plus n m p ? ? ? g)
[ assumption
| assumption
| assumption
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 (distributive_times_plus_iter_p_gen nat plus O times n k p g)
[ apply symmetricIntPlus
| apply associative_plus
| intros.
]
qed.
+(*some properties of sigma_p invoked with an "always true" predicate (in this
+ way sigma_p just counts the elements, without doing any control) or with
+ the nat \to nat function which always returns (S O).
+ It 's not easily possible proving these theorems in a general form
+ in generic_sigma_p.ma
+ *)
+
+theorem sigma_p_true: \forall n:nat.
+(sigma_p n (\lambda x.true) (\lambda x.S O)) = n.
+intros.
+elim n
+[ simplify.
+ reflexivity
+| rewrite > (true_to_sigma_p_Sn n1 (\lambda x:nat.true) (\lambda x:nat.S O))
+ [ rewrite > H.
+ simplify.
+ reflexivity
+ | reflexivity
+ ]
+]
+qed.
+
+theorem sigma_P_SO_to_sigma_p_true: \forall n:nat. \forall g:nat \to bool.
+sigma_p n g (\lambda n:nat. (S O)) =
+sigma_p n (\lambda x:nat.true) (\lambda i:nat.bool_to_nat (g i)).
+intros.
+elim n
+[ simplify.
+ reflexivity
+| cut ((g n1) = true \lor (g n1) = false)
+ [ rewrite > true_to_sigma_p_Sn in \vdash (? ? ? %)
+ [ elim Hcut
+ [ rewrite > H1.
+ rewrite > true_to_sigma_p_Sn in \vdash (? ? % ?)
+ [ simplify.
+ apply eq_f.
+ assumption
+ | assumption
+ ]
+ | rewrite > H1.
+ rewrite > false_to_sigma_p_Sn in \vdash (? ? % ?)
+ [ simplify.
+ assumption
+ | assumption
+ ]
+ ]
+ | reflexivity
+ ]
+ | elim (g n1)
+ [ left.
+ reflexivity
+ | right.
+ reflexivity
+ ]
+ ]
+]
+qed.
+
+(* I introduce an equivalence in the form map_iter_i in order to use
+ * the existing result about permutation in that part of the library.
+ *)
+
+theorem eq_map_iter_i_sigma_p_alwaysTrue: \forall n:nat.\forall g:nat \to nat.
+map_iter_i n g plus O = sigma_p (S n) (\lambda c:nat.true) g.
+intros.
+elim n
+[ simplify.
+ rewrite < plus_n_O.
+ reflexivity
+| rewrite > true_to_sigma_p_Sn
+ [ simplify in \vdash (? ? % ?).
+ rewrite < plus_n_O.
+ apply eq_f.
+ assumption
+ | reflexivity
+ ]
+]
+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.
+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
+ ]
+]
+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)
+ (\lambda a.(sigma_p n (\lambda x:nat.true) (\lambda b.f (b*m + a)))).
+intro.
+elim n
+[ simplify.
+ elim m
+ [ simplify.
+ reflexivity
+ | rewrite > true_to_sigma_p_Sn
+ [ rewrite < H.
+ reflexivity
+ | reflexivity
+ ]
+ ]
+| change in \vdash (? ? ? (? ? ? (\lambda a:?.%))) with ((f ((n1*m)+a)) +
+ (sigma_p n1 (\lambda x:nat.true) (\lambda b:nat.f (b*m +a)))).
+ rewrite > sigma_p_plus_1 in \vdash (? ? ? %).
+ rewrite > (sym_times (S n1) m).
+ rewrite < (times_n_Sm m n1).
+ rewrite > sigma_p_plus in \vdash (? ? % ?).
+ apply eq_f2
+ [ rewrite < (sym_times m n1).
+ apply eq_sigma_p
+ [ intros.
+ reflexivity
+ | intros.
+ rewrite < (sym_plus ? (m * n1)).
+ reflexivity
+ ]
+ | rewrite > (sym_times m n1).
+ apply H
+ ]
+]
+qed.
+
+theorem eq_sigma_p_sigma_p_times2 : \forall n,m:nat.\forall f:nat \to nat.
+sigma_p (n *m) (\lambda c:nat.true) f =
+sigma_p n (\lambda c:nat.true)
+ (\lambda a.(sigma_p m (\lambda c:nat.true) (\lambda b:nat.f (b* n + a)))).
+intros.
+rewrite > sym_times.
+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.
