src/HOL/Deriv.thy
 author huffman Wed, 18 Feb 2009 17:02:00 -0800 changeset 29982 6ec97eba1ee3 parent 29975 28c5322f0df3 child 29985 57975b45ab70 permissions -rw-r--r--
finish converting Deriv.thy to new polynomial library
```
(*  Title       : Deriv.thy
ID          : \$Id\$
Author      : Jacques D. Fleuriot
Copyright   : 1998  University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
GMVT by Benjamin Porter, 2005
*)

theory Deriv
imports Lim Polynomial
begin

text{*Standard Definitions*}

definition
deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
--{*Differentiation: D is derivative of function f at x*}
("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
"DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"

consts
Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
primrec
"Bolzano_bisect P a b 0 = (a,b)"
"Bolzano_bisect P a b (Suc n) =
(let (x,y) = Bolzano_bisect P a b n
in if P(x, (x+y)/2) then ((x+y)/2, y)
else (x, (x+y)/2))"

subsection {* Derivatives *}

lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"

lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"

lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"

lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
by (simp add: deriv_def cong: LIM_cong)

fixes a b c d :: "'a::ab_group_add"
shows "(a + c) - (b + d) = (a - b) + (c - d)"
by simp

"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"

lemma DERIV_minus:
"DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
by (simp only: deriv_def minus_diff_minus divide_minus_left LIM_minus)

lemma DERIV_diff:
"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
by (simp only: diff_def DERIV_add DERIV_minus)

"\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"

lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
proof (unfold isCont_iff)
assume "DERIV f x :> D"
hence "(\<lambda>h. (f(x+h) - f(x)) / h) -- 0 --> D"
by (rule DERIV_D)
hence "(\<lambda>h. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
by (intro LIM_mult LIM_ident)
hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
by simp
hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
by (simp cong: LIM_cong)
thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
qed

lemma DERIV_mult_lemma:
fixes a b c d :: "'a::real_field"
shows "(a * b - c * d) / h = a * ((b - d) / h) + ((a - c) / h) * d"

lemma DERIV_mult':
assumes f: "DERIV f x :> D"
assumes g: "DERIV g x :> E"
shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
proof (unfold deriv_def)
from f have "isCont f x"
by (rule DERIV_isCont)
hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
by (simp only: isCont_iff)
hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) / h) +
((f(x+h) - f x) / h) * g x)
-- 0 --> f x * E + D * g x"
by (intro LIM_add LIM_mult LIM_const DERIV_D f g)
thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) / h)
-- 0 --> f x * E + D * g x"
by (simp only: DERIV_mult_lemma)
qed

lemma DERIV_mult:
"[| DERIV f x :> Da; DERIV g x :> Db |]
==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
by (drule (1) DERIV_mult', simp only: mult_commute add_commute)

lemma DERIV_unique:
"[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
apply (blast intro: LIM_unique)
done

text{*Differentiation of finite sum*}

lemma DERIV_sumr [rule_format (no_asm)]:
"(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
--> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
apply (induct "n")
done

text{*Alternative definition for differentiability*}

lemma DERIV_LIM_iff:
"((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
apply (rule iffI)
apply (drule_tac k="- a" in LIM_offset)
apply (drule_tac k="a" in LIM_offset)
done

lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)

lemma inverse_diff_inverse:
"\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
\<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"

lemma DERIV_inverse_lemma:
"\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
\<Longrightarrow> (inverse a - inverse b) / h
= - (inverse a * ((a - b) / h) * inverse b)"

lemma DERIV_inverse':
assumes der: "DERIV f x :> D"
assumes neq: "f x \<noteq> 0"
shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
(is "DERIV _ _ :> ?E")
proof (unfold DERIV_iff2)
from der have lim_f: "f -- x --> f x"
by (rule DERIV_isCont [unfolded isCont_def])

from neq have "0 < norm (f x)" by simp
with LIM_D [OF lim_f] obtain s
where s: "0 < s"
and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
\<Longrightarrow> norm (f z - f x) < norm (f x)"
by fast

show "(\<lambda>z. (inverse (f z) - inverse (f x)) / (z - x)) -- x --> ?E"
proof (rule LIM_equal2 [OF s])
fix z
assume "z \<noteq> x" "norm (z - x) < s"
hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
hence "f z \<noteq> 0" by auto
thus "(inverse (f z) - inverse (f x)) / (z - x) =
- (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x))"
using neq by (rule DERIV_inverse_lemma)
next
from der have "(\<lambda>z. (f z - f x) / (z - x)) -- x --> D"
by (unfold DERIV_iff2)
thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) / (z - x)) * inverse (f x)))
-- x --> ?E"
by (intro LIM_mult LIM_inverse LIM_minus LIM_const lim_f neq)
qed
qed

lemma DERIV_divide:
"\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
\<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
apply (erule subst)
apply (unfold divide_inverse)
apply (erule DERIV_mult')
apply (erule (1) DERIV_inverse')
done

lemma DERIV_power_Suc:
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
