Theory Arcwise_Connected
section ‹Arcwise-Connected Sets›
theory Arcwise_Connected
imports Path_Connected Ordered_Euclidean_Space "HOL-Computational_Algebra.Primes"
begin
lemma path_connected_interval [simp]:
fixes a b::"'a::ordered_euclidean_space"
shows "path_connected {a..b}"
using is_interval_cc is_interval_path_connected by blast
lemma segment_to_closest_point:
fixes S :: "'a :: euclidean_space set"
shows "⟦closed S; S ≠ {}⟧ ⟹ open_segment a (closest_point S a) ∩ S = {}"
unfolding disjoint_iff
by (metis closest_point_le dist_commute dist_in_open_segment not_le)
lemma segment_to_point_exists:
fixes S :: "'a :: euclidean_space set"
assumes "closed S" "S ≠ {}"
obtains b where "b ∈ S" "open_segment a b ∩ S = {}"
by (metis assms segment_to_closest_point closest_point_exists that)
subsection ‹The Brouwer reduction theorem›
theorem Brouwer_reduction_theorem_gen:
fixes S :: "'a::euclidean_space set"
assumes "closed S" "φ S"
and φ: "⋀F. ⟦⋀n. closed(F n); ⋀n. φ(F n); ⋀n. F(Suc n) ⊆ F n⟧ ⟹ φ(⋂(range F))"
obtains T where "T ⊆ S" "closed T" "φ T" "⋀U. ⟦U ⊆ S; closed U; φ U⟧ ⟹ ¬ (U ⊂ T)"
proof -
obtain B :: "nat ⇒ 'a set"
where "inj B" "⋀n. open(B n)" and open_cov: "⋀S. open S ⟹ ∃K. S = ⋃(B ` K)"
by (metis Setcompr_eq_image that univ_second_countable_sequence)
define A where "A ≡ rec_nat S (λn a. if ∃U. U ⊆ a ∧ closed U ∧ φ U ∧ U ∩ (B n) = {}
then SOME U. U ⊆ a ∧ closed U ∧ φ U ∧ U ∩ (B n) = {}
else a)"
have [simp]: "A 0 = S"
by (simp add: A_def)
have ASuc: "A(Suc n) = (if ∃U. U ⊆ A n ∧ closed U ∧ φ U ∧ U ∩ (B n) = {}
then SOME U. U ⊆ A n ∧ closed U ∧ φ U ∧ U ∩ (B n) = {}
else A n)" for n
by (auto simp: A_def)
have sub: "⋀n. A(Suc n) ⊆ A n"
by (auto simp: ASuc dest!: someI_ex)
have subS: "A n ⊆ S" for n
by (induction n) (use sub in auto)
have clo: "closed (A n) ∧ φ (A n)" for n
by (induction n) (auto simp: assms ASuc dest!: someI_ex)
show ?thesis
proof
show "⋂(range A) ⊆ S"
using ‹⋀n. A n ⊆ S› by blast
show "closed (⋂(A ` UNIV))"
using clo by blast
show "φ (⋂(A ` UNIV))"
by (simp add: clo φ sub)
show "¬ U ⊂ ⋂(A ` UNIV)" if "U ⊆ S" "closed U" "φ U" for U
proof -
have "∃y. x ∉ A y" if "x ∉ U" and Usub: "U ⊆ (⋂x. A x)" for x
proof -
obtain e where "e > 0" and e: "ball x e ⊆ -U"
using ‹closed U› ‹x ∉ U› openE [of "-U"] by blast
moreover obtain K where K: "ball x e = ⋃(B ` K)"
using open_cov [of "ball x e"] by auto
ultimately have "⋃(B ` K) ⊆ -U"
by blast
have "K ≠ {}"
using ‹0 < e› ‹ball x e = ⋃(B ` K)› by auto
then obtain n where "n ∈ K" "x ∈ B n"
by (metis K UN_E ‹0 < e› centre_in_ball)
then have "U ∩ B n = {}"
using K e by auto
show ?thesis
proof (cases "∃U⊆A n. closed U ∧ φ U ∧ U ∩ B n = {}")
case True
then show ?thesis
apply (rule_tac x="Suc n" in exI)
apply (simp add: ASuc)
apply (erule someI2_ex)
using ‹x ∈ B n› by blast
next
case False
then show ?thesis
by (meson Inf_lower Usub ‹U ∩ B n = {}› ‹φ U› ‹closed U› range_eqI subset_trans)
qed
qed
with that show ?thesis
by (meson Inter_iff psubsetE rangeI subsetI)
qed
qed
qed
corollary Brouwer_reduction_theorem:
fixes S :: "'a::euclidean_space set"
assumes "compact S" "φ S" "S ≠ {}"
and φ: "⋀F. ⟦⋀n. compact(F n); ⋀n. F n ≠ {}; ⋀n. φ(F n); ⋀n. F(Suc n) ⊆ F n⟧ ⟹ φ(⋂(range F))"
obtains T where "T ⊆ S" "compact T" "T ≠ {}" "φ T"
"⋀U. ⟦U ⊆ S; closed U; U ≠ {}; φ U⟧ ⟹ ¬ (U ⊂ T)"
proof (rule Brouwer_reduction_theorem_gen [of S "λT. T ≠ {} ∧ T ⊆ S ∧ φ T"])
fix F
assume cloF: "⋀n. closed (F n)"
and F: "⋀n. F n ≠ {} ∧ F n ⊆ S ∧ φ (F n)" and Fsub: "⋀n. F (Suc n) ⊆ F n"
show "⋂(F ` UNIV) ≠ {} ∧ ⋂(F ` UNIV) ⊆ S ∧ φ (⋂(F ` UNIV))"
proof (intro conjI)
show "⋂(F ` UNIV) ≠ {}"
by (metis F Fsub ‹compact S› cloF closed_Int_compact compact_nest inf.orderE lift_Suc_antimono_le)
show " ⋂(F ` UNIV) ⊆ S"
using F by blast
show "φ (⋂(F ` UNIV))"
by (metis F Fsub φ ‹compact S› cloF closed_Int_compact inf.orderE)
qed
next
show "S ≠ {} ∧ S ⊆ S ∧ φ S"
by (simp add: assms)
qed (meson assms compact_imp_closed seq_compact_closed_subset seq_compact_eq_compact)+
subsection‹Arcwise Connections›
subsection‹Density of points with dyadic rational coordinates›
proposition closure_dyadic_rationals:
"closure (⋃k. ⋃f ∈ Basis → ℤ.
