X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fneper.ma;h=9378bd8dee4685787088fb23a6820134b6172753;hb=10f29fdd78ee089a9a94446207b543d33d6c851c;hp=f3db405d42f09b63c8a9378b481075934d667d59;hpb=1f214ad28b490ed66602eb2d80359c01ba55ee05;p=helm.git

diff --git a/helm/software/matita/library/nat/neper.ma b/helm/software/matita/library/nat/neper.ma
index f3db405d4..9378bd8de 100644
--- a/helm/software/matita/library/nat/neper.ma
+++ b/helm/software/matita/library/nat/neper.ma
@@ -17,6 +17,8 @@ set "baseuri" "cic:/matita/nat/neper".
 include "nat/iteration2.ma".
 include "nat/div_and_mod_diseq.ma".
 include "nat/binomial.ma".
+include "nat/log.ma".
+include "nat/chebyshev.ma".
 
 theorem sigma_p_div_exp: \forall n,m.
 sigma_p n (\lambda i.true) (\lambda i.m/(exp (S(S O)) i)) \le 
@@ -194,8 +196,552 @@ apply lt_sigma_p
     ]
   ]
 qed.
+
+theorem le_exp_SSO_fact:\forall n. 
+exp (S(S O)) (pred n) \le n!.
+intro.elim n
+  [apply le_n
+  |change with ((S(S O))\sup n1 ≤(S n1)*n1!).
+   apply (nat_case1 n1);intros
+    [apply le_n
+    |change in ⊢ (? % ?) with ((S(S O))*exp (S(S O)) (pred (S m))).
+     apply le_times
+      [apply le_S_S.apply le_S_S.apply le_O_n
+      |rewrite < H1.assumption
+      ]
+    ]
+  ]
+qed.
        
-   
+theorem lt_SO_to_lt_exp_Sn_n_SSSO: \forall n. S O < n \to 
+(exp (S n) n) < (S(S(S O)))*(exp n n).
+intros.  
+apply (lt_to_le_to_lt ? (sigma_p (S n) (\lambda k.true) (\lambda k.(exp n n)/k!)))
+  [apply lt_exp_sigma_p_exp.assumption
+  |apply (trans_le ? (sigma_p (S n) (\lambda i.true) (\lambda i.(exp n n)/(exp (S(S O)) (pred i)))))
+    [apply le_sigma_p.intros.
+     apply le_times_to_le_div
+      [apply lt_O_exp.
+       apply lt_O_S
+      |apply (trans_le ? ((S(S O))\sup (pred i)* n \sup n/i!))
+        [apply le_times_div_div_times.
+         apply lt_O_fact
+        |apply le_times_to_le_div2
+          [apply lt_O_fact
+          |rewrite < sym_times.
+           apply le_times_r.
+           apply le_exp_SSO_fact
+          ]
+        ]
+      ]
+    |rewrite > eq_sigma_p_pred
+      [rewrite > div_SO.
+       rewrite > sym_plus.
+       change in ⊢ (? ? %) with ((exp n n)+(S(S O)*(exp n n))).
+       apply le_plus_r.
+       apply (trans_le ? (((S(S O))*(exp n n)*(exp (S(S O)) n) - (S(S O))*(exp n n))/(exp (S(S O)) n)))
+        [apply sigma_p_div_exp
+        |apply le_times_to_le_div2
+          [apply lt_O_exp.
+           apply lt_O_S.
+          |apply le_minus_m
+          ]
+        ]
+      |reflexivity
+      ]
+    ]
+  ]
+qed.     
     
+theorem lt_exp_Sn_n_SSSO: \forall n.
+(exp (S n) n) < (S(S(S O)))*(exp n n).
+intro.cases n;intros
+  [simplify.apply le_S.apply le_n
+  |cases n1;intros
+    [simplify.apply le_n
+    |apply lt_SO_to_lt_exp_Sn_n_SSSO.
