apply le_log;
[assumption
|simplify;rewrite < times_n_SO;assumption]]]
-qed.
+qed.
+
+lemma neper_sigma_p1 : \forall n,a.n \divides a \to
+exp (a * S n) n =
+sigma_p (S n) (\lambda x.true) (\lambda k.(bc n k)*(exp n (n-k))*(exp a n)).
+intros;rewrite < times_exp;rewrite > exp_S_sigma_p;
+rewrite > distributive_times_plus_sigma_p;
+apply eq_sigma_p;intros;
+ [reflexivity
+ |rewrite > sym_times;reflexivity;]
+qed.
+
+lemma eq_exp_pi_p : \forall a,n.(exp a n) = pi_p n (\lambda x.true) (\lambda x.a).
+intros;elim n
+ [simplify;reflexivity
+ |change in \vdash (? ? % ?) with (a*exp a n1);rewrite > true_to_pi_p_Sn
+ [apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]
+qed.
+lemma eq_fact_pi_p : \forall n.n! = pi_p n (\lambda x.true) (\lambda x.S x).
+intros;elim n
+ [simplify;reflexivity
+ |rewrite > true_to_pi_p_Sn
+ [change in \vdash (? ? % ?) with (S n1*n1!);apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]
+qed.
+
+lemma divides_pi_p : \forall m,n,p,f.m \leq n \to pi_p m p f \divides pi_p n p f.
+intros;elim H
+ [apply divides_n_n
+ |apply (bool_elim ? (p n1));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > sym_times;rewrite > times_n_SO;apply divides_times
+ [assumption
+ |apply divides_SO_n]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]
+qed.
+
+lemma divides_fact_fact : \forall m,n.m \leq n \to m! \divides n!.
+intros;do 2 rewrite > eq_fact_pi_p;apply divides_pi_p;assumption.
+qed.
+
+lemma divides_times_to_eq : \forall a,b,c.O < c \to c \divides a \to a*b/c = a/c*b.
+intros;elim H1;rewrite > H2;cases H;rewrite > assoc_times;do 2 rewrite > div_times;
+reflexivity;
+qed.
+
+lemma divides_pi_p_to_eq : \forall k,p,f,g.(\forall x.p x = true \to O < g x \land (g x \divides f x)) \to
+pi_p k p f/pi_p k p g = pi_p k p (\lambda x.(f x)/(g x)).
+intros;
+cut (\forall k1.(pi_p k1 p g \divides pi_p k1 p f))
+ [|intro;elim k1
+ [simplify;apply divides_n_n
+ |apply (bool_elim ? (p n));intro;
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [elim (H n)
+ [elim H4;elim H1;rewrite > H5;rewrite > H6;
+ rewrite < assoc_times;rewrite > assoc_times in ⊢ (? ? (? % ?));
+ rewrite > sym_times in ⊢ (? ? (? (? ? %) ?));
+ rewrite > assoc_times;rewrite > assoc_times;
+ apply divides_times
+ [apply divides_n_n
+ |rewrite > times_n_SO in \vdash (? % ?);apply divides_times
+ [apply divides_n_n
+ |apply divides_SO_n]]
+ |assumption]
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn
+ [assumption
+ |assumption]
+ |assumption]]]]
+elim k
+ [simplify;reflexivity
+ |apply (bool_elim ? (p n))
+ [intro;rewrite > true_to_pi_p_Sn;
+ [rewrite > true_to_pi_p_Sn;
+ [rewrite > true_to_pi_p_Sn;
+ [elim (H n);
+ [elim H4;rewrite > H5;rewrite < eq_div_div_div_times;
+ [cases H3
+ [rewrite > assoc_times;do 2 rewrite > div_times;
+ elim (Hcut n);rewrite > H6;rewrite < assoc_times;
+ rewrite < sym_times in \vdash (? ? (? (? % ?) ?) ?);
+ cut (O < pi_p n p g)
+ [rewrite < H1;rewrite > H6;cases Hcut1;
+ rewrite > assoc_times;do 2 rewrite > div_times;reflexivity
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n3));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n3);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]
+ |rewrite > assoc_times;do 2 rewrite > div_times;
+ elim (Hcut n);rewrite > H7;rewrite < assoc_times;
+ rewrite < sym_times in \vdash (? ? (? (? % ?) ?) ?);
+ cut (O < pi_p n p g)
+ [rewrite < H1;rewrite > H7;cases Hcut1;
+ rewrite > assoc_times;do 2 rewrite > div_times;reflexivity
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n3));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n3);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]]
+ |assumption
+ |(*già usata 2 volte: fattorizzare*)
+ elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n1));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n1);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]
+ |assumption]
+ |assumption]
+ |assumption]
+ |assumption]
+ |intro;rewrite > (false_to_pi_p_Sn ? ? ? H2);
+ rewrite > (false_to_pi_p_Sn ? ? ? H2);rewrite > (false_to_pi_p_Sn ? ? ? H2);
+ assumption]]
+qed.