{ ∑i :: 'a :: euclidean_space ∈ Basis. (f i / 2^k) *⇩R i }) = UNIV"
proof -
have "x ∈ closure (⋃k. ⋃f ∈ Basis → ℤ. {∑i ∈ Basis. (f i / 2^k) *⇩R i})" for x::'a
proof (clarsimp simp: closure_approachable)
fix e::real
assume "e > 0"
then obtain k where k: "(1/2)^k < e/DIM('a)"
by (meson DIM_positive divide_less_eq_1_pos of_nat_0_less_iff one_less_numeral_iff real_arch_pow_inv semiring_norm(76) zero_less_divide_iff zero_less_numeral)
have "dist (∑i∈Basis. (real_of_int ⌊2^k*(x ∙ i)⌋ / 2^k) *⇩R i) x =
dist (∑i∈Basis. (real_of_int ⌊2^k*(x ∙ i)⌋ / 2^k) *⇩R i) (∑i∈Basis. (x ∙ i) *⇩R i)"
by (simp add: euclidean_representation)
also have "... = norm ((∑i∈Basis. (real_of_int ⌊2^k*(x ∙ i)⌋ / 2^k) *⇩R i - (x ∙ i) *⇩R i))"
by (simp add: dist_norm sum_subtractf)
also have "... ≤ DIM('a)*((1/2)^k)"
proof (rule sum_norm_bound, simp add: algebra_simps)
fix i::'a
assume "i ∈ Basis"
then have "norm ((real_of_int ⌊x ∙ i*2^k⌋ / 2^k) *⇩R i - (x ∙ i) *⇩R i) =
¦real_of_int ⌊x ∙ i*2^k⌋ / 2^k - x ∙ i¦"
by (simp add: scaleR_left_diff_distrib [symmetric])
also have "... ≤ (1/2) ^ k"
by (simp add: divide_simps) linarith
finally show "norm ((real_of_int ⌊x ∙ i*2^k⌋ / 2^k) *⇩R i - (x ∙ i) *⇩R i) ≤ (1/2) ^ k" .
qed
also have "... < DIM('a)*(e/DIM('a))"
using DIM_positive k linordered_comm_semiring_strict_class.comm_mult_strict_left_mono of_nat_0_less_iff by blast
also have "... = e"
by simp
finally have "dist (∑i∈Basis. (⌊2^k*(x ∙ i)⌋ / 2^k) *⇩R i) x < e" .
with Ints_of_int
show "∃k. ∃f ∈ Basis → ℤ. dist (∑b∈Basis. (f b / 2^k) *⇩R b) x < e"
by fastforce
qed
then show ?thesis by auto
qed
corollary closure_rational_coordinates:
"closure (⋃f ∈ Basis → ℚ. { ∑i :: 'a :: euclidean_space ∈ Basis. f i *⇩R i }) = UNIV"
proof -
have *: "(⋃k. ⋃f ∈ Basis → ℤ. { ∑i::'a ∈ Basis. (f i / 2^k) *⇩R i })
⊆ (⋃f ∈ Basis → ℚ. { ∑i ∈ Basis. f i *⇩R i })"
proof clarsimp
fix k and f :: "'a ⇒ real"
assume f: "f ∈ Basis → ℤ"
show "∃x ∈ Basis → ℚ. (∑i ∈ Basis. (f i / 2^k) *⇩R i) = (∑i ∈ Basis. x i *⇩R i)"
apply (rule_tac x="λi. f i / 2^k" in bexI)
using Ints_subset_Rats f by auto
qed
show ?thesis
using closure_dyadic_rationals closure_mono [OF *] by blast
qed
lemma closure_dyadic_rationals_in_convex_set:
"⟦convex S; interior S ≠ {}⟧
⟹ closure(S ∩
(⋃k. ⋃f ∈ Basis → ℤ.
{ ∑i :: 'a :: euclidean_space ∈ Basis. (f i / 2^k) *⇩R i })) =
closure S"
by (simp add: closure_dyadic_rationals closure_convex_Int_superset)
lemma closure_rationals_in_convex_set:
"⟦convex S; interior S ≠ {}⟧
⟹ closure(S ∩ (⋃f ∈ Basis → ℚ. { ∑i :: 'a :: euclidean_space ∈ Basis. f i *⇩R i })) =
closure S"
by (simp add: closure_rational_coordinates closure_convex_Int_superset)
text‹ Every path between distinct points contains an arc, and hence
path connection is equivalent to arcwise connection for distinct points.
The proof is based on Whyburn's "Topological Analysis".›
lemma closure_dyadic_rationals_in_convex_set_pos_1:
fixes S :: "real set"
assumes "convex S" and intnz: "interior S ≠ {}" and pos: "⋀x. x ∈ S ⟹ 0 ≤ x"
shows "closure(S ∩ (⋃k m. {of_nat m / 2^k})) = closure S"
proof -
have "∃m. f 1/2^k = real m / 2^k" if "(f 1) / 2^k ∈ S" "f 1 ∈ ℤ" for k and f :: "real ⇒ real"
using that by (force simp: Ints_def zero_le_divide_iff power_le_zero_eq dest: pos zero_le_imp_eq_int)
then have "S ∩ (⋃k m. {real m / 2^k}) = S ∩
(⋃k. ⋃f∈Basis → ℤ. {∑i∈Basis. (f i / 2^k) *⇩R i})"
by force
then show ?thesis
using closure_dyadic_rationals_in_convex_set [OF ‹convex S› intnz] by simp
qed
definition dyadics :: "'a::field_char_0 set" where "dyadics ≡ ⋃k m. {of_nat m / 2^k}"
lemma real_in_dyadics [simp]: "real m ∈ dyadics"
by (simp add: dyadics_def) (metis divide_numeral_1 numeral_One power_0)
lemma nat_neq_4k1: "of_nat m ≠ (4 * of_nat k + 1) / (2 * 2^n :: 'a::field_char_0)"
proof
assume "of_nat m = (4 * of_nat k + 1) / (2 * 2^n :: 'a)"
then have "of_nat (m * (2 * 2^n)) = (of_nat (Suc (4 * k)) :: 'a)"
by (simp add: field_split_simps)
then have "m * (2 * 2^n) = Suc (4 * k)"
using of_nat_eq_iff by blast
then have "odd (m * (2 * 2^n))"
by simp
then show False
by simp
qed
lemma nat_neq_4k3: "of_nat m ≠ (4 * of_nat k + 3) / (2 * 2^n :: 'a::field_char_0)"
proof
assume "of_nat m = (4 * of_nat k + 3) / (2 * 2^n :: 'a)"
then have "of_nat (m * (2 * 2^n)) = (of_nat (4 * k + 3) :: 'a)"
by (simp add: field_split_simps)
then have "m * (2 * 2^n) = (4 * k) + 3"
using of_nat_eq_iff by blast
then have "odd (m * (2 * 2^n))"
by simp
then show False
by simp
qed
lemma iff_4k:
