Main inequalities for e.

diff --git a/helm/software/matita/library/nat/binomial.ma b/helm/software/matita/library/nat/binomial.ma
index 4434f769b29e526b0de535836e4abfd9f6a0fdd0..9505761488e3dd9568d51036ff877471c21215de 100644 (file)
--- a/helm/software/matita/library/nat/binomial.ma
+++ b/helm/software/matita/library/nat/binomial.ma
@@ -213,4 +213,17 @@ intros.elim n
             |reflexivity]]
       |reflexivity]
    |reflexivity]]
+qed.
+
+theorem exp_S_sigma_p:\forall a,n.
+exp (S a) n = sigma_p (S n) (\lambda k.true) (\lambda k.(bc n k)*(exp a (n-k))).
+intros.
+rewrite > plus_n_SO.
+rewrite > exp_plus_sigma_p.
+apply eq_sigma_p;intros
+  [reflexivity
+  |rewrite < exp_SO_n.
+   rewrite < times_n_SO.
+   reflexivity
+  ]
 qed.
\ No newline at end of file

diff --git a/helm/software/matita/library/nat/exp.ma b/helm/software/matita/library/nat/exp.ma
index 74a3be71f63d202ed5118663088ac238688d142d..c94e713b1b0381684f8ec00528a7ae5f6ce04b77 100644 (file)
--- a/helm/software/matita/library/nat/exp.ma
+++ b/helm/software/matita/library/nat/exp.ma
@@ -40,6 +40,13 @@ theorem exp_n_SO : \forall n:nat. n = n \sup (S O).
 intro.simplify.rewrite < times_n_SO.reflexivity.
 qed.
 
+theorem exp_SO_n : \forall n:nat. S O = (S O) \sup n.
+intro.elim n
+  [reflexivity
+  |simplify.rewrite < plus_n_O.assumption
+  ]
+qed.
+
 theorem exp_SSO: \forall n. exp n (S(S O)) = n*n.
 intro.simplify.
 rewrite < times_n_SO.

diff --git a/helm/software/matita/library/nat/iteration2.ma b/helm/software/matita/library/nat/iteration2.ma
index 0230362e72aaa9610ffb25034c21f0028d4e8b69..ea306f4554a6c2c83e809dace78d034d9534bcc8 100644 (file)
--- a/helm/software/matita/library/nat/iteration2.ma
+++ b/helm/software/matita/library/nat/iteration2.ma
@@ -191,7 +191,151 @@ apply (eq_iter_p_gen_gh nat O plus ? ? ? g h h1 n n1 p1 p2)
 ]
 qed.
 
+(* monotonicity *)
+theorem le_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 
+sigma_p n p g1 \le sigma_p n p g2.
+intros.
+generalize in match H.
+elim n
+  [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.
+           intros.
+           apply H2[apply le_S.assumption|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
+        ]
+      |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
+(\exists i. i < n \and (p i = true) \and (g1 i < g2 i)) \to
+sigma_p n p g1 < sigma_p n p g2.
+intros 4.
+elim n
+  [elim H1.clear H1.
+   elim H2.clear H2.
+   elim H1.clear H1.
+   apply False_ind.
+   apply (lt_to_not_le ? ? H2).
+   apply le_O_n
+  |apply (bool_elim ? (p n1));intros
+    [apply (bool_elim ? (leb (S (g1 n1)) (g2 n1)));intros
+      [rewrite > true_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+          [change with 
+           (S (g1 n1)+sigma_p n1 p g1 \le g2 n1+sigma_p n1 p g2).
+           apply le_plus
+            [apply leb_true_to_le.assumption
+            |apply le_sigma_p.intros.
+             apply H1
+              [apply lt_to_le.apply le_S_S.assumption
+              |assumption
+              ]
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      |rewrite > true_to_sigma_p_Sn
+        [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
+          [unfold lt.
+           rewrite > plus_n_Sm.
+           apply le_plus
+            [apply H1
+              [apply le_n
+              |assumption
+              ]
+            |apply H
+              [intros.apply H1
+                [apply lt_to_le.apply le_S_S.assumption
+                |assumption
+                ]
+              |elim H2.clear H2.
+               elim H5.clear H5.
+               elim H2.clear H2.
+               apply (ex_intro ? ? a).
+               split
+                [split
+                  [elim (le_to_or_lt_eq a n1)
+                    [assumption
+                    |absurd (g1 a < g2 a)
+                      [assumption
+                      |apply leb_false_to_not_le.
+                       rewrite > H2.
+                       assumption
+                      ]
+                    |apply le_S_S_to_le.
+                     assumption
+                    ]
+                  |assumption
+                  ]
+                |assumption
+                ]
+              ]
+            ]
+          |assumption
+          ]
+        |assumption
+        ]
+      ]
+    |rewrite > false_to_sigma_p_Sn
+      [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
+        [apply H
+          [intros.apply H1
+            [apply lt_to_le.apply le_S_S.assumption
+            |assumption
+            ]
+          |elim H2.clear H2.
+           elim H4.clear H4.
+           elim H2.clear H2.
+           apply (ex_intro ? ? a).
+           split
+            [split
+              [elim (le_to_or_lt_eq a n1)
+                [assumption
+                |apply False_ind.
+                 apply not_eq_true_false.
+                 rewrite < H6.
+                 rewrite < H3.
+                 rewrite < H2. 
+                 reflexivity
+                |apply le_S_S_to_le.
+                 assumption
+                ]
+              |assumption
+              ]
+            |assumption
+            ]
+          ]
+        |assumption
+        ]
+      |assumption
+      ]
+    ]
+  ]
+qed.
+          
 theorem sigma_p_divides: 
 \forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to 
 \forall g: nat \to nat.

diff --git a/helm/software/matita/library/nat/neper.ma b/helm/software/matita/library/nat/neper.ma
index e618ec5687bf8cd944eff252c00c71f117261fac..f3db405d42f09b63c8a9378b481075934d667d59 100644 (file)
--- a/helm/software/matita/library/nat/neper.ma
+++ b/helm/software/matita/library/nat/neper.ma
@@ -16,8 +16,9 @@ set "baseuri" "cic:/matita/nat/neper".
 
 include "nat/iteration2.ma".
 include "nat/div_and_mod_diseq.ma".
+include "nat/binomial.ma".
 