+     apply le_S_S.apply le_S_S.apply le_O_n
+    ]
+  ]
+qed.
+
+theorem lt_exp_Sn_m_SSSO: \forall n,m. O < m \to
+divides n m \to
+(exp (S n) m) < (exp (S(S(S O))) (m/n)) *(exp n m).
+intros.
+elim H1.rewrite > H2.
+rewrite < exp_exp_times.
+rewrite < exp_exp_times.
+rewrite > sym_times in ⊢ (? ? (? (? ? (? % ?)) ?)).
+rewrite > lt_O_to_div_times
+  [rewrite > times_exp.
+   apply lt_exp1
+    [apply (O_lt_times_to_O_lt ? n).
+     rewrite > sym_times.
+     rewrite < H2.
+     assumption
+    |apply lt_exp_Sn_n_SSSO
+    ]
+  |apply (O_lt_times_to_O_lt ? n2).
+   rewrite < H2.
+   assumption
+  ]
+qed.
+
+theorem le_log_exp_Sn_log_exp_n: \forall n,m,p. O < m \to S O < p \to
+divides n m \to
+log p (exp (S n) m) \le S((m/n)*S(log p (S(S(S O))))) + log p (exp n m).
+intros.
+apply (trans_le ? (log p (((S(S(S O))))\sup(m/n)*((n)\sup(m)))))
+  [apply le_log
+    [assumption
+    |apply lt_to_le.
+     apply lt_exp_Sn_m_SSSO;assumption
+    ]
+  |apply (trans_le ? (S(log p (exp (S(S(S O))) (m/n)) + log p (exp n m))))
+    [apply log_times.
+     assumption
+    |change in ⊢ (? ? %) with (S (m/n*S (log p (S(S(S O))))+log p ((n)\sup(m)))).
+     apply le_S_S.
+     apply le_plus_l.
+     apply log_exp1.
+     assumption
+    ]
+  ]
+qed.
+
+theorem le_log_exp_fact_sigma_p: \forall a,b,n,p. S O < p \to
+O < a \to a \le b \to b \le n \to
+log p (exp b n!) - log p (exp a n!) \le
+sigma_p b (\lambda i.leb a i) (\lambda i.S((n!/i)*S(log p (S(S(S O)))))).
+intros 7.
+elim b
+  [simplify.
+   apply (lt_O_n_elim ? (lt_O_fact n)).intro.
+   apply le_n.
+  |apply (bool_elim ? (leb a n1));intro
+    [rewrite > true_to_sigma_p_Sn
+      [apply (trans_le ? (S (n!/n1*S (log p (S(S(S O)))))+(log p ((n1)\sup(n!))-log p ((a)\sup(n!)))))
+        [rewrite > sym_plus. 
+         rewrite > plus_minus
+          [apply le_plus_to_minus_r.
+           rewrite < plus_minus_m_m
+            [rewrite > sym_plus.
+             apply le_log_exp_Sn_log_exp_n
+              [apply lt_O_fact
+              |assumption
+              |apply divides_fact;
+                 [apply (trans_le ? ? ? H1);apply leb_true_to_le;assumption
+                 |apply lt_to_le;assumption]]
+            |apply le_log
+              [assumption
+              |cut (O = exp O n!)
+                 [rewrite > Hcut;apply monotonic_exp1;constructor 2;
+                  apply leb_true_to_le;assumption
+                 |elim n
+                    [reflexivity
+                    |simplify;rewrite > exp_plus_times;rewrite < H6;
+                     rewrite > sym_times;rewrite < times_n_O;reflexivity]]]]
+        |apply le_log
+          [assumption
+          |apply monotonic_exp1;apply leb_true_to_le;assumption]]
+      |rewrite > sym_plus;rewrite > sym_plus in \vdash (? ? %);apply le_minus_to_plus;
+       rewrite < minus_plus_m_m;apply H3;apply lt_to_le;assumption]
+    |assumption]
+  |lapply (not_le_to_lt ? ? (leb_false_to_not_le ? ? H5));
+   rewrite > eq_minus_n_m_O
+    [apply le_O_n
+    |apply le_log
+       [assumption
+       |apply monotonic_exp1;assumption]]]]
+qed.