assumes "r = real k" "odd k"
shows "(4 * real m + r) / (2 * 2^n) = (4 * real m' + r) / (2 * 2 ^ n') ⟷ m=m' ∧ n=n'"
proof -
{ assume "(4 * real m + r) / (2 * 2^n) = (4 * real m' + r) / (2 * 2 ^ n')"
then have "real ((4 * m + k) * (2 * 2 ^ n')) = real ((4 * m' + k) * (2 * 2^n))"
using assms by (auto simp: field_simps)
then have "(4 * m + k) * (2 * 2 ^ n') = (4 * m' + k) * (2 * 2^n)"
using of_nat_eq_iff by blast
then have "(4 * m + k) * (2 ^ n') = (4 * m' + k) * (2^n)"
by linarith
then obtain "4*m + k = 4*m' + k" "n=n'"
using prime_power_cancel2 [OF two_is_prime_nat] assms
by (metis even_mult_iff even_numeral odd_add)
then have "m=m'" "n=n'"
by auto
}
then show ?thesis by blast
qed
lemma neq_4k1_k43: "(4 * real m + 1) / (2 * 2^n) ≠ (4 * real m' + 3) / (2 * 2 ^ n')"
proof
assume "(4 * real m + 1) / (2 * 2^n) = (4 * real m' + 3) / (2 * 2 ^ n')"
then have "real (Suc (4 * m) * (2 * 2 ^ n')) = real ((4 * m' + 3) * (2 * 2^n))"
by (auto simp: field_simps)
then have "Suc (4 * m) * (2 * 2 ^ n') = (4 * m' + 3) * (2 * 2^n)"
using of_nat_eq_iff by blast
then have "Suc (4 * m) * (2 ^ n') = (4 * m' + 3) * (2^n)"
by linarith
then have "Suc (4 * m) = (4 * m' + 3)"
by (rule prime_power_cancel2 [OF two_is_prime_nat]) auto
then have "1 + 2 * m' = 2 * m"
using ‹Suc (4 * m) = 4 * m' + 3› by linarith
then show False
using even_Suc by presburger
qed
lemma dyadic_413_cases:
obtains "(of_nat m::'a::field_char_0) / 2^k ∈ Nats"
| m' k' where "k' < k" "(of_nat m:: 'a) / 2^k = of_nat (4*m' + 1) / 2^Suc k'"
| m' k' where "k' < k" "(of_nat m:: 'a) / 2^k = of_nat (4*m' + 3) / 2^Suc k'"
proof (cases "m>0")
case False
then have "m=0" by simp
with that show ?thesis by auto
next
case True
obtain k' m' where m': "odd m'" and k': "m = m' * 2^k'"
using prime_power_canonical [OF two_is_prime_nat True] by blast
then obtain q r where q: "m' = 4*q + r" and r: "r < 4"
by (metis not_add_less2 split_div zero_neq_numeral)
show ?thesis
proof (cases "k ≤ k'")
case True
have "(of_nat m:: 'a) / 2^k = of_nat m' * (2 ^ k' / 2^k)"
using k' by (simp add: field_simps)
also have "... = (of_nat m'::'a) * 2 ^ (k'-k)"
using k' True by (simp add: power_diff)
also have "... ∈ ℕ"
by (metis Nats_mult of_nat_in_Nats of_nat_numeral of_nat_power)
finally show ?thesis by (auto simp: that)
next
case False
then obtain kd where kd: "Suc kd = k - k'"
using Suc_diff_Suc not_less by blast
have "(of_nat m:: 'a) / 2^k = of_nat m' * (2 ^ k' / 2^k)"
using k' by (simp add: field_simps)
also have "... = (of_nat m'::'a) / 2 ^ (k-k')"
using k' False by (simp add: power_diff)
also have "... = ((of_nat r + 4 * of_nat q)::'a) / 2 ^ (k-k')"
using q by force
finally have meq: "(of_nat m:: 'a) / 2^k = (of_nat r + 4 * of_nat q) / 2 ^ (k - k')" .
have "r ≠ 0" "r ≠ 2"
using q m' by presburger+
with r consider "r = 1" | "r = 3"
by linarith
then show ?thesis
proof cases
assume "r = 1"
with meq kd that(2) [of kd q] show ?thesis
by simp
next
assume "r = 3"
with meq kd that(3) [of kd q] show ?thesis
by simp
qed
qed
qed
lemma dyadics_iff:
"(dyadics :: 'a::field_char_0 set) =
Nats ∪ (⋃k m. {of_nat (4*m + 1) / 2^Suc k}) ∪ (⋃k m. {of_nat (4*m + 3) / 2^Suc k})"
(is "_ = ?rhs")
proof
show "dyadics ⊆ ?rhs"
unfolding dyadics_def
apply clarify
apply (rule dyadic_413_cases, force+)
done
next
have "range of_nat ⊆ (⋃k m. {(of_nat m::'a) / 2 ^ k})"
by clarsimp (metis divide_numeral_1 numeral_One power_0)
moreover have "⋀k m. ∃k' m'. ((1::'a) + 4 * of_nat m) / 2 ^ Suc k = of_nat m' / 2 ^ k'"
by (metis (no_types) of_nat_Suc of_nat_mult of_nat_numeral)
moreover have "⋀k m. ∃k' m'. (4 * of_nat m + (3::'a)) / 2 ^ Suc k = of_nat m' / 2 ^ k'"
by (metis of_nat_add of_nat_mult of_nat_numeral)
ultimately show "?rhs ⊆ dyadics"
by (auto simp: dyadics_def Nats_def)
qed
function (domintros) dyad_rec :: "[nat ⇒ 'a, 'a⇒'a, 'a⇒'a, real] ⇒ 'a" where
"dyad_rec b l r (real m) = b m"
| "dyad_rec b l r ((4 * real m + 1) / 2 ^ (Suc n)) = l (dyad_rec b l r ((2*m + 1) / 2^n))"
| "dyad_rec b l r ((4 * real m + 3) / 2 ^ (Suc n)) = r (dyad_rec b l r ((2*m + 1) / 2^n))"
| "x ∉ dyadics ⟹ dyad_rec b l r x = undefined"
using iff_4k [of _ 1] iff_4k [of _ 3]
apply (simp_all add: nat_neq_4k1 nat_neq_4k3 neq_4k1_k43 dyadics_iff Nats_def)
by (fastforce simp: field_simps)+
lemma dyadics_levels: "dyadics = (⋃K. ⋃k<K. ⋃ m. {of_nat m / 2^k})"
unfolding dyadics_def by auto
lemma dyad_rec_level_termination:
assumes "k < K"
shows "dyad_rec_dom(b, l, r, real m / 2^k)"
using assms
proof (induction K arbitrary: k m)