+\forall g1,g2: nat \to nat.
+(\forall a,b:nat. a < (S n) \to b < (S m) \to (g b a) < (S n)*(S m)) \to
+(\forall a,b:nat. a < (S n) \to b < (S m) \to (g1 (g b a)) = a) \to
+(\forall a,b:nat. a < (S n) \to b < (S m) \to (g2 (g b a)) = b) \to
+(\forall a,b:nat. a < (S n) \to b < (S m) \to f (g b a) = andb (f2 b) (f1 a)) \to
+(sigma_p ((S n) * (S m)) f (\lambda c:nat.(S O))) =
+sigma_p (S n) f1 (\lambda c:nat.(S O)) * sigma_p (S m) f2 (\lambda c:nat.(S O)).
+intros.
+
+rewrite > (sigma_P_SO_to_sigma_p_true ).
+rewrite > (S_pred ((S n)*(S m))) in \vdash (? ? (? % ? ?) ?)
+[ rewrite < (eq_map_iter_i_sigma_p_alwaysTrue (pred ((S n)* (S m)))).
+ rewrite > (permut_to_eq_map_iter_i plus assoc_plus sym_plus ? ? ?
+ (\lambda i.g (div i (S n)) (mod i (S n))))
+ [ rewrite > eq_map_iter_i_sigma_p_alwaysTrue.
+ rewrite < S_pred
+ [ rewrite > eq_sigma_p_sigma_p_times2.
+ apply (trans_eq ? ? (sigma_p (S n) (\lambda c:nat.true)
+ (\lambda a. sigma_p (S m) (\lambda c:nat.true)
+ (\lambda b.(bool_to_nat (f2 b))*(bool_to_nat (f1 a))))))
+ [ apply eq_sigma_p;intros
+ [ reflexivity
+ | apply eq_sigma_p;intros
+ [ reflexivity
+ |
+ rewrite > (div_mod_spec_to_eq (x1*(S n) + x) (S n) ((x1*(S n) + x)/(S n))
+ ((x1*(S n) + x) \mod (S n)) x1 x)
+ [ rewrite > (div_mod_spec_to_eq2 (x1*(S n) + x) (S n) ((x1*(S n) + x)/(S n))
+ ((x1*(S n) + x) \mod (S n)) x1 x)
+ [ rewrite > H3
+ [ apply bool_to_nat_andb
+ | assumption
+ | assumption
+ ]
+ | apply div_mod_spec_div_mod.
+ apply lt_O_S
+ | constructor 1
+ [ assumption
+ | reflexivity
+ ]
+ ]
+ | apply div_mod_spec_div_mod.
+ apply lt_O_S
+ | constructor 1
+ [ assumption
+ | reflexivity
+ ]
+ ]
+ ]
+ ]
+ | apply (trans_eq ? ?
+ (sigma_p (S n) (\lambda c:nat.true) (\lambda n.((bool_to_nat (f1 n)) *
+ (sigma_p (S m) (\lambda c:nat.true) (\lambda n.bool_to_nat (f2 n)))))))
+ [ apply eq_sigma_p;intros
+ [ reflexivity
+ | rewrite > distributive_times_plus_sigma_p.
+ apply eq_sigma_p;intros
+ [ reflexivity
+ | rewrite > sym_times.
+ reflexivity
+ ]
+ ]
+ | apply sym_eq.
+ rewrite > sigma_P_SO_to_sigma_p_true.
+ rewrite > sigma_P_SO_to_sigma_p_true in \vdash (? ? (? ? %) ?).
+ rewrite > sym_times.
+ rewrite > distributive_times_plus_sigma_p.
+ apply eq_sigma_p;intros
+ [ reflexivity
+ | rewrite > distributive_times_plus_sigma_p.
+ rewrite < sym_times.
+ rewrite > distributive_times_plus_sigma_p.
+ apply eq_sigma_p;
+ intros; reflexivity
+ ]
+ ]
+ ]
+ | apply lt_O_times_S_S
+ ]
+
+ | unfold permut.
+ split
+ [ intros.
+ rewrite < plus_n_O.
+ apply le_S_S_to_le.
+ rewrite < S_pred in \vdash (? ? %)
+ [ change with ((g (i/(S n)) (i \mod (S n))) \lt (S n)*(S m)).
+ apply H
+ [ apply lt_mod_m_m.
+ unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ | apply (lt_times_to_lt_l n).