+
+theorem le_exp_div:\forall n,m,q. O < m \to 
+exp (n/m) q \le (exp n q)/(exp m q).
+intros.
+apply le_times_to_le_div
+  [apply lt_O_exp.
+   assumption
+  |rewrite > times_exp.
+   rewrite < sym_times.
+   apply monotonic_exp1.
+   rewrite > (div_mod n m) in ⊢ (? ? %)
+    [apply le_plus_n_r
+    |assumption
+    ]
+  ]
+qed.
+
+theorem le_log_div_sigma_p: \forall a,b,n,p. S O < p \to
+O < a \to a \le b \to b \le n \to
+log p (b/a) \le
+(sigma_p b (\lambda i.leb a i) (\lambda i.S((n!/i)*S(log p (S(S(S O)))))))/n!.
+intros.
+apply le_times_to_le_div
+  [apply lt_O_fact
+  |apply (trans_le ? (log p (exp (b/a) n!)))
+    [apply log_exp2
+      [assumption
+      |apply le_times_to_le_div
+        [assumption
+        |rewrite < times_n_SO.
+         assumption
+        ]
+      ]
+    |apply (trans_le ? (log p ((exp b n!)/(exp a n!)))) 
+      [apply le_log
+        [assumption
+        |apply le_exp_div.assumption
+        ]
+      |apply (trans_le ? (log p (exp b n!) - log p (exp a n!)))
+        [apply log_div
+          [assumption
+          |apply lt_O_exp.
+           assumption
+          |apply monotonic_exp1.
+           assumption
+          ]
+        |apply le_log_exp_fact_sigma_p;assumption
+        ]
+      ]
+    ]
+  ]
+qed.
+      
+theorem sigma_p_log_div1: \forall n,b. S O < b \to
+sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
+\le sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)*S(log b (S(S(S O)))))))/n!
+).
+intros.
+apply le_sigma_p.intros.
+apply le_log_div_sigma_p
+  [assumption
+  |apply prime_to_lt_O.
+   apply primeb_true_to_prime.
+   apply (andb_true_true ? ? H2)
+  |apply le_S_S_to_le.
+   assumption
+  |apply le_n
+  ]
+qed.
+
+theorem sigma_p_log_div2: \forall n,b. S O < b \to
+sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
+\le 
+(sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)))))*S(log b (S(S(S O))))/n!).
+intros.
+apply (trans_le ? (sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)*S(log b (S(S(S O)))))))/n!
+)))
+  [apply sigma_p_log_div1.assumption
+  |apply le_times_to_le_div
+    [apply lt_O_fact
+    |rewrite > distributive_times_plus_sigma_p.
+     rewrite < sym_times in ⊢ (? ? %).
+     rewrite > distributive_times_plus_sigma_p.
+     apply le_sigma_p.intros.
+     apply (trans_le ? ((n!*(sigma_p n (λj:nat.leb i j) (λi:nat.S (n!/i*S (log b (S(S(S O)))))))/n!)))
+      [apply le_times_div_div_times.
+       apply lt_O_fact
+      |rewrite > sym_times.
+       rewrite > lt_O_to_div_times
+        [rewrite > distributive_times_plus_sigma_p.
+         apply le_sigma_p.intros.
+         rewrite < times_n_Sm in ⊢ (? ? %).
+         rewrite > plus_n_SO.
+         rewrite > sym_plus.
+         apply le_plus
+          [apply le_S_S.apply le_O_n
+          |rewrite < sym_times.
+           apply le_n
+          ]
+        |apply lt_O_fact
+        ]
+      ]
+    ]
+  ]
+qed.