+
+lemma divides_times_to_divides_div : \forall a,b,c.O < b \to
+ a*b \divides c \to a \divides c/b.
+intros;elim H1;rewrite > H2;rewrite > sym_times in \vdash (? ? (? (? % ?) ?));
+rewrite > assoc_times;cases H;rewrite > div_times;rewrite > times_n_SO in \vdash (? % ?);
+apply divides_times
+ [1,3:apply divides_n_n
+ |*:apply divides_SO_n]
+qed.
+
+lemma neper_sigma_p2 : \forall n,a.O < n \to n \divides a \to
+sigma_p (S n) (\lambda x.true) (\lambda k.((bc n k)*(exp a n)*(exp n (n-k)))/(exp n n))
+= sigma_p (S n) (\lambda x.true)
+(\lambda k.(exp a (n-k))*(pi_p k (\lambda y.true) (\lambda i.a - (a*i/n)))/k!).
+intros;apply eq_sigma_p;intros;
+ [reflexivity
+ |transitivity (bc n x*exp a n/exp n x)
+ [rewrite > minus_n_O in ⊢ (? ? (? ? (? ? %)) ?);
+ rewrite > (minus_n_n x);
+ rewrite < (eq_plus_minus_minus_minus n x x);
+ [rewrite > exp_plus_times;
+ rewrite > sym_times;rewrite > sym_times in \vdash (? ? (? ? %) ?);
+ rewrite < eq_div_div_times;
+ [reflexivity
+ |*:apply lt_O_exp;assumption]
+ |apply le_n
+ |apply le_S_S_to_le;assumption]
+ |rewrite > minus_n_O in ⊢ (? ? (? (? ? (? ? %)) ?) ?);
+ rewrite > (minus_n_n x);
+ rewrite < (eq_plus_minus_minus_minus n x x);
+ [rewrite > exp_plus_times;
+ unfold bc;
+ elim (bc2 n x)
+ [rewrite > H3;cut (x!*n2 = pi_p x (\lambda i.true) (\lambda i.(n - i)))
+ [rewrite > sym_times in ⊢ (? ? (? (? (? (? % ?) ?) ?) ?) ?);
+ rewrite > assoc_times;rewrite > sym_times in ⊢ (? ? (? (? (? ? %) ?) ?) ?);
+ rewrite < eq_div_div_times
+ [rewrite > Hcut;rewrite < assoc_times;
+ cut (pi_p x (λi:nat.true) (λi:nat.n-i)/x!*(a)\sup(x)
+ = pi_p x (λi:nat.true) (λi:nat.(n-i))*pi_p x (\lambda i.true) (\lambda i.a)/x!)
+ [rewrite > Hcut1;rewrite < times_pi_p;
+ rewrite > divides_times_to_eq in \vdash (? ? % ?);
+ [rewrite > eq_div_div_div_times;
+ [rewrite > sym_times in ⊢ (? ? (? (? ? %) ?) ?);
+ rewrite < eq_div_div_div_times;
+ [cut (exp n x = pi_p x (\lambda i.true) (\lambda i.n))
+ [rewrite > Hcut2;rewrite > divides_pi_p_to_eq
+ [rewrite > sym_times in \vdash (? ? ? %);
+ rewrite > divides_times_to_eq in \vdash (? ? ? %);
+ [apply eq_f2
+ [apply eq_f2
+ [apply eq_pi_p;intros
+ [reflexivity
+ |rewrite > sym_times;
+ rewrite > distr_times_minus;elim H1;
+ rewrite > H5;(* in ⊢ (? ? (? (? ? (? % ?)) ?) ?);*)
+ rewrite > sym_times;rewrite > assoc_times;
+ rewrite < distr_times_minus;
+ generalize in match H;cases n;intros
+ [elim (not_le_Sn_O ? H6)
+ |do 2 rewrite > div_times;reflexivity]]
+ |reflexivity]
+ |reflexivity]
+ |apply lt_O_fact
+ |cut (pi_p x (λy:nat.true) (λi:nat.a-a*i/n) = (exp a x)/(exp n x)*(n!/(n-x)!))