+ apply (le_to_lt_to_lt ? i)
+ [ rewrite > (div_mod i (S n)) in \vdash (? ? %)
+ [ rewrite > sym_plus.
+ apply le_plus_n
+ | unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ ]
+ | unfold lt.
+ rewrite > S_pred in \vdash (? ? %)
+ [ apply le_S_S.
+ rewrite > plus_n_O in \vdash (? ? %).
+ rewrite > sym_times.
+ assumption
+ | apply lt_O_times_S_S
+ ]
+ ]
+ ]
+ | apply lt_O_times_S_S
+ ]
+ | rewrite < plus_n_O.
+ unfold injn.
+ intros.
+ cut (i < (S n)*(S m))
+ [ cut (j < (S n)*(S m))
+ [ cut ((i \mod (S n)) < (S n))
+ [ cut ((i/(S n)) < (S m))
+ [ cut ((j \mod (S n)) < (S n))
+ [ cut ((j/(S n)) < (S m))
+ [ rewrite > (div_mod i (S n))
+ [ rewrite > (div_mod j (S n))
+ [ rewrite < (H1 (i \mod (S n)) (i/(S n)) Hcut2 Hcut3).
+ rewrite < (H2 (i \mod (S n)) (i/(S n)) Hcut2 Hcut3) in \vdash (? ? (? % ?) ?).
+ rewrite < (H1 (j \mod (S n)) (j/(S n)) Hcut4 Hcut5).
+ rewrite < (H2 (j \mod (S n)) (j/(S n)) Hcut4 Hcut5) in \vdash (? ? ? (? % ?)).
+ rewrite > H6.
+ reflexivity
+ | unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ ]
+ | unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ ]
+ | apply (lt_times_to_lt_l n).
+ apply (le_to_lt_to_lt ? j)
+ [ rewrite > (div_mod j (S n)) in \vdash (? ? %)
+ [ rewrite > sym_plus.
+ apply le_plus_n
+ | unfold lt. apply le_S_S.
+ apply le_O_n
+ ]
+ | rewrite < sym_times.
+ assumption
+ ]
+ ]
+ | apply lt_mod_m_m.
+ unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ ]
+ | apply (lt_times_to_lt_l n).
+ apply (le_to_lt_to_lt ? i)
+ [ rewrite > (div_mod i (S n)) in \vdash (? ? %)
+ [ rewrite > sym_plus.
+ apply le_plus_n
+ | unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ ]
+ | rewrite < sym_times.
+ assumption
+ ]
+ ]
+ | apply lt_mod_m_m.
+ unfold lt.
+ apply le_S_S.
+ apply le_O_n
+ ]
+ | unfold lt.
+ rewrite > S_pred in \vdash (? ? %)
+ [ apply le_S_S.
+ assumption
+ | apply lt_O_times_S_S
+ ]
+ ]
+ | unfold lt.
+ rewrite > S_pred in \vdash (? ? %)
+ [ apply le_S_S.
+ assumption
+ | apply lt_O_times_S_S
+ ]
+ ]
+ ]
+ | intros.
+ apply False_ind.
+ apply (not_le_Sn_O m1 H4)
+ ]
+| apply lt_O_times_S_S
+]
+qed.
+
+theorem sigma_p_knm:
+\forall g: nat \to nat.
+\forall h2:nat \to nat \to nat.
+\forall h11,h12:nat \to nat.
+\forall k,n,m.
+\forall p1,p21:nat \to bool.
+\forall p22:nat \to nat \to bool.
+(\forall x. x < k \to p1 x = true \to
+p21 (h11 x) = true \land p22 (h11 x) (h12 x) = true
+\land h2 (h11 x) (h12 x) = x
+\land (h11 x) < n \land (h12 x) < m) \to
+(\forall i,j. i < n \to j < m \to p21 i = true \to p22 i j = true \to
+p1 (h2 i j) = true \land
+h11 (h2 i j) = i \land h12 (h2 i j) = j
+\land h2 i j < k) \to
+sigma_p k p1 g=
+sigma_p n p21 (\lambda x:nat.sigma_p m (p22 x) (\lambda y. g (h2 x y))).
+intros.
+unfold sigma_p.
+unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)).
+apply iter_p_gen_knm
+ [apply symmetricIntPlus
+ |apply associative_plus
+ |intro.rewrite < plus_n_O.reflexivity
+ |exact h11
+ |exact h12
+ |assumption
+ |assumption
+ ]
+qed.
+
+
+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.
\ No newline at end of file
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