(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/factorial2".
-
include "nat/exp.ma".
include "nat/factorial.ma".
intros.reflexivity.
qed.
-lemma times_SSO: \forall n.(S(S O))*(S n) = S(S((S(S O))*n)).
-intro.simplify.rewrite < plus_n_Sm.reflexivity.
-qed.
-
-theorem fact1: \forall n.
-fact ((S(S O))*n) \le (exp (S(S O)) ((S(S O))*n))*(fact n)*(fact n).
-intro.elim n
- [rewrite < times_n_O.apply le_n
+theorem lt_O_to_fact1: \forall n.O<n \to
+fact (2*n) \le (exp 2 (pred (2*n)))*(fact n)*(fact n).
+intros.elim H
+ [apply le_n
|rewrite > times_SSO.
rewrite > factS.
rewrite > factS.
rewrite < assoc_times.
rewrite > factS.
- apply (trans_le ? (((S(S O))*(S n1))*((S(S O))*(S n1))*(fact (((S(S O))*n1)))))
+ apply (trans_le ? ((2*(S n1))*(2*(S n1))*(fact (2*n1))))
[apply le_times_l.
rewrite > times_SSO.
apply le_times_r.
|rewrite > assoc_times.
rewrite > assoc_times.
rewrite > assoc_times in ⊢ (? ? %).
+ change in ⊢ (? ? (? (? ? %) ?)) with (S(2*n1)).
rewrite > exp_S.
rewrite > assoc_times in ⊢ (? ? %).
apply le_times_r.
rewrite < sym_times in ⊢ (? (? (? % ?) ?) ?).
rewrite > assoc_times.
rewrite > assoc_times.
- rewrite > exp_S.
- rewrite > assoc_times in ⊢ (? ? %).
- apply le_times_r.
- rewrite > sym_times in ⊢ (? ? %).
- rewrite > assoc_times in ⊢ (? ? %).
- rewrite > assoc_times in ⊢ (? ? %).
- apply le_times_r.
- rewrite < assoc_times in ⊢ (? ? %).
- rewrite < assoc_times in ⊢ (? ? %).
- rewrite < sym_times in ⊢ (? ? (? (? % ?) ?)).
- rewrite > assoc_times in ⊢ (? ? %).
- rewrite > assoc_times in ⊢ (? ? %).
- apply le_times_r.
- rewrite > sym_times in ⊢ (? ? (? ? %)).
- rewrite > sym_times in ⊢ (? ? %).
- assumption
+ rewrite > S_pred in ⊢ (? ? (? (? ? %) ?))
+ [rewrite > exp_S.
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite > sym_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < sym_times in ⊢ (? ? (? (? % ?) ?)).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite > sym_times in ⊢ (? ? (? ? %)).
+ rewrite > sym_times in ⊢ (? ? %).
+ assumption
+ |unfold.rewrite > times_n_SO in ⊢ (? % ?).
+ apply le_times
+ [apply le_n_Sn
+ |assumption
+ ]
+ ]
]
]
qed.
+theorem fact1: \forall n.
+fact (2*n) \le (exp 2 (pred (2*n)))*(fact n)*(fact n).
+intro.cases n
+ [apply le_n
+ |apply lt_O_to_fact1.
+ apply lt_O_S
+ ]
+qed.
+
theorem lt_O_fact: \forall n. O < fact n.
intro.elim n
[simplify.apply lt_O_S
qed.
theorem fact2: \forall n.O < n \to
-(exp (S(S O)) ((S(S O))*n))*(fact n)*(fact n) < fact (S((S(S O))*n)).
+(exp 2 (2*n))*(fact n)*(fact n) < fact (S(2*n)).
intros.elim H
[simplify.apply le_S.apply le_n
|rewrite > times_SSO.
]
qed.
+(* a slightly better result *)
+theorem fact3: \forall n.O < n \to
+(exp 2 (2*n))*(exp (fact n) 2) \le 2*n*fact (2*n).
+intros.
+elim H
+ [simplify.apply le_n
+ |rewrite > times_SSO.
+ rewrite > factS.
+ rewrite < times_exp.
+ change in ⊢ (? (? % ?) ?) with ((S(S O))*((S(S O))*(exp (S(S O)) ((S(S O))*n1)))).
+ rewrite > assoc_times.