+ [rewrite > Hcut3;rewrite > times_n_SO;
+ rewrite > sym_times;apply divides_times
+ [apply divides_SO_n;
+ |apply divides_times_to_divides_div;
+ [apply lt_O_fact
+ |apply bc2;apply le_S_S_to_le;assumption]]
+ |cut (pi_p x (\lambda y.true) (\lambda i. a - a*i/n) =
+ pi_p x (\lambda y.true) (\lambda i. a*(n-i)/n))
+ [rewrite > Hcut3;
+ rewrite < (divides_pi_p_to_eq ? ? (\lambda i.(a*(n-i))) (\lambda i.n))
+ [rewrite > (times_pi_p ? ? (\lambda i.a) (\lambda i.(n-i)));
+ rewrite > divides_times_to_eq;
+ [apply eq_f2
+ [apply eq_f2;rewrite < eq_exp_pi_p;reflexivity
+ |rewrite < Hcut;rewrite > H3;
+ rewrite < sym_times in ⊢ (? ? ? (? (? % ?) ?));
+ rewrite > (S_pred ((n-x)!));
+ [rewrite > assoc_times;
+ rewrite > div_times;reflexivity
+ |apply lt_O_fact]]
+ |unfold lt;cut (1 = pi_p x (\lambda y.true) (\lambda y.1))
+ [rewrite > Hcut4;apply le_pi_p;intros;assumption
+ |elim x
+ [simplify;reflexivity
+ |rewrite > true_to_pi_p_Sn;
+ [rewrite < H4;reflexivity
+ |reflexivity]]]
+ |elim x
+ [simplify;apply divides_SO_n
+ |rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [apply divides_times;assumption
+ |reflexivity]
+ |reflexivity]]]
+ |intros;split
+ [assumption
+ |rewrite > times_n_SO;apply divides_times
+ [assumption
+ |apply divides_SO_n]]]
+ |apply eq_pi_p;intros;
+ [reflexivity
+ |elim H1;rewrite > H5;rewrite > (S_pred n);
+ [rewrite > assoc_times;
+ rewrite > assoc_times;
+ rewrite > div_times;
+ rewrite > div_times;
+ rewrite > distr_times_minus;
+ rewrite > sym_times;
+ reflexivity
+ |assumption]]]]]
+ |intros;split
+ [assumption
+ |rewrite > sym_times;rewrite > times_n_SO;
+ apply divides_times
+ [assumption
+ |apply divides_SO_n]]]
+ |rewrite < eq_exp_pi_p;reflexivity]
+ |apply lt_O_exp;assumption
+ |apply lt_O_fact]
+ |apply lt_O_fact
+ |apply lt_O_exp;assumption]
+ |apply lt_O_exp;assumption
+ |rewrite > (times_pi_p ? ? (\lambda x.(n-x)) (\lambda x.a));
+ rewrite > divides_times_to_eq
+ [rewrite > times_n_SO;rewrite > sym_times;apply divides_times
+ [apply divides_SO_n
+ |elim x
+ [simplify;apply divides_SO_n
+ |change in \vdash (? % ?) with (n*(exp n n1));
+ rewrite > true_to_pi_p_Sn
+ [apply divides_times;assumption
+ |reflexivity]]]
+ |apply lt_O_fact
+ |apply (witness ? ? n2);symmetry;assumption]]
+ |rewrite > divides_times_to_eq;
+ [apply eq_f2
+ [reflexivity
+ |elim x
+ [simplify;reflexivity
+ |change in \vdash (? ? % ?) with (a*(exp a n1));
+ rewrite > true_to_pi_p_Sn
+ [apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]]
+ |apply lt_O_fact
+ |apply (witness ? ? n2);symmetry;assumption]]
+ |apply lt_O_fact
+ |apply lt_O_fact]
+ |apply (inj_times_r (pred ((n-x)!)));
+ rewrite < (S_pred ((n-x)!))
