Main inequalities for e.

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

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.
+       
+   
+    
+
+