case 0
then show ?case by auto
next
case (Suc K)
then consider "k = K" | "k < K"
using less_antisym by blast
then show ?case
proof cases
assume "k = K"
show ?case
proof (rule dyadic_413_cases [of m k, where 'a=real])
show "real m / 2^k ∈ ℕ ⟹ dyad_rec_dom (b, l, r, real m / 2^k)"
by (force simp: Nats_def nat_neq_4k1 nat_neq_4k3 intro: dyad_rec.domintros)
show ?case if "k' < k" and eq: "real m / 2^k = real (4 * m' + 1) / 2^Suc k'" for m' k'
proof -
have "dyad_rec_dom (b, l, r, (4 * real m' + 1) / 2^Suc k')"
proof (rule dyad_rec.domintros)
fix m n
assume "(4 * real m' + 1) / (2 * 2 ^ k') = (4 * real m + 1) / (2 * 2^n)"
then have "m' = m" "k' = n" using iff_4k [of _ 1]
by auto
have "dyad_rec_dom (b, l, r, real (2 * m + 1) / 2 ^ k')"
using Suc.IH ‹k = K› ‹k' < k› by blast
then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)"
using ‹k' = n› by (auto simp: algebra_simps)
next
fix m n
assume "(4 * real m' + 1) / (2 * 2 ^ k') = (4 * real m + 3) / (2 * 2^n)"
then have "False"
by (metis neq_4k1_k43)
then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)" ..
qed
then show ?case by (simp add: eq add_ac)
qed
show ?case if "k' < k" and eq: "real m / 2^k = real (4 * m' + 3) / 2^Suc k'" for m' k'
proof -
have "dyad_rec_dom (b, l, r, (4 * real m' + 3) / 2^Suc k')"
proof (rule dyad_rec.domintros)
fix m n
assume "(4 * real m' + 3) / (2 * 2 ^ k') = (4 * real m + 1) / (2 * 2^n)"
then have "False"
by (metis neq_4k1_k43)
then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)" ..
next
fix m n
assume "(4 * real m' + 3) / (2 * 2 ^ k') = (4 * real m + 3) / (2 * 2^n)"
then have "m' = m" "k' = n" using iff_4k [of _ 3]
by auto
have "dyad_rec_dom (b, l, r, real (2 * m + 1) / 2 ^ k')"
using Suc.IH ‹k = K› ‹k' < k› by blast
then show "dyad_rec_dom (b, l, r, (2 * real m + 1) / 2^n)"
using ‹k' = n› by (auto simp: algebra_simps)
qed
then show ?case by (simp add: eq add_ac)
qed
qed
next
assume "k < K"
then show ?case
using Suc.IH by blast
qed
qed
lemma dyad_rec_termination: "x ∈ dyadics ⟹ dyad_rec_dom(b,l,r,x)"
by (auto simp: dyadics_levels intro: dyad_rec_level_termination)
lemma dyad_rec_of_nat [simp]: "dyad_rec b l r (real m) = b m"
by (simp add: dyad_rec.psimps dyad_rec_termination)
lemma dyad_rec_41 [simp]: "dyad_rec b l r ((4 * real m + 1) / 2 ^ (Suc n)) = l (dyad_rec b l r ((2*m + 1) / 2^n))"
proof (rule dyad_rec.psimps)
show "dyad_rec_dom (b, l, r, (4 * real m + 1) / 2 ^ Suc n)"
by (metis add.commute dyad_rec_level_termination lessI of_nat_Suc of_nat_mult of_nat_numeral)
qed
lemma dyad_rec_43 [simp]: "dyad_rec b l r ((4 * real m + 3) / 2 ^ (Suc n)) = r (dyad_rec b l r ((2*m + 1) / 2^n))"
proof (rule dyad_rec.psimps)
show "dyad_rec_dom (b, l, r, (4 * real m + 3) / 2 ^ Suc n)"
by (metis dyad_rec_level_termination lessI of_nat_add of_nat_mult of_nat_numeral)
qed
lemma dyad_rec_41_times2:
assumes "n > 0"
shows "dyad_rec b l r (2 * ((4 * real m + 1) / 2^Suc n)) = l (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
proof -
obtain n' where n': "n = Suc n'"
using assms not0_implies_Suc by blast
have "dyad_rec b l r (2 * ((4 * real m + 1) / 2^Suc n)) = dyad_rec b l r ((2 * (4 * real m + 1)) / (2 * 2^n))"
by auto
also have "... = dyad_rec b l r ((4 * real m + 1) / 2^n)"
by (subst mult_divide_mult_cancel_left) auto
also have "... = l (dyad_rec b l r ((2 * real m + 1) / 2 ^ n'))"
by (simp add: add.commute [of 1] n' del: power_Suc)
also have "... = l (dyad_rec b l r ((2 * (2 * real m + 1)) / (2 * 2 ^ n')))"
by (subst mult_divide_mult_cancel_left) auto
also have "... = l (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
by (simp add: add.commute n')
finally show ?thesis .
qed
lemma dyad_rec_43_times2:
assumes "n > 0"
shows "dyad_rec b l r (2 * ((4 * real m + 3) / 2^Suc n)) = r (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
proof -
obtain n' where n': "n = Suc n'"
using assms not0_implies_Suc by blast
have "dyad_rec b l r (2 * ((4 * real m + 3) / 2^Suc n)) = dyad_rec b l r ((2 * (4 * real m + 3)) / (2 * 2^n))"
by auto
also have "... = dyad_rec b l r ((4 * real m + 3) / 2^n)"
by (subst mult_divide_mult_cancel_left) auto
also have "... = r (dyad_rec b l r ((2 * real m + 1) / 2 ^ n'))"
by (simp add: n' del: power_Suc)
also have "... = r (dyad_rec b l r ((2 * (2 * real m + 1)) / (2 * 2 ^ n')))"
by (subst mult_divide_mult_cancel_left) auto
also have "... = r (dyad_rec b l r (2 * (2 * real m + 1) / 2^n))"
by (simp add: n')
finally show ?thesis .