+ [rewrite < assoc_times;rewrite < sym_times in \vdash (? ? (? % ?) ?);
+ rewrite < H3;generalize in match H2;elim x
+ [rewrite < minus_n_O;simplify;rewrite < times_n_SO;reflexivity
+ |rewrite < fact_minus in H4;
+ [rewrite > true_to_pi_p_Sn
+ [rewrite < assoc_times;rewrite > sym_times in \vdash (? ? ? (? % ?));
+ apply H4;apply lt_to_le;assumption
+ |reflexivity]
+ |apply le_S_S_to_le;assumption]]
+ |apply lt_O_fact]]
+ |apply le_S_S_to_le;assumption]
+ |apply le_n
+ |apply le_S_S_to_le;assumption]]]
+qed.
+
+lemma divides_sigma_p_to_eq : \forall k,p,f,b.O < b \to
+ (\forall x.p x = true \to b \divides f x) \to
+ (sigma_p k p f)/b = sigma_p k p (\lambda x. (f x)/b).
+intros;cut (\forall k1.b \divides sigma_p k1 p f)
+ [|intro;elim k1
+ [simplify;apply (witness ? ? O);rewrite < times_n_O;reflexivity
+ |apply (bool_elim ? (p n));intro
+ [rewrite > true_to_sigma_p_Sn;
+ [elim (H1 n);
+ [elim H2;rewrite > H4;rewrite > H5;rewrite < distr_times_plus;
+ rewrite > times_n_SO in \vdash (? % ?);apply divides_times
+ [apply divides_n_n
+ |apply divides_SO_n]
+ |assumption]
+ |assumption]
+ |rewrite > false_to_sigma_p_Sn;assumption]]]
+elim k
+ [cases H;simplify;reflexivity
+ |apply (bool_elim ? (p n));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [elim (H1 n);
+ [elim (Hcut n);rewrite > H4;rewrite > H5;rewrite < distr_times_plus;
+ rewrite < H2;rewrite > H5;cases H;do 3 rewrite > div_times;
+ reflexivity
+ |assumption]
+ |assumption]
+ |assumption]
+ |do 2 rewrite > false_to_sigma_p_Sn;assumption]]
+qed.
+
+lemma neper_sigma_p3 : \forall a,n.O < a \to O < n \to n \divides a \to (exp (S n) n)/(exp n n) =
+sigma_p (S n) (\lambda x.true)
+(\lambda k.(exp a (n-k))*(pi_p k (\lambda y.true) (\lambda i.a - (a*i/n)))/k!)/(exp a n).
+intros;transitivity ((exp a n)*(exp (S n) n)/(exp n n)/(exp a n))
+ [rewrite > eq_div_div_div_times
+ [rewrite > sym_times in \vdash (? ? ? (? ? %));rewrite < eq_div_div_times;
+ [reflexivity
+ |apply lt_O_exp;assumption
+ |apply lt_O_exp;assumption]
+ |apply lt_O_exp;assumption
+ |apply lt_O_exp;assumption]
+ |apply eq_f2;
+ [rewrite > times_exp;rewrite > neper_sigma_p1
+ [transitivity (sigma_p (S n) (λx:nat.true) (λk:nat.bc n k*(a)\sup(n)*(exp n (n-k))/(exp n n)))
+ [elim H2;rewrite > H3;rewrite < times_exp;rewrite > sym_times in ⊢ (? ? (? (? ? ? (λ_:?.? ? %)) ?) ?);
+ rewrite > sym_times in ⊢ (? ? ? (? ? ? (λ_:?.? (? (? ? %) ?) ?)));
+ transitivity (sigma_p (S n) (λx:nat.true)
+(λk:nat.(exp n n)*(bc n k*(n)\sup(n-k)*(n2)\sup(n)))/exp n n)
+ [apply eq_f2
+ [apply eq_sigma_p;intros;
+ [reflexivity
+ |rewrite < assoc_times;rewrite > sym_times;reflexivity]
+ |reflexivity]
+ |rewrite < (distributive_times_plus_sigma_p ? ? ? (\lambda k.bc n k*(exp n (n-k))*(exp n2 n)));
+ transitivity (sigma_p (S n) (λx:nat.true)
+ (λk:nat.bc n k*(n2)\sup(n)*(n)\sup(n-k)))
+ [rewrite > (S_pred (exp n n))
+ [rewrite > div_times;apply eq_sigma_p;intros
+ [reflexivity
+ |rewrite > sym_times;rewrite > sym_times in ⊢ (? ? ? (? % ?));
+ rewrite > assoc_times in \vdash (? ? ? %);reflexivity]
+ |apply lt_O_exp;assumption]
+ |apply eq_sigma_p;intros
+ [reflexivity
+ |rewrite < assoc_times;rewrite > assoc_times in \vdash (? ? ? %);
+ rewrite > sym_times in \vdash (? ? ? (? (? ? %) ?));
+ rewrite < assoc_times;rewrite > sym_times in \vdash (? ? ? %);
+ rewrite > (S_pred (exp n n))
+ [rewrite > div_times;reflexivity
+ |apply lt_O_exp;assumption]]]]
+ |rewrite > neper_sigma_p2;
+ [reflexivity
+ |assumption
+ |assumption]]
+ |assumption]
+ |reflexivity]]
+qed.