assumes f: "DERIV f x :> D"
shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (1 + of_nat n) * (D * f x ^ n)"
proof (induct n)
case 0
show ?case by (simp add: power_Suc f)
case (Suc k)
from DERIV_mult' [OF f Suc] show ?case
apply (simp only: of_nat_Suc ring_distribs mult_1_left)
apply (simp only: power_Suc algebra_simps)
done
qed

lemma DERIV_power:
fixes f :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
assumes f: "DERIV f x :> D"
shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n * (D * f x ^ (n - Suc 0))"
by (cases "n", simp, simp add: DERIV_power_Suc f)

text {* Caratheodory formulation of derivative at a point *}

lemma CARAT_DERIV:
"(DERIV f x :> l) =
(\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
(is "?lhs = ?rhs")
proof
assume der: "DERIV f x :> l"
show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
proof (intro exI conjI)
let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
show "isCont ?g x" using der
by (simp add: isCont_iff DERIV_iff diff_minus
cong: LIM_equal [rule_format])
show "?g x = l" by simp
qed
next
assume "?rhs"
then obtain g where
"(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
thus "(DERIV f x :> l)"
by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
qed

lemma DERIV_chain':
assumes f: "DERIV f x :> D"
assumes g: "DERIV g (f x) :> E"
shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
proof (unfold DERIV_iff2)
obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
using CARAT_DERIV [THEN iffD1, OF g] by fast
from f have "f -- x --> f x"
by (rule DERIV_isCont [unfolded isCont_def])
with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
by (rule isCont_LIM_compose)
hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
-- x --> d (f x) * D"
by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
by (simp add: d dfx real_scaleR_def)
qed

(* let's do the standard proof though theorem *)
(* LIM_mult2 follows from a NS proof          *)

lemma DERIV_cmult:
"DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
by (drule DERIV_mult' [OF DERIV_const], simp)

(* standard version *)
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)

lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
by (auto dest: DERIV_chain simp add: o_def)

(*derivative of linear multiplication*)
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)

lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
apply (cut_tac DERIV_power [OF DERIV_ident])
done

text{*Power of -1*}

lemma DERIV_inverse:
fixes x :: "'a::{real_normed_field,recpower}"
shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)

text{*Derivative of inverse*}
lemma DERIV_inverse_fun:
fixes x :: "'a::{real_normed_field,recpower}"
shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)

text{*Derivative of quotient*}
lemma DERIV_quotient:
fixes x :: "'a::{real_normed_field,recpower}"
shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)

lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
by auto

subsection {* Differentiability predicate *}

definition
differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
(infixl "differentiable" 60) where
"f differentiable x = (\<exists>D. DERIV f x :> D)"

lemma differentiableE [elim?]:
assumes "f differentiable x"
obtains df where "DERIV f x :> df"
using prems unfolding differentiable_def ..

lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"

lemma differentiableI: "DERIV f x :> D ==> f differentiable x"

lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
by (rule DERIV_ident [THEN differentiableI])

lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
by (rule DERIV_const [THEN differentiableI])

lemma differentiable_compose:
assumes f: "f differentiable (g x)"
assumes g: "g differentiable x"
shows "(\<lambda>x. f (g x)) differentiable x"
proof -
from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
moreover
from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
ultimately
have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
thus ?thesis by (rule differentiableI)
qed

lemma differentiable_sum [simp]:
assumes "f differentiable x"
and "g differentiable x"
shows "(\<lambda>x. f x + g x) differentiable x"
proof -
from `f differentiable x` obtain df where "DERIV f x :> df" ..
moreover
from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
ultimately
have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
thus ?thesis by (rule differentiableI)
qed

lemma differentiable_minus [simp]:
assumes "f differentiable x"
shows "(\<lambda>x. - f x) differentiable x"
proof -
from `f differentiable x` obtain df where "DERIV f x :> df" ..
hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
thus ?thesis by (rule differentiableI)
qed

lemma differentiable_diff [simp]:
assumes "f differentiable x"
assumes "g differentiable x"
shows "(\<lambda>x. f x - g x) differentiable x"
unfolding diff_minus using prems by simp

lemma differentiable_mult [simp]:
assumes "f differentiable x"
assumes "g differentiable x"
shows "(\<lambda>x. f x * g x) differentiable x"
proof -
from `f differentiable x` obtain df where "DERIV f x :> df" ..
moreover
from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
ultimately
have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
thus ?thesis by (rule differentiableI)
qed

lemma differentiable_inverse [simp]:
assumes "f differentiable x" and "f x \<noteq> 0"
shows "(\<lambda>x. inverse (f x)) differentiable x"
proof -
from `f differentiable x` obtain df where "DERIV f x :> df" ..
hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
using `f x \<noteq> 0` by (rule DERIV_inverse')
thus ?thesis by (rule differentiableI)
qed

lemma differentiable_divide [simp]:
assumes "f differentiable x"
assumes "g differentiable x" and "g x \<noteq> 0"
shows "(\<lambda>x. f x / g x) differentiable x"
unfolding divide_inverse using prems by simp

lemma differentiable_power [simp]:
fixes f :: "'a::{recpower,real_normed_field} \<Rightarrow> 'a"
assumes "f differentiable x"
shows "(\<lambda>x. f x ^ n) differentiable x"
by (induct n, simp, simp add: power_Suc prems)

subsection {* Nested Intervals and Bisection *}

text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
All considerably tidied by lcp.*}

lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
apply (induct "no")
apply (auto intro: order_trans)
done

lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
\<forall>n. g(Suc n) \<le> g(n);
\<forall>n. f(n) \<le> g(n) |]
==> Bseq (f :: nat \<Rightarrow> real)"
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
apply (induct_tac "n")
apply (auto intro: order_trans)
apply (rule_tac y = "g (Suc na)" in order_trans)
apply (induct_tac [2] "na")
apply (auto intro: order_trans)
done

lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
\<forall>n. g(Suc n) \<le> g(n);
\<forall>n. f(n) \<le> g(n) |]
==> Bseq (g :: nat \<Rightarrow> real)"
apply (subst Bseq_minus_iff [symmetric])
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
apply auto
done

lemma f_inc_imp_le_lim:
fixes f :: "nat \<Rightarrow> real"
shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
apply (rule linorder_not_less [THEN iffD1])
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
apply (drule real_less_sum_gt_zero)
apply (drule_tac x = "f n + - lim f" in spec, safe)
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
apply (subgoal_tac "lim f \<le> f (no + n) ")
apply (drule_tac no=no and m=n in lemma_f_mono_add)
apply (induct_tac "no")
apply simp
apply (auto intro: order_trans simp add: diff_minus abs_if)
done

lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
apply (rule LIMSEQ_minus [THEN limI])
done

lemma g_dec_imp_lim_le:
fixes g :: "nat \<Rightarrow> real"
shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
done

lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
\<forall>n. g(Suc n) \<le> g(n);
\<forall>n. f(n) \<le> g(n) |]
==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
((\<forall>n. m \<le> g(n)) & g ----> m)"
apply (subgoal_tac "monoseq f & monoseq g")
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
apply (subgoal_tac "Bseq f & Bseq g")
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
apply (rule_tac x = "lim f" in exI)
apply (rule_tac x = "lim g" in exI)
apply (auto intro: LIMSEQ_le)
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
done

lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
\<forall>n. g(Suc n) \<le> g(n);
\<forall>n. f(n) \<le> g(n);
(%n. f(n) - g(n)) ----> 0 |]
==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
((\<forall>n. l \<le> g(n)) & g ----> l)"
apply (drule lemma_nest, auto)
apply (subgoal_tac "l = m")
apply (drule_tac [2] X = f in LIMSEQ_diff)
apply (auto intro: LIMSEQ_unique)
done

text{*The universal quantifiers below are required for the declaration
of @{text Bolzano_nest_unique} below.*}

lemma Bolzano_bisect_le:
"a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
apply (rule allI)
apply (induct_tac "n")
apply (auto simp add: Let_def split_def)
done

lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
\<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
apply (rule allI)
apply (induct_tac "n")
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
done

lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
\<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
apply (rule allI)
apply (induct_tac "n")
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
done

lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
apply (auto)
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
apply (simp)
done

lemma Bolzano_bisect_diff:
"a \<le> b ==>
snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
(b-a) / (2 ^ n)"
apply (induct "n")
done

lemmas Bolzano_nest_unique =
lemma_nest_unique
[OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]

lemma not_P_Bolzano_bisect:
assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
and notP: "~ P(a,b)"
and le:   "a \<le> b"
shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
proof (induct n)
case 0 show ?case using notP by simp
next
case (Suc n)
thus ?case
by (auto simp del: surjective_pairing [symmetric]
simp add: Let_def split_def Bolzano_bisect_le [OF le]
P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
qed

(*Now we re-package P_prem as a formula*)
lemma not_P_Bolzano_bisect':
"[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
~ P(a,b);  a \<le> b |] ==>
\<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])

lemma lemma_BOLZANO:
"[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
\<forall>x. \<exists>d::real. 0 < d &
(\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
a \<le> b |]
==> P(a,b)"
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
apply (rule LIMSEQ_minus_cancel)
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
apply (rule ccontr)
apply (drule not_P_Bolzano_bisect', assumption+)
apply (rename_tac "l")
apply (drule_tac x = l in spec, clarify)
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
apply (drule real_less_half_sum, auto)
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
apply safe
apply (simp_all (no_asm_simp))
apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
apply (rule real_sum_of_halves [THEN subst])
done

lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
(\<forall>x. \<exists>d::real. 0 < d &
(\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
--> (\<forall>a b. a \<le> b --> P(a,b))"
apply clarify
apply (blast intro: lemma_BOLZANO)
done

subsection {* Intermediate Value Theorem *}

text {*Prove Contrapositive by Bisection*}

lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
a \<le> b;
(\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
apply (rule contrapos_pp, assumption)
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
apply safe
apply simp_all
apply (rule ccontr)
apply (subgoal_tac "a \<le> x & x \<le> b")
prefer 2
apply simp
apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
apply (drule_tac x = x in spec)+
apply simp
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
apply safe
apply simp
apply (drule_tac x = s in spec, clarify)
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
apply (drule_tac x = "ba-x" in spec)
apply (drule_tac x = "aa-x" in spec)
apply (case_tac "x \<le> aa", simp_all)
done

lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
a \<le> b;
(\<forall>x. a \<le> x & x \<le> b --> isCont f x)
|] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
apply (drule IVT [where f = "%x. - f x"], assumption)
apply (auto intro: isCont_minus)
done

(*HOL style here: object-level formulations*)
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
(\<forall>x. a \<le> x & x \<le> b --> isCont f x))
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
apply (blast intro: IVT)
done

lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
(\<forall>x. a \<le> x & x \<le> b --> isCont f x))
--> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
apply (blast intro: IVT2)
done

subsection {* Boundedness of continuous functions *}

text{*By bisection, function continuous on closed interval is bounded above*}

lemma isCont_bounded:
"[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