qed
definition dyad_rec2
where "dyad_rec2 u v lc rc x =
dyad_rec (λz. (u,v)) (λ(a,b). (a, lc a b (midpoint a b))) (λ(a,b). (rc a b (midpoint a b), b)) (2*x)"
abbreviation leftrec where "leftrec u v lc rc x ≡ fst (dyad_rec2 u v lc rc x)"
abbreviation rightrec where "rightrec u v lc rc x ≡ snd (dyad_rec2 u v lc rc x)"
lemma leftrec_base: "leftrec u v lc rc (real m / 2) = u"
by (simp add: dyad_rec2_def)
lemma leftrec_41: "n > 0 ⟹ leftrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) = leftrec u v lc rc ((2 * real m + 1) / 2^n)"
unfolding dyad_rec2_def dyad_rec_41_times2
by (simp add: case_prod_beta)
lemma leftrec_43: "n > 0 ⟹
leftrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) =
rc (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n))
(midpoint (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n)))"
unfolding dyad_rec2_def dyad_rec_43_times2
by (simp add: case_prod_beta)
lemma rightrec_base: "rightrec u v lc rc (real m / 2) = v"
by (simp add: dyad_rec2_def)
lemma rightrec_41: "n > 0 ⟹
rightrec u v lc rc ((4 * real m + 1) / 2 ^ (Suc n)) =
lc (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n))
(midpoint (leftrec u v lc rc ((2 * real m + 1) / 2^n)) (rightrec u v lc rc ((2 * real m + 1) / 2^n)))"
unfolding dyad_rec2_def dyad_rec_41_times2
by (simp add: case_prod_beta)
lemma rightrec_43: "n > 0 ⟹ rightrec u v lc rc ((4 * real m + 3) / 2 ^ (Suc n)) = rightrec u v lc rc ((2 * real m + 1) / 2^n)"
unfolding dyad_rec2_def dyad_rec_43_times2
by (simp add: case_prod_beta)
lemma dyadics_in_open_unit_interval:
"{0<..<1} ∩ (⋃k m. {real m / 2^k}) = (⋃k. ⋃m ∈ {0<..<2^k}. {real m / 2^k})"
by (auto simp: field_split_simps)
lemma padic_rational_approximation_straddle:
assumes "ε > 0" "p > 1"
obtains n q r
where "of_int q / p^n < x" "x < of_int r / p^n" "¦q / p^n - r / p^n ¦ < ε"
proof -
obtain n where n: "2 / ε < p ^ n"
using ‹p>1› real_arch_pow by blast
define q where "q ≡ ⌊p ^ n * x⌋ - 1"
show thesis
proof
show "q / p ^ n < x" "x < real_of_int (q+2) / p ^ n"
using assms by (simp_all add: q_def divide_simps floor_less_cancel mult.commute)
show "¦q / p ^ n - real_of_int (q+2) / p ^ n¦ < ε"
using assms n by (simp add: q_def divide_simps mult.commute)
qed
qed
lemma padic_rational_approximation_straddle_pos:
assumes "ε > 0" "p > 1" "x > 0"
obtains n q r
where "of_nat q / p^n < x" "x < of_nat r / p^n" "¦q / p^n - r / p^n ¦ < ε"
proof -
obtain n q r
where *: "of_int q / p^n < x" "x < of_int r / p^n" "¦q / p^n - r / p^n ¦ < ε"
using padic_rational_approximation_straddle assms by metis
then have "r ≥ 0"
using assms by (smt (verit, best) divide_nonpos_pos of_int_0_le_iff zero_less_power)
show thesis
proof
show "real (max 0 (nat q)) / p ^ n < x"
using * by (metis assms(3) div_0 max_nat.left_neutral nat_eq_iff2 of_nat_0 of_nat_nat)
show "x < real (nat r) / p ^ n"
using ‹r ≥ 0› * by force
show "¦real (max 0 (nat q)) / p ^ n - real (nat r) / p ^ n¦ < ε"
using * assms by (simp add: divide_simps)
qed
qed
lemma padic_rational_approximation_straddle_pos_le:
assumes "ε > 0" "p > 1" "x ≥ 0"
obtains n q r
where "of_nat q / p^n ≤ x" "x < of_nat r / p^n" "¦q / p^n - r / p^n ¦ < ε"
proof -
obtain n q r
where *: "of_int q / p^n < x" "x < of_int r / p^n" "¦q / p^n - r / p^n ¦ < ε"
using padic_rational_approximation_straddle assms by metis
then have "r ≥ 0"
using assms by (smt (verit, best) divide_nonpos_pos of_int_0_le_iff zero_less_power)
show thesis
proof
show "real (max 0 (nat q)) / p ^ n ≤ x"
using * assms(3) nle_le by fastforce
show "x < real (nat r) / p ^ n"
using ‹r ≥ 0› * by force
show "¦real (max 0 (nat q)) / p ^ n - real (nat r) / p ^ n¦ < ε"
using * assms by (simp add: divide_simps)
qed
qed
subsubsection ‹Definition by recursion on dyadic rationals in [0,1]›
lemma recursion_on_dyadic_fractions:
assumes base: "R a b"
and step: "⋀x y. R x y ⟹ ∃z. R x z ∧ R z y" and trans: "⋀x y z. ⟦R x y; R y z⟧ ⟹ R x z"
shows "∃f :: real ⇒ 'a. f 0 = a ∧ f 1 = b ∧
(∀x ∈ dyadics ∩ {0..1}. ∀y ∈ dyadics ∩ {0..1}. x < y ⟶ R (f x) (f y))"
proof -
obtain mid where mid: "R x y ⟹ R x (mid x y)" "R x y ⟹ R (mid x y) y" for x y
using step by metis
define g where "g ≡ rec_nat (λk. if k = 0 then a else b) (λn r k. if even k then r (k div 2) else mid (r ((k - 1) div 2)) (r ((Suc k) div 2)))"
have g0 [simp]: "g 0 = (λk. if k = 0 then a else b)"
by (simp add: g_def)