+
+theorem neper_monotonic : \forall n. O < n \to
+(exp (S n) n)/(exp n n) \leq (exp (S (S n)) (S n))/(exp (S n) (S n)).
+intros;rewrite > (neper_sigma_p3 (n*S n) n)
+ [rewrite > (neper_sigma_p3 (n*S n) (S n))
+ [change in ⊢ (? ? (? ? %)) with ((n * S n)*(exp (n * S n) n));
+ rewrite < eq_div_div_div_times
+ [apply monotonic_div;
+ [apply lt_O_exp;rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |apply le_times_to_le_div
+ [rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |rewrite > distributive_times_plus_sigma_p;
+ apply (trans_le ? (sigma_p (S n) (λx:nat.true)
+ (λk:nat.((n*S n))\sup(S n-k)*pi_p k (λy:nat.true) (λi:nat.n*S n-n*S n*i/S n)/k!)))
+ [apply le_sigma_p;intros;rewrite > sym_times in ⊢ (? (? ? %) ?);
+ rewrite > sym_times in \vdash (? ? (? % ?));
+ rewrite > divides_times_to_eq in \vdash (? ? %)
+ [rewrite > divides_times_to_eq in \vdash (? % ?)
+ [rewrite > sym_times in \vdash (? (? ? %) ?);
+ rewrite < assoc_times;
+ rewrite > sym_times in \vdash (? ? %);
+ rewrite > minus_Sn_m;
+ [apply le_times_r;apply monotonic_div
+ [apply lt_O_fact
+ |apply le_pi_p;intros;apply monotonic_le_minus_r;
+ rewrite > assoc_times in \vdash (? % ?);
+ rewrite > sym_times in \vdash (? % ?);
+ rewrite > assoc_times in \vdash (? % ?);
+ rewrite > div_times;
+ rewrite > (S_pred n) in \vdash (? ? %);
+ [rewrite > assoc_times;rewrite > div_times;
+ rewrite < S_pred
+ [rewrite > sym_times;apply le_times_l;apply le_S;apply le_n
+ |assumption]
+ |assumption]]
+ |apply le_S_S_to_le;assumption]
+ |apply lt_O_fact
+ |cut (pi_p i (λy:nat.true) (λi:nat.n*S n-n*S n*i/n) =
+ pi_p i (\lambda y.true) (\lambda i.S n) *
+ pi_p i (\lambda y.true) (\lambda i.(n-i)))
+ [rewrite > Hcut;rewrite > times_n_SO;
+ rewrite > sym_times;apply divides_times
+ [apply divides_SO_n
+ |elim (bc2 n i);
+ [apply (witness ? ? n2);
+ cut (pi_p i (\lambda y.true) (\lambda i.n-i) = (n!/(n-i)!))
+ [rewrite > Hcut1;rewrite > H3;rewrite > sym_times in ⊢ (? ? (? (? % ?) ?) ?);
+ rewrite > (S_pred ((n-i)!))
