beginning proof of chebyshev's bound on prim.

[helm.git] / helm / software / matita / library / nat / iteration2.ma

diff --git a/helm/software/matita/library/nat/iteration2.ma b/helm/software/matita/library/nat/iteration2.ma
index c5236ff4a6f8678907c1143bf47ffd18d5c1dcf5..211df69d0fa0d9941a9dcf14bebca51e1812ffda 100644 (file)
--- a/helm/software/matita/library/nat/iteration2.ma
+++ b/helm/software/matita/library/nat/iteration2.ma
@@ -107,6 +107,84 @@ apply (false_to_eq_iter_p_gen);
   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.
@@ -240,6 +318,90 @@ elim n
   ]
 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.
+generalize in match H.
+elim n
+  [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.
+               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
+            ]
+          |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 (H2 n1) as H5
+              [rewrite > H3 in H5.
+               rewrite > H4 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
+            ]
+          |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 (H2 n1) as H5
+              [rewrite > H3 in H5.
+               rewrite > H4 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
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > false_to_sigma_p_Sn
+        [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+          [apply H1.intros.
+           apply H2.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
@@ -467,32 +629,33 @@ elim n
 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) 
@@ -539,7 +702,6 @@ 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. 
@@ -801,4 +963,21 @@ apply(iter_p_gen_2_eq nat O plus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12
 | 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.
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