apply safe
apply simp_all
apply (rename_tac x xa ya M Ma)
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
apply (rule_tac x = Ma in exI, clarify)
apply (cut_tac x = xb and y = xa in linorder_linear, force)
apply (rule_tac x = M in exI, clarify)
apply (cut_tac x = xb and y = xa in linorder_linear, force)
apply (case_tac "a \<le> x & x \<le> b")
apply (rule_tac [2] x = 1 in exI)
prefer 2 apply force
apply (drule_tac x = x in spec, auto)
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
apply (drule_tac x = 1 in spec, auto)
apply (rule_tac x = s in exI, clarify)
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
apply (drule_tac x = "xa-x" in spec)
done

text{*Refine the above to existence of least upper bound*}

lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
(\<exists>t. isLub UNIV S t)"
by (blast intro: reals_complete)

lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
(\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
in lemma_reals_complete)
apply auto
apply (drule isCont_bounded, assumption)
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
apply (rule exI, auto)
apply (auto dest!: spec simp add: linorder_not_less)
done

text{*Now show that it attains its upper bound*}

lemma isCont_eq_Ub:
assumes le: "a \<le> b"
and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
proof -
from isCont_has_Ub [OF le con]
obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
show ?thesis
proof (intro exI, intro conjI)
show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
proof (rule ccontr)
assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
by (fastsimp simp add: linorder_not_le [symmetric])
hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
by (auto simp add: isCont_inverse isCont_diff con)
from isCont_bounded [OF le this]
obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
by (auto intro: order_le_less_trans [of _ k])
with Minv
have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
by (intro strip less_imp_inverse_less, simp_all)
hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
by simp
have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
by (simp, arith)
from M2 [OF this]
obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
thus False using invlt [of x] by force
qed
qed
qed

text{*Same theorem for lower bound*}

lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
(\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
prefer 2 apply (blast intro: isCont_minus)
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
apply safe
apply auto
done

text{*Another version.*}

lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
(\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
apply (frule isCont_eq_Lb)
apply (frule_tac [2] isCont_eq_Ub)
apply (assumption+, safe)
apply (rule_tac x = "f x" in exI)
apply (rule_tac x = "f xa" in exI, simp, safe)
apply (cut_tac x = x and y = xa in linorder_linear, safe)
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
apply (rule_tac [2] x = xb in exI)
apply (rule_tac [4] x = xb in exI, simp_all)
done

subsection {* Local extrema *}

text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}

lemma DERIV_left_inc:
fixes f :: "real => real"
assumes der: "DERIV f x :> l"
and l:   "0 < l"
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
proof -
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
then obtain s
where s:   "0 < s"
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
by auto
thus ?thesis
proof (intro exI conjI strip)
show "0<s" using s .
fix h::real
assume "0 < h" "h < s"
with all [of h] show "f x < f (x+h)"
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
with l
have "0 < (f (x+h) - f x) / h" by arith
thus "f x < f (x+h)"
qed
qed
qed

lemma DERIV_left_dec:
fixes f :: "real => real"
assumes der: "DERIV f x :> l"
and l:   "l < 0"
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
proof -
from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
then obtain s
where s:   "0 < s"
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
by auto
thus ?thesis
proof (intro exI conjI strip)
show "0<s" using s .
fix h::real
assume "0 < h" "h < s"
with all [of "-h"] show "f x < f (x-h)"
proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
with l
have "0 < (f (x-h) - f x) / h" by arith
thus "f x < f (x-h)"
qed
qed
qed

lemma DERIV_local_max:
fixes f :: "real => real"
assumes der: "DERIV f x :> l"
and d:   "0 < d"
and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
shows "l = 0"
proof (cases rule: linorder_cases [of l 0])
case equal thus ?thesis .
next
case less
from DERIV_left_dec [OF der less]
obtain d' where d': "0 < d'"
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
from real_lbound_gt_zero [OF d d']
obtain e where "0 < e \<and> e < d \<and> e < d'" ..
with lt le [THEN spec [where x="x-e"]]
show ?thesis by (auto simp add: abs_if)
next
case greater
from DERIV_left_inc [OF der greater]
obtain d' where d': "0 < d'"
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
from real_lbound_gt_zero [OF d d']
obtain e where "0 < e \<and> e < d \<and> e < d'" ..
with lt le [THEN spec [where x="x+e"]]
show ?thesis by (auto simp add: abs_if)
qed

text{*Similar theorem for a local minimum*}
lemma DERIV_local_min:
fixes f :: "real => real"
shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
by (drule DERIV_minus [THEN DERIV_local_max], auto)

text{*In particular, if a function is locally flat*}
lemma DERIV_local_const:
fixes f :: "real => real"
shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
by (auto dest!: DERIV_local_max)

subsection {* Rolle's Theorem *}

text{*Lemma about introducing open ball in open interval*}
lemma lemma_interval_lt:
"[| a < x;  x < b |]
==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"

apply (insert linorder_linear [of "x-a" "b-x"], safe)
apply (rule_tac x = "x-a" in exI)
apply (rule_tac [2] x = "b-x" in exI, auto)
done

lemma lemma_interval: "[| a < x;  x < b |] ==>
\<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
apply (drule lemma_interval_lt, auto)
apply (auto intro!: exI)
done

text{*Rolle's Theorem.
If @{term f} is defined and continuous on the closed interval
@{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
and @{term "f(a) = f(b)"},
then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
theorem Rolle:
assumes lt: "a < b"
and eq: "f(a) = f(b)"
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
proof -
have le: "a \<le> b" using lt by simp
from isCont_eq_Ub [OF le con]
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
and alex: "a \<le> x" and xleb: "x \<le> b"
by blast
from isCont_eq_Lb [OF le con]
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
and alex': "a \<le> x'" and x'leb: "x' \<le> b"
by blast
show ?thesis
proof cases
assume axb: "a < x & x < b"
--{*@{term f} attains its maximum within the interval*}
hence ax: "a<x" and xb: "x<b" by arith +
from lemma_interval [OF ax xb]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
by blast
from differentiableD [OF dif [OF axb]]
obtain l where der: "DERIV f x :> l" ..
have "l=0" by (rule DERIV_local_max [OF der d bound'])
--{*the derivative at a local maximum is zero*}
thus ?thesis using ax xb der by auto
next
assume notaxb: "~ (a < x & x < b)"
hence xeqab: "x=a | x=b" using alex xleb by arith
hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
show ?thesis
proof cases
assume ax'b: "a < x' & x' < b"
--{*@{term f} attains its minimum within the interval*}
hence ax': "a<x'" and x'b: "x'<b" by arith+
from lemma_interval [OF ax' x'b]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
by blast
from differentiableD [OF dif [OF ax'b]]
obtain l where der: "DERIV f x' :> l" ..