 
+theorem sigma_p_log_div: \forall n,b. S O < b \to
+sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
+\le (sigma_p n (\lambda i.leb (S n) (i*i)) (\lambda i.(prim i)*S(n!/i)))*S(log b (S(S(S O))))/n!
+.
+intros.
+apply (trans_le ? (sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)))))*S(log b (S(S(S O))))/n!))
+  [apply sigma_p_log_div2.assumption
+  |apply monotonic_div
+    [apply lt_O_fact
+    |apply le_times_l.
+     unfold prim.
+     cut
+     (sigma_p (S n) (λp:nat.primeb p∧leb (S n) (p*p))
+      (λp:nat.sigma_p n (λi:nat.leb p i) (λi:nat.S (n!/i)))
+     = sigma_p n (λi:nat.leb (S n) (i*i))
+       (λi:nat.sigma_p (S n) (\lambda p.primeb p \land leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i))))
+      [rewrite > Hcut.
+       apply le_sigma_p.intros.
+       rewrite < sym_times.
+       rewrite > distributive_times_plus_sigma_p.
+       rewrite < times_n_SO.
+       cut 
+        (sigma_p (S n) (λp:nat.primeb p∧leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i))
+         = sigma_p (S i) (\lambda p.primeb p \land leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i)))
+        [rewrite > Hcut1.
+         apply le_sigma_p1.intros.
+         rewrite < andb_sym.
+         rewrite < andb_sym in ⊢ (? (? (? (? ? %)) ?) ?).
+         apply (bool_elim  ? (leb i1 i));intros
+          [apply (bool_elim  ? (leb (S n) (i1*i1)));intros
+            [apply le_n
+            |apply le_O_n
+            ]
+          |apply le_O_n
+          ]
+        |apply or_false_to_eq_sigma_p
+          [apply le_S.assumption
+          |intros.
+           left.rewrite > (lt_to_leb_false i1 i)
+            [rewrite > andb_sym.reflexivity
+            |assumption
+            ]
+          ]
+        ]
+      |apply sigma_p_sigma_p.intros.
+       apply (bool_elim ? (leb x y));intros
+        [apply (bool_elim ? (leb (S n) (x*x)));intros
+          [rewrite > le_to_leb_true in ⊢ (? ? ? (? % ?))
+            [reflexivity
+            |apply (trans_le ? (x*x))
+              [apply leb_true_to_le.assumption
+              |apply le_times;apply leb_true_to_le;assumption
+              ]
+            ]
+          |rewrite < andb_sym in ⊢ (? ? (? % ?) ?).
+           rewrite < andb_sym in ⊢ (? ? ? (? ? (? % ?))).
+           rewrite < andb_sym in ⊢ (? ? ? %).
+           reflexivity
+          ]
+        |rewrite < andb_sym.
+         rewrite > andb_assoc in ⊢ (? ? ? %).
+         rewrite < andb_sym in ⊢ (? ? ? (? % ?)).
+         reflexivity
+        ]
+      ]
+    ]
+  ]
+qed.
+
+theorem le_sigma_p_div_log_div_pred_log : \forall n,b,m. S O < b \to b*b \leq n \to
+sigma_p (S n) (\lambda i.leb (S n) (i*i)) (\lambda i.m/(log b i))
+\leq ((S (S O)) * n * m)/(pred (log b n)).
+intros.