have gSuc [simp]: "⋀n. g(Suc n) = (λk. if even k then g n (k div 2) else mid (g n ((k - 1) div 2)) (g n ((Suc k) div 2)))"
by (auto simp: g_def)
have g_eq_g: "2 ^ d * k = k' ⟹ g n k = g (n + d) k'" for n d k k'
by (induction d arbitrary: k k') auto
have "g n k = g n' k'" if "real k / 2^n = real k' / 2^n'" "n' ≤ n" for k n k' n'
proof -
have "real k = real k' * 2 ^ (n-n')"
using that by (simp add: power_diff divide_simps)
then have "k = k' * 2 ^ (n-n')"
using of_nat_eq_iff by fastforce
with g_eq_g show ?thesis
by (metis le_add_diff_inverse mult.commute that(2))
qed
then have g_eq_g: "g n k = g n' k'" if "real k / 2 ^ n = real k' / 2 ^ n'" for k n k' n'
by (metis nat_le_linear that)
then obtain f where "(λ(k,n). g n k) = f ∘ (λ(k,n). k / 2 ^ n)"
using function_factors_left by (smt (verit, del_insts) case_prod_beta')
then have f_eq_g: "⋀k n. f(real k / 2 ^ n) = g n k"
by (simp add: fun_eq_iff)
show ?thesis
proof (intro exI conjI strip)
show "f 0 = a"
by (metis f_eq_g g0 div_0 of_nat_0)
show "f 1 = b"
by (metis f_eq_g g0 div_by_1 of_nat_1_eq_iff power_0 zero_neq_one)
show "R (f x) (f y)"
if x: "x ∈ dyadics ∩ {0..1}" and y: "y ∈ dyadics ∩ {0..1}" and "x < y" for x y
proof -
obtain n1 k1 where xeq: "x = real k1 / 2^n1" "k1 ≤ 2^n1"
using x by (auto simp: dyadics_def)
obtain n2 k2 where yeq: "y = real k2 / 2^n2" "k2 ≤ 2^n2"
using y by (auto simp: dyadics_def)
have xcommon: "x = real(2^n2 * k1) / 2 ^ (n1+n2)"
using xeq by (simp add: power_add)
have ycommon: "y = real(2^n1 * k2) / 2 ^ (n1+n2)"
using yeq by (simp add: power_add)
have *: "R (g n j) (g n k)" if "j < k" "k ≤ 2^n" for n j k
using that
proof (induction n arbitrary: j k)
case 0
then show ?case
by (simp add: base)
next
case (Suc n)
show ?case
proof (cases "even j")
case True
then obtain a where [simp]: "j = 2*a"
by blast
show ?thesis
proof (cases "even k")
case True
with Suc show ?thesis
by (auto elim!: evenE)
next
case False
then obtain b where [simp]: "k = Suc (2*b)"
using oddE by fastforce
show ?thesis
using Suc
apply simp
by (smt (verit, ccfv_SIG) less_Suc_eq linorder_not_le local.trans mid(1) nat_mult_less_cancel1 pos2)
qed
next
case False
then obtain a where [simp]: "j = Suc (2*a)"
using oddE by fastforce
show ?thesis
proof (cases "even k")
case True
then obtain b where [simp]: "k = 2*b"
by blast
show ?thesis
using Suc
apply simp
by (smt (verit, ccfv_SIG) Suc_leI Suc_lessD le_trans lessI linorder_neqE_nat linorder_not_le local.trans mid(2) nat_mult_less_cancel1 pos2)
next
case False
then obtain b where [simp]: "k = Suc (2*b)"
using oddE by fastforce
show ?thesis
using Suc
apply simp
by (smt (verit) Suc_leI le_trans lessI less_or_eq_imp_le linorder_neqE_nat linorder_not_le local.trans mid(1) mid(2) nat_mult_less_cancel1 pos2)
qed
qed
qed
show ?thesis
unfolding xcommon ycommon f_eq_g
proof (rule *)
show "2 ^ n2 * k1 < 2 ^ n1 * k2"
using of_nat_less_iff ‹x < y› by (fastforce simp: xeq yeq field_simps)
show "2 ^ n1 * k2 ≤ 2 ^ (n1 + n2)"
by (simp add: power_add yeq)
qed
qed
qed
qed
lemma dyadics_add:
assumes "x ∈ dyadics" "y ∈ dyadics"
shows "x+y ∈ dyadics"
proof -
obtain i j m n where x: "x = of_nat i / 2 ^ m" and y: "y = of_nat j / 2 ^ n"
using assms by (auto simp: dyadics_def)
have xcommon: "x = of_nat(2^n * i) / 2 ^ (m+n)"
using x by (simp add: power_add)
moreover
have ycommon: "y = of_nat(2^m * j) / 2 ^ (m+n)"
using y by (simp add: power_add)
ultimately have "x+y = (of_nat(2^n * i + 2^m * j)) / 2 ^ (m+n)"
by (simp add: field_simps)
then show ?thesis
unfolding dyadics_def by blast
qed
lemma dyadics_diff:
fixes x :: "'a::linordered_field"
assumes "x ∈ dyadics" "y ∈ dyadics" "y ≤ x"
shows "x-y ∈ dyadics"
proof -
obtain i j m n where x: "x = of_nat i / 2 ^ m" and y: "y = of_nat j / 2 ^ n"
using assms by (auto simp: dyadics_def)
have j_le_i: "j * 2 ^ m ≤ i * 2 ^ n"
using of_nat_le_iff ‹y ≤ x› unfolding x y by (fastforce simp add: divide_simps)
have xcommon: "x = of_nat(2^n * i) / 2 ^ (m+n)"
using x by (simp add: power_add)
moreover
have ycommon: "y = of_nat(2^m * j) / 2 ^ (m+n)"
using y by (simp add: power_add)
ultimately have "x-y = (of_nat(2^n * i - 2^m * j)) / 2 ^ (m+n)"
by (simp add: xcommon ycommon field_simps j_le_i of_nat_diff)
then show ?thesis
unfolding dyadics_def by blast
qed
theorem homeomorphic_monotone_image_interval:
fixes f :: "real ⇒ 'a::{real_normed_vector,complete_space}"
assumes cont_f: "continuous_on {0..1} f"