have "l=0" by (rule DERIV_local_min [OF der d bound'])
--{*the derivative at a local minimum is zero*}
thus ?thesis using ax' x'b der by auto
next
assume notax'b: "~ (a < x' & x' < b)"
--{*@{term f} is constant througout the interval*}
hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
from dense [OF lt]
obtain r where ar: "a < r" and rb: "r < b" by blast
from lemma_interval [OF ar rb]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
proof (clarify)
fix z::real
assume az: "a \<le> z" and zb: "z \<le> b"
show "f z = f b"
proof (rule order_antisym)
show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
qed
qed
have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
proof (intro strip)
fix y::real
assume lt: "\<bar>r-y\<bar> < d"
hence "f y = f b" by (simp add: eq_fb bound)
thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
qed
from differentiableD [OF dif [OF conjI [OF ar rb]]]
obtain l where der: "DERIV f r :> l" ..
have "l=0" by (rule DERIV_local_const [OF der d bound'])
--{*the derivative of a constant function is zero*}
thus ?thesis using ar rb der by auto
qed
qed
qed

subsection{*Mean Value Theorem*}

lemma lemma_MVT:
"f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
proof cases
assume "a=b" thus ?thesis by simp
next
assume "a\<noteq>b"
hence ba: "b-a \<noteq> 0" by arith
show ?thesis
by (rule real_mult_left_cancel [OF ba, THEN iffD1],
qed

theorem MVT:
assumes lt:  "a < b"
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
(f(b) - f(a) = (b-a) * l)"
proof -
let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
proof (clarify)
fix x::real
assume ax: "a < x" and xb: "x < b"
from differentiableD [OF dif [OF conjI [OF ax xb]]]
obtain l where der: "DERIV f x :> l" ..
show "?F differentiable x"
by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
blast intro: DERIV_diff DERIV_cmult_Id der)
qed
from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
by blast
have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
by (rule DERIV_cmult_Id)
hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
:> 0 + (f b - f a) / (b - a)"
show ?thesis
proof (intro exI conjI)
show "a < z" using az .
show "z < b" using zb .
show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
qed
qed

lemma MVT2:
"[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
apply (drule MVT)
apply (blast intro: DERIV_isCont)
apply (force dest: order_less_imp_le simp add: differentiable_def)
apply (blast dest: DERIV_unique order_less_imp_le)
done

text{*A function is constant if its derivative is 0 over an interval.*}

lemma DERIV_isconst_end:
fixes f :: "real => real"
shows "[| a < b;
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
\<forall>x. a < x & x < b --> DERIV f x :> 0 |]
==> f b = f a"
apply (drule MVT, assumption)
apply (blast intro: differentiableI)
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
done

lemma DERIV_isconst1:
fixes f :: "real => real"
shows "[| a < b;
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
\<forall>x. a < x & x < b --> DERIV f x :> 0 |]
==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
apply safe
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
apply (drule_tac b = x in DERIV_isconst_end, auto)
done

lemma DERIV_isconst2:
fixes f :: "real => real"
shows "[| a < b;
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
\<forall>x. a < x & x < b --> DERIV f x :> 0;
a \<le> x; x \<le> b |]
==> f x = f a"
apply (blast dest: DERIV_isconst1)
done

lemma DERIV_isconst3: fixes a b x y :: real
assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
shows "f x = f y"
proof (cases "x = y")
case False
let ?a = "min x y"
let ?b = "max x y"

have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
proof (rule allI, rule impI)
fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hence "z \<in> {a<..<b}" by auto
thus "DERIV f z :> 0" by (rule derivable)
qed
hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto

have "?a < ?b" using `x \<noteq> y` by auto
from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
show ?thesis by auto
qed auto

lemma DERIV_isconst_all:
fixes f :: "real => real"
shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
apply (rule linorder_cases [of x y])
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
done

lemma DERIV_const_ratio_const:
fixes f :: "real => real"
shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
apply (rule linorder_cases [of a b], auto)
apply (drule_tac [!] f = f in MVT)
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
done

lemma DERIV_const_ratio_const2:
fixes f :: "real => real"
shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
done

lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
by (simp)

lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
by (simp)

text{*Gallileo's "trick": average velocity = av. of end velocities*}

lemma DERIV_const_average:
fixes v :: "real => real"
assumes neq: "a \<noteq> (b::real)"
and der: "\<forall>x. DERIV v x :> k"
shows "v ((a + b)/2) = (v a + v b)/2"
proof (cases rule: linorder_cases [of a b])
case equal with neq show ?thesis by simp
next
case less
have "(v b - v a) / (b - a) = k"
by (rule DERIV_const_ratio_const2 [OF neq der])
hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
ultimately show ?thesis using neq by force
next
case greater
have "(v b - v a) / (b - a) = k"
by (rule DERIV_const_ratio_const2 [OF neq der])
hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
qed

subsection {* Continuous injective functions *}

text{*Dull lemma: an continuous injection on an interval must have a
strict maximum at an end point, not in the middle.*}

lemma lemma_isCont_inj:
fixes f :: "real \<Rightarrow> real"
assumes d: "0 < d"
and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
proof (rule ccontr)
assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
show False
proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
case le
from d cont all [of "x+d"]
have flef: "f(x+d) \<le> f x"
and xlex: "x - d \<le> x"
and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
from IVT [OF le flef xlex cont']
obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
moreover
hence "g(f x') = g (f(x+d))" by simp
ultimately show False using d inj [of x'] inj [of "x+d"]
next
case ge
from d cont all [of "x-d"]
have flef: "f(x-d) \<le> f x"
and xlex: "x \<le> x+d"
and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
from IVT2 [OF ge flef xlex cont']
obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
moreover
hence "g(f x') = g (f(x-d))" by simp
ultimately show False using d inj [of x'] inj [of "x-d"]
qed
qed

text{*Similar version for lower bound.*}

lemma lemma_isCont_inj2:
fixes f g :: "real \<Rightarrow> real"
shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
apply (insert lemma_isCont_inj
[where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
done

text{*Show there's an interval surrounding @{term "f(x)"} in
@{text "f[[x - d, x + d]]"} .*}

lemma isCont_inj_range:
fixes f :: "real \<Rightarrow> real"
assumes d: "0 < d"
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
proof -
have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
from isCont_Lb_Ub [OF this]
obtain L M
where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
and all2 [rule_format]:
"\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
by auto
with d have "L \<le> f x & f x \<le> M" by simp
moreover have "L \<noteq> f x"
proof -
from lemma_isCont_inj2 [OF d inj cont]
obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
thus ?thesis using all1 [of u] by arith
qed
moreover have "f x \<noteq> M"
proof -
from lemma_isCont_inj [OF d inj cont]
obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
thus ?thesis using all1 [of u] by arith
qed
ultimately have "L < f x & f x < M" by arith
hence "0 < f x - L" "0 < M - f x" by arith+
from real_lbound_gt_zero [OF this]
obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
thus ?thesis
proof (intro exI conjI)
show "0<e" using e(1) .