+apply (trans_le ? (sigma_p (S n) 
+             (\lambda i.leb (S n) (i*i)) (\lambda i.(S (S O))*m/(pred (log b n)))))
+  [apply le_sigma_p;intros;apply le_times_to_le_div
+     [rewrite > minus_n_O in ⊢ (? ? (? %));rewrite < eq_minus_S_pred;
+      apply le_plus_to_minus_r;simplify;
+      rewrite < (eq_log_exp b ? H);
+      apply le_log;
+          [assumption
+          |simplify;rewrite < times_n_SO;assumption]
+     |apply (trans_le ? ((pred (log b n) * m)/log b i))
+        [apply le_times_div_div_times;apply lt_O_log
+          [elim (le_to_or_lt_eq ? ? (le_O_n i))
+            [assumption
+            |apply False_ind;apply not_eq_true_false;rewrite < H3;rewrite < H4;
+             reflexivity]
+          |apply (le_exp_to_le1 ? ? (S (S O)))
+            [apply lt_O_S;
+            |apply (trans_le ? (S n))
+               [apply le_S;simplify;rewrite < times_n_SO;assumption
+               |rewrite > exp_SSO;apply leb_true_to_le;assumption]]]
+        |apply le_times_to_le_div2
+          [apply lt_O_log
+            [elim (le_to_or_lt_eq ? ? (le_O_n i))
+              [assumption
+              |apply False_ind;apply not_eq_true_false;rewrite < H3;rewrite < H4;
+               reflexivity]
+          |apply (le_exp_to_le1 ? ? (S (S O)))
+            [apply lt_O_S;
+            |apply (trans_le ? (S n))
+               [apply le_S;simplify;rewrite < times_n_SO;assumption
+               |rewrite > exp_SSO;apply leb_true_to_le;assumption]]]
+          |rewrite > sym_times in \vdash (? ? %);rewrite < assoc_times;
+           apply le_times_l;rewrite > sym_times;
+           rewrite > minus_n_O in \vdash (? (? %) ?);
+           rewrite < eq_minus_S_pred;apply le_plus_to_minus;
+           simplify;rewrite < plus_n_O;apply (trans_le ? (log b (i*i)))
+             [apply le_log
+                [assumption
+                |apply lt_to_le;apply leb_true_to_le;assumption]
+             |rewrite > sym_plus;simplify;apply log_times;assumption]]]]
+        |rewrite > times_n_SO in \vdash (? (? ? ? (\lambda i:?.%)) ?);
+         rewrite < distributive_times_plus_sigma_p;
+         apply (trans_le ? ((((S (S O))*m)/(pred (log b n)))*n))
+           [apply le_times_r;apply (trans_le ? (sigma_p (S n) (\lambda i:nat.leb (S O) (i*i)) (\lambda Hbeta1:nat.S O)))
+             [apply le_sigma_p1;intros;do 2 rewrite < times_n_SO;
+              apply (bool_elim ? (leb (S n) (i*i)))
+                [intro;cut (leb (S O) (i*i) = true)
+                  [rewrite > Hcut;apply le_n
+                  |apply le_to_leb_true;apply (trans_le ? (S n))
+                     [apply le_S_S;apply le_O_n
+                     |apply leb_true_to_le;assumption]]
+                |intro;simplify in \vdash (? % ?);apply le_O_n]
+             |elim n
+                [simplify;apply le_n
+                |apply (bool_elim ? (leb (S O) ((S n1)*(S n1))));intro
+                   [rewrite > true_to_sigma_p_Sn
+                      [change in \vdash (? % ?) with (S (sigma_p (S n1) (\lambda i:nat.leb (S O) (i*i)) (\lambda Hbeta1:nat.S O)));
+                       apply le_S_S;assumption
+                      |assumption]
+                   |rewrite > false_to_sigma_p_Sn
+                      [apply le_S;assumption
+                      |assumption]]]]
+          |rewrite > sym_times in \vdash (? % ?);
+           rewrite > sym_times in \vdash (? ? (? (? % ?) ?));
+           rewrite > assoc_times;
+           apply le_times_div_div_times;
+           rewrite > minus_n_O in ⊢ (? ? (? %));rewrite < eq_minus_S_pred;
+           apply le_plus_to_minus_r;simplify;
+           rewrite < (eq_log_exp b ? H);
+           apply le_log;
+             [assumption
+             |simplify;rewrite < times_n_SO;assumption]]]
+qed.         
+
+(* theorem le_log_exp_Sn_log_exp_n: \forall n,m,a,p. O < m \to S O < p \to
+divides n m \to
+log p (exp n m) - log p (exp a m) \le
+sigma_p (S n) (\lambda i.leb (S a) i) (\lambda i.S((m/i)*S(log p (S(S(S O)))))).