and conn: "⋀y. connected ({0..1} ∩ f -` {y})"
and f_1not0: "f 1 ≠ f 0"
shows "(f ` {0..1}) homeomorphic {0..1::real}"
proof -
have "∃c d. a ≤ c ∧ c ≤ m ∧ m ≤ d ∧ d ≤ b ∧
(∀x ∈ {c..d}. f x = f m) ∧
(∀x ∈ {a..<c}. (f x ≠ f m)) ∧
(∀x ∈ {d<..b}. (f x ≠ f m)) ∧
(∀x ∈ {a..<c}. ∀y ∈ {d<..b}. f x ≠ f y)"
if m: "m ∈ {a..b}" and ab01: "{a..b} ⊆ {0..1}" for a b m
proof -
have comp: "compact (f -` {f m} ∩ {0..1})"
by (simp add: compact_eq_bounded_closed bounded_Int closed_vimage_Int cont_f)
obtain c0 d0 where cd0: "{0..1} ∩ f -` {f m} = {c0..d0}"
using connected_compact_interval_1 [of "{0..1} ∩ f -` {f m}"] conn comp
by (metis Int_commute)
with that have "m ∈ cbox c0 d0"
by auto
obtain c d where cd: "{a..b} ∩ f -` {f m} = {c..d}"
using ab01 cd0
by (rule_tac c="max a c0" and d="min b d0" in that) auto
then have cdab: "{c..d} ⊆ {a..b}"
by blast
show ?thesis
proof (intro exI conjI ballI)
show "a ≤ c" "d ≤ b"
using cdab cd m by auto
show "c ≤ m" "m ≤ d"
using cd m by auto
show "⋀x. x ∈ {c..d} ⟹ f x = f m"
using cd by blast
show "f x ≠ f m" if "x ∈ {a..<c}" for x
using that m cd [THEN equalityD1, THEN subsetD] ‹c ≤ m› by force
show "f x ≠ f m" if "x ∈ {d<..b}" for x
using that m cd [THEN equalityD1, THEN subsetD, of x] ‹m ≤ d› by force
show "f x ≠ f y" if "x ∈ {a..<c}" "y ∈ {d<..b}" for x y
proof (cases "f x = f m ∨ f y = f m")
case True
then show ?thesis
using ‹⋀x. x ∈ {a..<c} ⟹ f x ≠ f m› that by auto
next
case False
have False if "f x = f y"
proof -
have "x ≤ m" "m ≤ y"
using ‹c ≤ m› ‹x ∈ {a..<c}› ‹m ≤ d› ‹y ∈ {d<..b}› by auto
then have "x ∈ ({0..1} ∩ f -` {f y})" "y ∈ ({0..1} ∩ f -` {f y})"
using ‹x ∈ {a..<c}› ‹y ∈ {d<..b}› ab01 by (auto simp: that)
then have "m ∈ ({0..1} ∩ f -` {f y})"
by (meson ‹m ≤ y› ‹x ≤ m› is_interval_connected_1 conn [of "f y"] is_interval_1)
with False show False by auto
qed
then show ?thesis by auto
qed
qed
qed
then obtain leftcut rightcut where LR:
"⋀a b m. ⟦m ∈ {a..b}; {a..b} ⊆ {0..1}⟧ ⟹
(a ≤ leftcut a b m ∧ leftcut a b m ≤ m ∧ m ≤ rightcut a b m ∧ rightcut a b m ≤ b ∧
(∀x ∈ {leftcut a b m..rightcut a b m}. f x = f m) ∧
(∀x ∈ {a..<leftcut a b m}. f x ≠ f m) ∧
(∀x ∈ {rightcut a b m<..b}. f x ≠ f m) ∧
(∀x ∈ {a..<leftcut a b m}. ∀y ∈ {rightcut a b m<..b}. f x ≠ f y))"
apply atomize
apply (clarsimp simp only: imp_conjL [symmetric] choice_iff choice_iff')
apply (rule that, blast)
done
then have left_right: "⋀a b m. ⟦m ∈ {a..b}; {a..b} ⊆ {0..1}⟧ ⟹ a ≤ leftcut a b m ∧ rightcut a b m ≤ b"
and left_right_m: "⋀a b m. ⟦m ∈ {a..b}; {a..b} ⊆ {0..1}⟧ ⟹ leftcut a b m ≤ m ∧ m ≤ rightcut a b m"
by auto
have left_neq: "⟦a ≤ x; x < leftcut a b m; a ≤ m; m ≤ b; {a..b} ⊆ {0..1}⟧ ⟹ f x ≠ f m"
and right_neq: "⟦rightcut a b m < x; x ≤ b; a ≤ m; m ≤ b; {a..b} ⊆ {0..1}⟧ ⟹ f x ≠ f m"
and left_right_neq: "⟦a ≤ x; x < leftcut a b m; rightcut a b m < y; y ≤ b; a ≤ m; m ≤ b; {a..b} ⊆ {0..1}⟧ ⟹ f x ≠ f m"
and feqm: "⟦leftcut a b m ≤ x; x ≤ rightcut a b m; a ≤ m; m ≤ b; {a..b} ⊆ {0..1}⟧
⟹ f x = f m" for a b m x y
by (meson atLeastAtMost_iff greaterThanAtMost_iff atLeastLessThan_iff LR)+
have f_eqI: "⋀a b m x y. ⟦leftcut a b m ≤ x; x ≤ rightcut a b m; leftcut a b m ≤ y; y ≤ rightcut a b m;
a ≤ m; m ≤ b; {a..b} ⊆ {0..1}⟧ ⟹ f x = f y"
by (metis feqm)
define u where "u ≡ rightcut 0 1 0"
have lc[simp]: "leftcut 0 1 0 = 0" and u01: "0 ≤ u" "u ≤ 1"
using LR [of 0 0 1] by (auto simp: u_def)
have f0u: "⋀x. x ∈ {0..u} ⟹ f x = f 0"
using LR [of 0 0 1] unfolding u_def [symmetric]
by (metis ‹leftcut 0 1 0 = 0› atLeastAtMost_iff order_refl zero_le_one)
have fu1: "⋀x. x ∈ {u<..1} ⟹ f x ≠ f 0"
using LR [of 0 0 1] unfolding u_def [symmetric] by fastforce
define v where "v ≡ leftcut u 1 1"
have rc[simp]: "rightcut u 1 1 = 1" and v01: "u ≤ v" "v ≤ 1"
using LR [of 1 u 1] u01 by (auto simp: v_def)
have fuv: "⋀x. x ∈ {u..<v} ⟹ f x ≠ f 1"
using LR [of 1 u 1] u01 v_def by fastforce
have f0v: "⋀x. x ∈ {0..<v} ⟹ f x ≠ f 1"
by (metis f_1not0 atLeastAtMost_iff atLeastLessThan_iff f0u fuv linear)
have fv1: "⋀x. x ∈ {v..1} ⟹ f x = f 1"
using LR [of 1 u 1] u01 v_def by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl rc)
define a where "a ≡ leftrec u v leftcut rightcut"
define b where "b ≡ rightrec u v leftcut rightcut"
define c where "c ≡ λx. midpoint (a x) (b x)"
have a_real [simp]: "a (real j) = u" for j
using a_def leftrec_base
by (metis nonzero_mult_div_cancel_right of_nat_mult of_nat_numeral zero_neq_numeral)