show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
proof (intro strip)
fix y::real
assume "\<bar>y - f x\<bar> \<le> e"
with e have "L \<le> y \<and> y \<le> M" by arith
from all2 [OF this]
obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
qed
qed
qed

text{*Continuity of inverse function*}

lemma isCont_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes d: "0 < d"
and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
shows "isCont g (f x)"
show "\<forall>r. 0 < r \<longrightarrow>
(\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
proof (intro strip)
fix r::real
assume r: "0<r"
from real_lbound_gt_zero [OF r d]
obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
with inj cont
have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
"\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
from isCont_inj_range [OF e this]
obtain e' where e': "0 < e'"
and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
by blast
show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
proof (intro exI conjI)
show "0<e'" using e' .
show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
proof (intro strip)
fix z::real
assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
with e e_lt e_simps all [rule_format, of "f x + z"]
show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
qed
qed
qed
qed

text {* Derivative of inverse function *}

lemma DERIV_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes der: "DERIV f (g x) :> D"
assumes neq: "D \<noteq> 0"
assumes a: "a < x" and b: "x < b"
assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
assumes cont: "isCont g x"
shows "DERIV g x :> inverse D"
unfolding DERIV_iff2
proof (rule LIM_equal2)
show "0 < min (x - a) (b - x)"
using a b by arith
next
fix y
assume "norm (y - x) < min (x - a) (b - x)"
hence "a < y" and "y < b"
thus "(g y - g x) / (y - x) =
inverse ((f (g y) - x) / (g y - g x))"
next
have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
by (rule der [unfolded DERIV_iff2])
hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
using inj a b by simp
have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
proof (safe intro!: exI)
show "0 < min (x - a) (b - x)"
using a b by simp
next
fix y
assume "norm (y - x) < min (x - a) (b - x)"
hence y: "a < y" "y < b"
assume "g y = g x"
hence "f (g y) = f (g x)" by simp
hence "y = x" using inj y a b by simp
also assume "y \<noteq> x"
finally show False by simp
qed
have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
using cont 1 2 by (rule isCont_LIM_compose2)
thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
-- x --> inverse D"
using neq by (rule LIM_inverse)
qed

subsection {* Generalized Mean Value Theorem *}

theorem GMVT:
fixes a b :: real
assumes alb: "a < b"
and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
proof -
let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
from prems have "a < b" by simp
moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
proof -
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
by (auto intro: isCont_mult)
moreover
have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
by (auto intro: isCont_mult)
ultimately show ?thesis
by (fastsimp intro: isCont_diff)
qed
moreover
have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
proof -
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
moreover
have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
ultimately show ?thesis by (simp add: differentiable_diff)
qed
ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..

from cdef have cint: "a < c \<and> c < b" by auto
with gd have "g differentiable c" by simp
hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
then obtain g'c where g'cdef: "DERIV g c :> g'c" ..

from cdef have "a < c \<and> c < b" by auto
with fd have "f differentiable c" by simp
hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
then obtain f'c where f'cdef: "DERIV f c :> f'c" ..