-theorem boh: \forall n,m.
+theorem sigma_p_div_exp: \forall n,m.
 sigma_p n (\lambda i.true) (\lambda i.m/(exp (S(S O)) i)) \le 
 ((S(S O))*m*(exp (S(S O)) n) - (S(S O))*m)/(exp (S(S O)) n).
 intros.
@@ -64,4 +65,137 @@ elim n
     ]
   ]
 qed.
-   
\ No newline at end of file
+   
+theorem le_fact_exp: \forall i,m. i \le m \to m!≤ m \sup i*(m-i)!.
+intro.elim i
+  [rewrite < minus_n_O.
+   simplify.rewrite < plus_n_O.
+   apply le_n
+  |simplify.
+   apply (trans_le ? ((m)\sup(n)*(m-n)!))
+    [apply H.
+     apply lt_to_le.assumption
+    |rewrite > sym_times in ⊢ (? ? (? % ?)).
+     rewrite > assoc_times.
+     apply le_times_r.
+     generalize in match H1.
+     cases m;intro
+      [apply False_ind.
+       apply (lt_to_not_le ? ? H2).
+       apply le_O_n
+      |rewrite > minus_Sn_m.
+       simplify.
+       apply le_plus_r.
+       apply le_times_l.
+       apply le_minus_m.
+       apply le_S_S_to_le.
+       assumption
+      ]
+    ]
+  ]
+qed.
+
+theorem le_fact_exp1: \forall n. S O < n \to (S(S O))*n!≤ n \sup n.
+intros.elim H
+  [apply le_n
+  |change with ((S(S O))*((S n1)*(fact n1)) \le (S n1)*(exp (S n1) n1)).   
+   rewrite < assoc_times.
+   rewrite < sym_times in ⊢ (? (? % ?) ?).
+   rewrite > assoc_times.
+   apply le_times_r.
+   apply (trans_le ? (exp n1 n1))
+    [assumption
+    |apply monotonic_exp1.
+     apply le_n_Sn
+    ]
+  ]
+qed.
+   
+theorem le_exp_sigma_p_exp: \forall n. (exp (S n) n) \le
+sigma_p (S n) (\lambda k.true) (\lambda k.(exp n n)/k!).
+intro.
+rewrite > exp_S_sigma_p.
+apply le_sigma_p.
+intros.unfold bc.
+apply (trans_le ? ((exp n (n-i))*((n \sup i)/i!)))
+  [rewrite > sym_times.
+   apply le_times_r.
+   rewrite > sym_times.
+   rewrite < eq_div_div_div_times
+    [apply monotonic_div
+      [apply lt_O_fact
+      |apply le_times_to_le_div2
+        [apply lt_O_fact
+        |apply le_fact_exp.
+         apply le_S_S_to_le.
+         assumption
+        ]
+      ]
+    |apply lt_O_fact
+    |apply lt_O_fact
+    ]
+  |rewrite > (plus_minus_m_m ? i) in ⊢ (? ? (? (? ? %) ?))
+    [rewrite > exp_plus_times.
+     apply le_times_div_div_times.
+     apply lt_O_fact
+    |apply le_S_S_to_le.
+     assumption
+    ]
+  ]
+qed.
+    
+theorem lt_exp_sigma_p_exp: \forall n. S O < n \to
+(exp (S n) n) <
+sigma_p (S n) (\lambda k.true) (\lambda k.(exp n n)/k!).
+intros.
+rewrite > exp_S_sigma_p.
+apply lt_sigma_p
+  [intros.unfold bc.
+   apply (trans_le ? ((exp n (n-i))*((n \sup i)/i!)))
+    [rewrite > sym_times.
+     apply le_times_r.
+     rewrite > sym_times.
+     rewrite < eq_div_div_div_times
+      [apply monotonic_div
+        [apply lt_O_fact
+        |apply le_times_to_le_div2
+          [apply lt_O_fact
+          |apply le_fact_exp.
+           apply le_S_S_to_le.
+           assumption
+          ]
+        ]
+      |apply lt_O_fact
+      |apply lt_O_fact
+      ]
+    |rewrite > (plus_minus_m_m ? i) in ⊢ (? ? (? (? ? %) ?))
+      [rewrite > exp_plus_times.
+       apply le_times_div_div_times.
+       apply lt_O_fact
+      |apply le_S_S_to_le.
+       assumption
+      ]
+    ]
+  |apply (ex_intro ? ? n).
+   split
+    [split
+      [apply le_n
+      |reflexivity
+      ]
+    |rewrite < minus_n_n.
+     rewrite > bc_n_n.
+     simplify.unfold lt.
+     apply le_times_to_le_div
+      [apply lt_O_fact
+      |rewrite > sym_times.
+       apply le_fact_exp1.
+       assumption
+      ]
+    ]
+  ]
+qed.
+       
+   
+    
+
+