(* *)
(**************************************************************************)
-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).
(\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.
-generalize in match H.
-elim n
+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 H2[apply le_n|assumption]
- |apply H1.
+ [apply H1[apply le_n|assumption]
+ |apply H.
intros.
- apply H2[apply le_S.assumption|assumption]
+ apply H1[apply le_S.assumption|assumption]
]
|assumption
]
]
|rewrite > false_to_sigma_p_Sn
[rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
- [apply H1.
+ [apply H.
intros.
- apply H2[apply le_S.assumption|assumption]
+ apply H1[apply le_S.assumption|assumption]
|assumption
]
|assumption
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.
-generalize in match H.
-elim n
+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 (H2 n1) as H5
- [rewrite > H3 in H5.
- rewrite > H4 in H5.
+ [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 H1.intros.
- apply H2.apply le_S.assumption
+ |apply H.intros.
+ apply H1.apply le_S.assumption
]
|assumption
]
[rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
[change in ⊢ (? ? %) with (O + sigma_p n1 p2 g2).
apply le_plus
- [lapply (H2 n1) as H5
- [rewrite > H3 in H5.
- rewrite > H4 in H5.
+ [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 H1.intros.
- apply H2.apply le_S.assumption
+ |apply H.intros.
+ apply H1.apply le_S.assumption
]
|assumption
]
[rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
[change in ⊢ (? % ?) with (O + sigma_p n1 p1 g1).
apply le_plus
- [lapply (H2 n1) as H5
- [rewrite > H3 in H5.
- rewrite > H4 in H5.
+ [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 H1.intros.
- apply H2.apply le_S.assumption
+ |apply H.intros.
+ apply H1.apply le_S.assumption
]
|assumption
]
]
|rewrite > false_to_sigma_p_Sn
[rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
- [apply H1.intros.
- apply H2.apply le_S.assumption
+ [apply H.intros.
+ apply H1.apply le_S.assumption
|assumption
]
|assumption
]
]
]
-qed.
+qed.
theorem lt_sigma_p:
\forall n:nat. \forall p:nat \to bool. \forall g1,g2: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)