from cdef have "DERIV ?h c :> l" by auto
moreover
{
have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
apply (insert DERIV_const [where k="f b - f a"])
apply (drule meta_spec [of _ c])
apply (drule DERIV_mult [OF _ g'cdef])
by simp
moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
apply (insert DERIV_const [where k="g b - g a"])
apply (drule meta_spec [of _ c])
apply (drule DERIV_mult [OF _ f'cdef])
by simp
ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
}
ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)

{
from cdef have "?h b - ?h a = (b - a) * l" by auto
also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
}
moreover
{
have "?h b - ?h a =
((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
hence "?h b - ?h a = 0" by auto
}
ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)

with g'cdef f'cdef cint show ?thesis by auto
qed

subsection {* Derivatives of univariate polynomials *}

definition
pderiv :: "'a::real_normed_field poly \<Rightarrow> 'a poly" where
"pderiv = poly_rec 0 (\<lambda>a p p'. p + pCons 0 p')"

lemma pderiv_0 [simp]: "pderiv 0 = 0"
unfolding pderiv_def by (simp add: poly_rec_0)

lemma pderiv_pCons: "pderiv (pCons a p) = p + pCons 0 (pderiv p)"
unfolding pderiv_def by (simp add: poly_rec_pCons)

lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
apply (induct p arbitrary: n, simp)
apply (simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
done

lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
apply (rule iffI)
apply (cases p, simp)
apply (simp add: expand_poly_eq coeff_pderiv del: of_nat_Suc)
apply (simp add: expand_poly_eq coeff_pderiv coeff_eq_0)
done

lemma degree_pderiv: "degree (pderiv p) = degree p - 1"
apply (rule order_antisym [OF degree_le])
apply (cases "degree p", simp)
apply (rule le_degree)
apply (simp add: coeff_pderiv del: of_nat_Suc)
apply (rule subst, assumption)
done

lemma pderiv_singleton [simp]: "pderiv [:a:] = 0"

lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
by (rule poly_ext, simp add: coeff_pderiv algebra_simps)

lemma pderiv_minus: "pderiv (- p) = - pderiv p"
by (rule poly_ext, simp add: coeff_pderiv)

lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
by (rule poly_ext, simp add: coeff_pderiv algebra_simps)

lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
by (rule poly_ext, simp add: coeff_pderiv algebra_simps)

lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
apply (induct p)
apply simp
done

lemma pderiv_power_Suc:
"pderiv (p ^ Suc n) = smult (of_nat (Suc n)) (p ^ n) * pderiv p"
apply (induct n)
apply simp
apply (subst power_Suc)
apply (subst pderiv_mult)
apply (erule ssubst)
done

lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
by (simp add: DERIV_cmult mult_commute [of _ c])

lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
by (rule lemma_DERIV_subst, rule DERIV_pow, simp)
declare DERIV_pow2 [simp] DERIV_pow [simp]

lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: 'a::real_normed_field) x :> D"
by (rule lemma_DERIV_subst, rule DERIV_add, auto)

lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
apply (induct p)
apply simp
apply (rule lemma_DERIV_subst)
apply (rule DERIV_add DERIV_mult DERIV_const DERIV_ident | assumption)+
apply simp
done

text{* Consequences of the derivative theorem above*}

lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (x::real)"
apply (blast intro: poly_DERIV)
done

lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
by (rule poly_DERIV [THEN DERIV_isCont])

lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
==> \<exists>x. a < x & x < b & (poly p x = 0)"
apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
done

lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
==> \<exists>x. a < x & x < b & (poly p x = 0)"
by (insert poly_IVT_pos [where p = "- p" ]) simp

lemma poly_MVT: "(a::real) < b ==>
\<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
apply (drule_tac f = "poly p" in MVT, auto)
apply (rule_tac x = z in exI)
apply (auto simp add: real_mult_left_cancel poly_DERIV [THEN DERIV_unique])
done

text{*Lemmas for Derivatives*}

lemma order_unique_lemma:
fixes p :: "'a::idom poly"
assumes "[:-a, 1:] ^ n dvd p \<and> \<not> [:-a, 1:] ^ Suc n dvd p"
shows "n = order a p"
unfolding Polynomial.order_def
apply (rule Least_equality [symmetric])
apply (rule assms [THEN conjunct2])
apply (erule contrapos_np)
apply (rule power_le_dvd)
apply (rule assms [THEN conjunct1])
apply simp
done

lemma lemma_order_pderiv1:
"pderiv ([:- a, 1:] ^ Suc n * q) = [:- a, 1:] ^ Suc n * pderiv q +
smult (of_nat (Suc n)) (q * [:- a, 1:] ^ n)"
apply (simp only: pderiv_mult pderiv_power_Suc)
apply (simp del: power_poly_Suc of_nat_Suc add: pderiv_pCons)
done

fixes a b c :: "'a::comm_ring_1"
shows "a dvd b + c \<Longrightarrow> a dvd b \<Longrightarrow> a dvd c"
by (drule (1) Ring_and_Field.dvd_diff, simp)

lemma lemma_order_pderiv [rule_format]:
"\<forall>p q a. 0 < n &
pderiv p \<noteq> 0 &
p = [:- a, 1:] ^ n * q & ~ [:- a, 1:] dvd q
--> n = Suc (order a (pderiv p))"
apply (cases "n", safe, rename_tac n p q a)
apply (rule order_unique_lemma)
apply (rule conjI)
apply (subst lemma_order_pderiv1)
apply (rule dvd_mult2)
apply (rule le_imp_power_dvd, simp)
apply (rule dvd_smult)
apply (rule dvd_mult)
apply (rule dvd_refl)
apply (subst lemma_order_pderiv1)
apply (erule contrapos_nn) back
apply (subgoal_tac "[:- a, 1:] ^ Suc n dvd q * [:- a, 1:] ^ n")
apply (simp del: mult_pCons_left)
apply (simp del: mult_pCons_left)
apply (drule dvd_smult_cancel, simp del: of_nat_Suc)
apply assumption
done

lemma order_decomp:
"p \<noteq> 0
==> \<exists>q. p = [:-a, 1:] ^ (order a p) * q &
~([:-a, 1:] dvd q)"
apply (drule order [where a=a])