+intros.
+elim n
+  [rewrite > false_to_sigma_p_Sn.
+   simplify.
+   apply (lt_O_n_elim ? H).intro.
+   simplify.apply le_O_n
+  |apply (bool_elim ? (leb a n1));intro
+    [rewrite > true_to_sigma_p_Sn
+      [apply (trans_le ? (S (m/S n1*S (log p (S(S(S O)))))+(log p ((n1)\sup(m))-log p ((a)\sup(m)))))
+        [rewrite > sym_plus. 
+         rewrite > plus_minus
+          [apply le_plus_to_minus_r.
+           rewrite < plus_minus_m_m
+            [rewrite > sym_plus.
+             apply le_log_exp_Sn_log_exp_n.
+
+
+* a generalization 
+theorem le_exp_sigma_p_exp_m: \forall m,n. (exp (S m) n) \le
+sigma_p (S n) (\lambda k.true) (\lambda k.((exp m (n-k))*(exp n k))/(k!)).
+intros.
+rewrite > exp_S_sigma_p.
+apply le_sigma_p.
+intros.unfold bc.
+apply (trans_le ? ((exp m (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
+    ]
+  |apply le_times_div_div_times.
+   apply lt_O_fact
+  ]
+qed.
+
+theorem lt_exp_sigma_p_exp_m: \forall m,n. S O < n \to
+(exp (S m) n) <
+sigma_p (S n) (\lambda k.true) (\lambda k.((exp m (n-k))*(exp n k))/(k!)).
+intros.
+rewrite > exp_S_sigma_p.
+apply lt_sigma_p
+  [intros.unfold bc.
+   apply (trans_le ? ((exp m (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
+      ]
+    |apply le_times_div_div_times.
+     apply lt_O_fact
+    ]
+  |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.
+       rewrite < plus_n_O.
+       apply le_fact_exp1.
+       assumption
+      ]
+    ]
+  ]
+qed.
    
+theorem lt_SO_to_lt_exp_Sn_n_SSSO_bof: \forall m,n. S O < n \to 
+(exp (S m) n) < (S(S(S O)))*(exp m n).
+intros.  
+apply (lt_to_le_to_lt ? (sigma_p (S n) (\lambda k.true) (\lambda k.((exp m (n-k))*(exp n k))/(k!))))
+  [apply lt_exp_sigma_p_exp_m.assumption
+  |apply (trans_le ? (sigma_p (S n) (\lambda i.true) (\lambda i.(exp n n)/(exp (S(S O)) (pred i)))))
+    [apply le_sigma_p.intros.
+     apply le_times_to_le_div
+      [apply lt_O_exp.
+       apply lt_O_S
+      |apply (trans_le ? ((S(S O))\sup (pred i)* n \sup n/i!))
+        [apply le_times_div_div_times.
+         apply lt_O_fact
+        |apply le_times_to_le_div2
+          [apply lt_O_fact
+          |rewrite < sym_times.
+           apply le_times_r.
+           apply le_exp_SSO_fact
+          ]
+        ]
+      ]
+    |rewrite > eq_sigma_p_pred
+      [rewrite > div_SO.
+       rewrite > sym_plus.
+       change in ⊢ (? ? %) with ((exp n n)+(S(S O)*(exp n n))).
+       apply le_plus_r.
+       apply (trans_le ? (((S(S O))*(exp n n)*(exp (S(S O)) n) - (S(S O))*(exp n n))/(exp (S(S O)) n)))
+        [apply sigma_p_div_exp
+        |apply le_times_to_le_div2
+          [apply lt_O_exp.
+           apply lt_O_S.
+          |apply le_minus_m
+          ]
+        ]
+      |reflexivity
+      ]
+    ]
+  ]
+qed.     
+*)
\ No newline at end of file