+ [rewrite > assoc_times;rewrite > div_times;
+ reflexivity
+ |apply lt_O_fact]
+ |generalize in match H1;elim i
+ [rewrite < minus_n_O;rewrite > div_n_n;
+ [reflexivity
+ |apply lt_O_fact]
+ |rewrite > true_to_pi_p_Sn
+ [rewrite > H4
+ [rewrite > sym_times;rewrite < divides_times_to_eq
+ [rewrite < fact_minus
+ [rewrite > sym_times;
+ rewrite < eq_div_div_times
+ [reflexivity
+ |apply lt_to_lt_O_minus;apply le_S_S_to_le;
+ assumption
+ |apply lt_O_fact;]
+ |apply le_S_S_to_le;assumption]
+ |apply lt_O_fact
+ |apply divides_fact_fact;
+ apply le_plus_to_minus;
+ rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n]
+ |apply lt_to_le;assumption]
+ |reflexivity]]]
+ |apply le_S_S_to_le;assumption]]
+ |rewrite < times_pi_p;apply eq_pi_p;intros;
+ [reflexivity
+ |rewrite > distr_times_minus;rewrite > assoc_times;
+ rewrite > (S_pred n) in \vdash (? ? (? ? (? (? % ?) %)) ?)
+ [rewrite > div_times;rewrite > sym_times;reflexivity
+ |assumption]]]]
+ |apply lt_O_fact
+ |cut (pi_p i (λy:nat.true) (λi:nat.n*S n-n*S n*i/S n) =
+ pi_p i (\lambda y.true) (\lambda i.n) *
+ pi_p i (\lambda y.true) (\lambda i.(S n-i)))
+ [rewrite > Hcut;rewrite > times_n_SO;rewrite > sym_times;
+ apply divides_times
+ [apply divides_SO_n
+ |elim (bc2 (S n) i);
+ [apply (witness ? ? n2);
+ cut (pi_p i (\lambda y.true) (\lambda i.S n-i) = ((S n)!/(S n-i)!))
+ [rewrite > Hcut1;rewrite > H3;rewrite > sym_times in ⊢ (? ? (? (? % ?) ?) ?);
+ rewrite > (S_pred ((S n-i)!))
+ [rewrite > assoc_times;rewrite > div_times;
+ reflexivity
+ |apply lt_O_fact]
+ |generalize in match H1;elim i
+ [rewrite < minus_n_O;rewrite > div_n_n;
+ [reflexivity
+ |apply lt_O_fact]
+ |rewrite > true_to_pi_p_Sn
+ [rewrite > H4
+ [rewrite > sym_times;rewrite < divides_times_to_eq
+ [rewrite < fact_minus
+ [rewrite > sym_times;
+ rewrite < eq_div_div_times
+ [reflexivity
+ |apply lt_to_lt_O_minus;apply lt_to_le;
+ assumption
+ |apply lt_O_fact]
+ |apply lt_to_le;assumption]
+ |apply lt_O_fact
+ |apply divides_fact_fact;
+ apply le_plus_to_minus;
+ rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n]
+ |apply lt_to_le;assumption]
+ |reflexivity]]]
+ |apply lt_to_le;assumption]]
+ |rewrite < times_pi_p;apply eq_pi_p;intros;
+ [reflexivity
+ |rewrite > distr_times_minus;rewrite > sym_times in \vdash (? ? (? ? (? (? % ?) ?)) ?);
+ rewrite > assoc_times;rewrite > div_times;reflexivity]]]
+ |rewrite > true_to_sigma_p_Sn in \vdash (? ? %)
+ [rewrite > sym_plus;rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n
+ |reflexivity]]]]
+ |rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |apply lt_O_exp;rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]]
+ |rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |apply lt_O_S
+ |apply (witness ? ? n);apply sym_times]
+ |rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |assumption
+ |apply (witness ? ? (S n));reflexivity]
+qed.
+
+theorem le_SSO_neper : \forall n.O < n \to (2 \leq (exp (S n) n)/(exp n n)).
+intros;elim H
+ [simplify;apply le_n
+ |apply (trans_le ? ? ? H2);apply neper_monotonic;assumption]
+qed.
+
+theorem le_SSO_exp_neper : \forall n.O < n \to 2*(exp n n) \leq exp (S n) n.
+intros;apply (trans_le ? ((exp (S n) n)/(exp n n)*(exp n n)))
+ [apply le_times_l;apply le_SSO_neper;assumption
+ |rewrite > sym_times;apply (trans_le ? ? ? (le_times_div_div_times ? ? ? ?))
+ [apply lt_O_exp;assumption
+ |cases (lt_O_exp ? n H);rewrite > div_times;apply le_n]]
+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
]
]
qed.
-*)
\ No newline at end of file
+*)
projects / helm.git / commitdiff