+ rewrite > assoc_times in ⊢ (? (? ? %) ?).
+ rewrite < assoc_times in ⊢ (? (? ? (? ? %)) ?).
+ rewrite < sym_times in ⊢ (? (? ? (? ? (? % ?))) ?).
+ rewrite > assoc_times in ⊢ (? (? ? (? ? %)) ?).
+ apply (trans_le ? (((S(S O))*((S(S O))*((S n1)\sup((S(S O)))*((S(S O))*n1*((S(S O))*n1)!))))))
+ [apply le_times_r.
+ apply le_times_r.
+ apply le_times_r.
+ assumption
+ |rewrite > factS.
+ rewrite > factS.
+ rewrite < times_SSO.
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite < assoc_times.
+ change in ⊢ (? (? (? ? %) ?) ?) with ((S n1)*((S n1)*(S O))).
+ rewrite < assoc_times in ⊢ (? (? % ?) ?).
+ rewrite < times_n_SO.
+ rewrite > sym_times in ⊢ (? (? (? % ?) ?) ?).
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < assoc_times in ⊢ (? ? (? % ?)).
+ apply le_times_r.
+ apply le_times_l.
+ apply le_S.apply le_n
+ ]
+ ]
+qed.
+
+theorem le_fact_10: fact (2*5) \le (exp 2 ((2*5)-2))*(fact 5)*(fact 5).
+simplify in \vdash (? (? %) ?).
+rewrite > factS in \vdash (? % ?).
+rewrite > factS in \vdash (? % ?).rewrite < assoc_times in \vdash(? % ?).
+rewrite > factS in \vdash (? % ?).rewrite < assoc_times in \vdash (? % ?).
+rewrite > factS in \vdash (? % ?).rewrite < assoc_times in \vdash (? % ?).
+rewrite > factS in \vdash (? % ?).rewrite < assoc_times in \vdash (? % ?).
+apply le_times_l.
+apply leb_true_to_le.reflexivity.
+qed.
+
+theorem ab_times_cd: \forall a,b,c,d.(a*b)*(c*d)=(a*c)*(b*d).
+intros.
+rewrite > assoc_times.
+rewrite > assoc_times.
+apply eq_f.
+rewrite < assoc_times.
+rewrite < assoc_times.
+rewrite > sym_times in \vdash (? ? (? % ?) ?).
+reflexivity.
+qed.
+
+(* an even better result *)
+theorem lt_SSSSO_to_fact: \forall n.4<n \to
+fact (2*n) \le (exp 2 ((2*n)-2))*(fact n)*(fact n).
+intros.elim H
+ [apply le_fact_10
+ |rewrite > times_SSO.
+ change in \vdash (? ? (? (? (? ? %) ?) ?)) with (2*n1 - O);
+ rewrite < minus_n_O.
+ rewrite > factS.
+ rewrite > factS.
+ rewrite < assoc_times.
+ rewrite > factS.
+ apply (trans_le ? ((2*(S n1))*(2*(S n1))*(fact (2*n1))))
+ [apply le_times_l.
+ rewrite > times_SSO.
+ apply le_times_r.
+ apply le_n_Sn
+ |apply (trans_le ? (2*S n1*(2*S n1)*(2\sup(2*n1-2)*n1!*n1!)))
+ [apply le_times_r.assumption
+ |rewrite > assoc_times.
+ rewrite > ab_times_cd in \vdash (? (? ? %) ?).
+ rewrite < assoc_times.
+ apply le_times_l.
+ rewrite < assoc_times in \vdash (? (? ? %) ?).
+ rewrite > ab_times_cd.
+ apply le_times_l.
+ rewrite < exp_S.
+ rewrite < exp_S.
+ apply le_exp
+ [apply lt_O_S
+ |rewrite > eq_minus_S_pred.
+ rewrite < S_pred
+ [rewrite > eq_minus_S_pred.
+ rewrite < S_pred
+ [rewrite < minus_n_O.
+ apply le_n
+ |elim H1;simplify
+ [apply lt_O_S
+ |apply lt_O_S
+ ]
+ ]
+ |elim H1;simplify
+ [apply lt_O_S
+ |rewrite < plus_n_Sm.
+ rewrite < minus_n_O.
+ apply lt_O_S
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+
(*
theorem stirling: \forall n,k.k \le n \to
log (fact n) < n*log n - n + k*log n.
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