apply (erule conjE)
apply (erule dvdE)
apply (rule exI)
apply (rule conjI, assumption)
apply (erule contrapos_nn)
apply (erule ssubst) back
apply (subst power_Suc2)
apply (erule mult_dvd_mono [OF dvd_refl])
done

lemma order_pderiv: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
==> (order a p = Suc (order a (pderiv p)))"
apply (case_tac "p = 0", simp)
apply (drule_tac a = a and p = p in order_decomp)
using neq0_conv
apply (blast intro: lemma_order_pderiv)
done

lemma order_mult: "p * q \<noteq> 0 \<Longrightarrow> order a (p * q) = order a p + order a q"
proof -
def i \<equiv> "order a p"
def j \<equiv> "order a q"
def t \<equiv> "[:-a, 1:]"
have t_dvd_iff: "\<And>u. t dvd u \<longleftrightarrow> poly u a = 0"
unfolding t_def by (simp add: dvd_iff_poly_eq_0)
assume "p * q \<noteq> 0"
then show "order a (p * q) = i + j"
apply clarsimp
apply (drule order [where a=a and p=p, folded i_def t_def])
apply (drule order [where a=a and p=q, folded j_def t_def])
apply clarify
apply (rule order_unique_lemma [symmetric], fold t_def)
apply (erule dvdE)+
done
qed

text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}

lemma order_divides: "[:-a, 1:] ^ n dvd p \<longleftrightarrow> p = 0 \<or> n \<le> order a p"
apply (cases "p = 0", auto)
apply (drule order_2 [where a=a and p=p])
apply (erule contrapos_np)
apply (erule power_le_dvd)
apply simp
apply (erule power_le_dvd [OF order_1])
done

lemma poly_squarefree_decomp_order:
assumes "pderiv p \<noteq> 0"
and p: "p = q * d"
and p': "pderiv p = e * d"
and d: "d = r * p + s * pderiv p"
shows "order a q = (if order a p = 0 then 0 else 1)"
proof (rule classical)
assume 1: "order a q \<noteq> (if order a p = 0 then 0 else 1)"
from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
with p have "order a p = order a q + order a d"
with 1 have "order a p \<noteq> 0" by (auto split: if_splits)
have "order a (pderiv p) = order a e + order a d"
using `pderiv p \<noteq> 0` `pderiv p = e * d` by (simp add: order_mult)
have "order a p = Suc (order a (pderiv p))"
using `pderiv p \<noteq> 0` `order a p \<noteq> 0` by (rule order_pderiv)
have "d \<noteq> 0" using `p \<noteq> 0` `p = q * d` by simp
have "([:-a, 1:] ^ (order a (pderiv p))) dvd d"
apply (rule dvd_mult)
apply (simp add: order_divides `p \<noteq> 0`
`order a p = Suc (order a (pderiv p))`)
apply (rule dvd_mult)
done
then have "order a (pderiv p) \<le> order a d"
using `d \<noteq> 0` by (simp add: order_divides)
show ?thesis
using `order a p = order a q + order a d`
using `order a (pderiv p) = order a e + order a d`
using `order a p = Suc (order a (pderiv p))`
using `order a (pderiv p) \<le> order a d`
by auto
qed

lemma poly_squarefree_decomp_order2: "[| pderiv p \<noteq> 0;
p = q * d;
pderiv p = e * d;
d = r * p + s * pderiv p
|] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
apply (blast intro: poly_squarefree_decomp_order)
done

lemma order_pderiv2: "[| pderiv p \<noteq> 0; order a p \<noteq> 0 |]
==> (order a (pderiv p) = n) = (order a p = Suc n)"
apply (auto dest: order_pderiv)
done

definition
rsquarefree :: "'a::idom poly => bool" where
"rsquarefree p = (p \<noteq> 0 & (\<forall>a. (order a p = 0) | (order a p = 1)))"

lemma pderiv_iszero: "pderiv p = 0 \<Longrightarrow> \<exists>h. p = [:h:]"
apply (case_tac p, auto split: if_splits)
done

lemma rsquarefree_roots:
"rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
apply (case_tac "p = 0", simp, simp)
apply (case_tac "pderiv p = 0")
apply simp
apply (drule pderiv_iszero, clarify)
apply simp
apply (rule allI)
apply (cut_tac p = "[:h:]" and a = a in order_root)
apply simp
apply (auto simp add: order_root order_pderiv2)
apply (erule_tac x="a" in allE, simp)
done

lemma poly_squarefree_decomp:
assumes "pderiv p \<noteq> 0"
and "p = q * d"
and "pderiv p = e * d"
and "d = r * p + s * pderiv p"
shows "rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
proof -
from `pderiv p \<noteq> 0` have "p \<noteq> 0" by auto
with `p = q * d` have "q \<noteq> 0" by simp
have "\<forall>a. order a q = (if order a p = 0 then 0 else 1)"
using assms by (rule poly_squarefree_decomp_order2)
with `p \<noteq> 0` `q \<noteq> 0` show ?thesis
qed

subsection {* Theorems about Limits *}

(* need to rename second isCont_inverse *)

lemma isCont_inv_fun:
fixes f g :: "real \<Rightarrow> real"
shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
\<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
==> isCont g (f x)"
by (rule isCont_inverse_function)

lemma isCont_inv_fun_inv:
fixes f g :: "real \<Rightarrow> real"
shows "[| 0 < d;
\<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
\<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
==> \<exists>e. 0 < e &
(\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
apply (drule isCont_inj_range)
prefer 2 apply (assumption, assumption, auto)
apply (rule_tac x = e in exI, auto)
apply (rotate_tac 2)
apply (drule_tac x = y in spec, auto)
done

text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
lemma LIM_fun_gt_zero:
"[| f -- c --> (l::real); 0 < l |]
==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
apply (drule_tac x = "l/2" in spec, safe, force)
apply (rule_tac x = s in exI)
apply (auto simp only: abs_less_iff)
done

lemma LIM_fun_less_zero:
"[| f -- c --> (l::real); l < 0 |]
==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
apply (drule_tac x = "-l/2" in spec, safe, force)
apply (rule_tac x = s in exI)
apply (auto simp only: abs_less_iff)
done

lemma LIM_fun_not_zero:
"[| f -- c --> (l::real); l \<noteq> 0 |]
==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
apply (drule LIM_fun_less_zero)
apply (drule_tac [3] LIM_fun_gt_zero)
apply force+
done

end
```