simplified version of a theorem.

[helm.git] / helm / software / matita / library / Z / sigma_p.ma

diff --git a/helm/software/matita/library/Z/sigma_p.ma b/helm/software/matita/library/Z/sigma_p.ma
index 5d85bc653bb6aa317ce3a0bd1362030ea0fb9877..b246b8444f0cafed9f6c69a4640b955f8de38b0b 100644 (file)
--- a/helm/software/matita/library/Z/sigma_p.ma
+++ b/helm/software/matita/library/Z/sigma_p.ma
@@ -12,16 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/Z/sigma_p.ma".
+set "baseuri" "cic:/matita/Z/sigma_p".
 
 include "Z/times.ma".
 include "nat/primes.ma".
 include "nat/ord.ma".
-include "nat/generic_sigma_p.ma".
+include "nat/generic_iter_p.ma".
 
-(* sigma_p in Z is a specialization of sigma_p_gen *)
+(* sigma_p in Z is a specialization of iter_p_gen *)
 definition sigma_p: nat \to (nat \to bool) \to (nat \to Z) \to Z \def
-\lambda n, p, g. (sigma_p_gen n p Z g OZ Zplus).
+\lambda n, p, g. (iter_p_gen n p Z g OZ Zplus).
 
 theorem symmetricZPlus: symmetric Z Zplus.
 change with (\forall a,b:Z. (Zplus a b) = (Zplus b a)).
@@ -36,7 +36,7 @@ p n = true \to sigma_p (S n) p g =
 (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.
    
@@ -45,7 +45,7 @@ theorem false_to_sigma_p_Sn:
 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.
 
@@ -56,7 +56,7 @@ theorem eq_sigma_p: \forall p1,p2:nat \to bool.
 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.
 
@@ -67,7 +67,7 @@ theorem eq_sigma_p1: \forall p1,p2:nat \to bool.
 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.
 
@@ -75,7 +75,7 @@ theorem sigma_p_false:
 \forall g: nat \to Z.\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.
@@ -84,7 +84,7 @@ 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.
 unfold sigma_p.
-apply (sigma_p_plusA_gen Z n k p g OZ Zplus)
+apply (iter_p_gen_plusA Z n k p g OZ Zplus)
 [ apply symmetricZPlus.
 | intros.
   apply cic:/matita/Z/plus/Zplus_z_OZ.con
@@ -98,7 +98,7 @@ theorem false_to_eq_sigma_p: \forall n,m:nat.n \le m \to
 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.
 
@@ -113,7 +113,7 @@ sigma_p n p1
   (\lambda x.sigma_p m p2 (g x)).
 intros.
 unfold sigma_p.
-apply (sigma_p2_gen n m p1 p2 Z g OZ Zplus)
+apply (iter_p_gen2 n m p1 p2 Z g OZ Zplus)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -135,7 +135,7 @@ 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 Z g OZ Zplus)
+apply (iter_p_gen2' n m p1 p2 Z g OZ Zplus)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -148,7 +148,7 @@ lemma sigma_p_gi: \forall g: nat \to Z.
 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 symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -168,10 +168,10 @@ theorem eq_sigma_p_gh:
 (\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 Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
+apply (eq_iter_p_gen_gh Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -186,6 +186,85 @@ apply (eq_sigma_p_gh_gen Z OZ Zplus ? ? ? g h h1 n n1 p1 p2)
 qed.
 
 
+theorem divides_exp_to_lt_ord:\forall n,m,j,p. O < n \to prime p \to
+p \ndivides n \to j \divides n*(exp p m) \to ord j p < S m.
+intros.
+cut (m = ord (n*(exp p m)) p)
+  [apply le_S_S.
+   rewrite > Hcut.
+   apply divides_to_le_ord
+    [elim (le_to_or_lt_eq ? ? (le_O_n j))
+      [assumption
+      |apply False_ind.
+       apply (lt_to_not_eq ? ? H).
+       elim H3.
+       rewrite < H4 in H5.simplify in H5.
+       elim (times_O_to_O ? ? H5)
+        [apply sym_eq.assumption
+        |apply False_ind.
+         apply (not_le_Sn_n O).
+         rewrite < H6 in \vdash (? ? %).
+         apply lt_O_exp.
+         elim H1.apply lt_to_le.assumption
+        ]
+      ]
+    |rewrite > (times_n_O O).
+     apply lt_times
+      [assumption|apply lt_O_exp.apply (prime_to_lt_O ? H1)]
+    |assumption
+    |assumption
+    ]
+  |unfold ord.
+   rewrite > (p_ord_exp1 p ? m n)
+    [reflexivity
+    |apply (prime_to_lt_O ? H1)
+    |assumption
+    |apply sym_times
+    ]
+  ]
+qed.
+
+theorem divides_exp_to_divides_ord_rem:\forall n,m,j,p. O < n \to prime p \to
+p \ndivides n \to j \divides n*(exp p m) \to ord_rem j p \divides n.
+intros.
+cut (O < j)
+  [cut (n = ord_rem (n*(exp p m)) p)
+    [rewrite > Hcut1.
+     apply divides_to_divides_ord_rem
+      [assumption   
+      |rewrite > (times_n_O O).
+       apply lt_times
+        [assumption|apply lt_O_exp.apply (prime_to_lt_O ? H1)]
+      |assumption
+      |assumption
+      ]
+    |unfold ord_rem.
+     rewrite > (p_ord_exp1 p ? m n)
+      [reflexivity
+      |apply (prime_to_lt_O ? H1)
+      |assumption
+      |apply sym_times
+      ]
+    ]
+  |elim (le_to_or_lt_eq ? ? (le_O_n j))
+    [assumption
+    |apply False_ind.
+     apply (lt_to_not_eq ? ? H).
+     elim H3.
+     rewrite < H4 in H5.simplify in H5.
+     elim (times_O_to_O ? ? H5)
+      [apply sym_eq.assumption
+      |apply False_ind.
+       apply (not_le_Sn_n O).
+       rewrite < H6 in \vdash (? ? %).
+       apply lt_O_exp.
+       elim H1.apply lt_to_le.assumption
+      ]
+    ]
+  ] 
+qed.
+
+
 theorem sigma_p_divides_b: 
 \forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
 \forall g: nat \to Z.
@@ -194,7 +273,7 @@ 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.
 unfold sigma_p.
-apply (sigma_p_divides_gen Z OZ Zplus n m p ? ? ? g)
+apply (iter_p_gen_divides Z OZ Zplus n m p ? ? ? g)
 [ assumption
 | assumption
 | assumption
@@ -210,7 +289,7 @@ qed.
 lemma Ztimes_sigma_pl: \forall z:Z.\forall n:nat.\forall p. \forall f.
 z * (sigma_p n p f) = sigma_p n p (\lambda i.z*(f i)).
 intros.
-apply (distributive_times_plus_sigma_p_generic Z Zplus OZ Ztimes n z p f)
+apply (distributive_times_plus_iter_p_gen Z Zplus OZ Ztimes n z p f)
 [ apply symmetricZPlus
 | apply associative_Zplus
 | intros.
@@ -232,4 +311,4 @@ apply eq_sigma_p
   [intros.reflexivity
   |intros.apply sym_Ztimes
   ]
-qed.
\ No newline at end of file
+qed.