have b_real [simp]: "b (real j) = v" for j
using b_def rightrec_base
by (metis nonzero_mult_div_cancel_right of_nat_mult of_nat_numeral zero_neq_numeral)
have a41: "a ((4 * real m + 1) / 2^Suc n) = a ((2 * real m + 1) / 2^n)" if "n > 0" for m n
using that a_def leftrec_41 by blast
have b41: "b ((4 * real m + 1) / 2^Suc n) =
leftcut (a ((2 * real m + 1) / 2^n))
(b ((2 * real m + 1) / 2^n))
(c ((2 * real m + 1) / 2^n))" if "n > 0" for m n
using that a_def b_def c_def rightrec_41 by blast
have a43: "a ((4 * real m + 3) / 2^Suc n) =
rightcut (a ((2 * real m + 1) / 2^n))
(b ((2 * real m + 1) / 2^n))
(c ((2 * real m + 1) / 2^n))" if "n > 0" for m n
using that a_def b_def c_def leftrec_43 by blast
have b43: "b ((4 * real m + 3) / 2^Suc n) = b ((2 * real m + 1) / 2^n)" if "n > 0" for m n
using that b_def rightrec_43 by blast
have uabv: "u ≤ a (real m / 2 ^ n) ∧ a (real m / 2 ^ n) ≤ b (real m / 2 ^ n) ∧ b (real m / 2 ^ n) ≤ v" for m n
proof (induction n arbitrary: m)
case 0
then show ?case by (simp add: v01)
next
case (Suc n p)
show ?case
proof (cases "even p")
case True
then obtain m where "p = 2*m" by (metis evenE)
then show ?thesis
by (simp add: Suc.IH)
next
case False
then obtain m where m: "p = 2*m + 1" by (metis oddE)
show ?thesis
proof (cases n)
case 0
then show ?thesis
by (simp add: a_def b_def leftrec_base rightrec_base v01)
next
case (Suc n')
then have "n > 0" by simp
have a_le_c: "a (real m / 2^n) ≤ c (real m / 2^n)" for m
unfolding c_def by (metis Suc.IH ge_midpoint_1)
have c_le_b: "c (real m / 2^n) ≤ b (real m / 2^n)" for m
unfolding c_def by (metis Suc.IH le_midpoint_1)
have c_ge_u: "c (real m / 2^n) ≥ u" for m
using Suc.IH a_le_c order_trans by blast
have c_le_v: "c (real m / 2^n) ≤ v" for m
using Suc.IH c_le_b order_trans by blast
have a_ge_0: "0 ≤ a (real m / 2^n)" for m
using Suc.IH order_trans u01(1) by blast
have b_le_1: "b (real m / 2^n) ≤ 1" for m
using Suc.IH order_trans v01(2) by blast
have left_le: "leftcut (a ((real m) / 2^n)) (b ((real m) / 2^n)) (c ((real m) / 2^n)) ≤ c ((real m) / 2^n)" for m
by (simp add: LR a_ge_0 a_le_c b_le_1 c_le_b)
have right_ge: "rightcut (a ((real m) / 2^n)) (b ((real m) / 2^n)) (c ((real m) / 2^n)) ≥ c ((real m) / 2^n)" for m
by (simp add: LR a_ge_0 a_le_c b_le_1 c_le_b)
show ?thesis
proof (cases "even m")
case True
then obtain r where r: "m = 2*r" by (metis evenE)
show ?thesis
using order_trans [OF left_le c_le_v, of "1+2*r"] Suc.IH [of "m+1"]
using a_le_c [of "m+1"] c_le_b [of "m+1"] a_ge_0 [of "m+1"] b_le_1 [of "m+1"] left_right ‹n > 0›
by (simp_all add: r m add.commute [of 1] a41 b41 del: power_Suc)
next
case False
then obtain r where r: "m = 2*r + 1" by (metis oddE)
show ?thesis
using order_trans [OF c_ge_u right_ge, of "1+2*r"] Suc.IH [of "m"]
using a_le_c [of "m"] c_le_b [of "m"] a_ge_0 [of "m"] b_le_1 [of "m"] left_right ‹n > 0›
apply (simp_all add: r m add.commute [of 3] a43 b43 del: power_Suc)
by (simp add: add.commute)
qed
qed
qed
qed
have a_ge_0 [simp]: "0 ≤ a(m / 2^n)" and b_le_1 [simp]: "b(m / 2^n) ≤ 1" for m::nat and n
using uabv order_trans u01 v01 by blast+
then have b_ge_0 [simp]: "0 ≤ b(m / 2^n)" and a_le_1 [simp]: "a(m / 2^n) ≤ 1" for m::nat and n
using uabv order_trans by blast+
have alec [simp]: "a(m / 2^n) ≤ c(m / 2^n)" and cleb [simp]: "c(m / 2^n) ≤ b(m / 2^n)" for m::nat and n
by (auto simp: c_def ge_midpoint_1 le_midpoint_1 uabv)
have c_ge_0 [simp]: "0 ≤ c(m / 2^n)" and c_le_1 [simp]: "c(m / 2^n) ≤ 1" for m::nat and n
using a_ge_0 alec b_le_1 cleb order_trans by blast+
have "⟦d = m-n; odd j; ¦real i / 2^m - real j / 2^n¦ < 1/2 ^ n⟧
⟹ (a(j / 2^n)) ≤ (c(i / 2^m)) ∧ (c(i / 2^m)) ≤ (b(j / 2^n))" for d i j m n
proof (induction d arbitrary: j n rule: less_induct)
case (less d j n)
show ?case
proof (cases "m ≤ n")
case True
have "¦2^n¦ * ¦real i / 2^m - real j / 2^n¦ = 0"
proof (rule Ints_nonzero_abs_less1)
have "(real i * 2^n - real j * 2^m) / 2^m = (real i * 2^n) / 2^m - (real j * 2^m) / 2^m"
using diff_divide_distrib by blast
also have "... = (real i * 2 ^ (n-m)) - (real j)"
using True by (auto simp: power_diff field_simps)
also have "... ∈ ℤ"
by simp
finally have "(real i * 2^n - real j * 2^m) / 2^m ∈ ℤ" .
with True Ints_abs show "¦2^n¦ * ¦real i / 2^m - real j / 2^n¦ ∈ ℤ"
by (fastforce simp: field_split_simps)
show "¦¦2^n¦ * ¦real i / 2^m - real j / 2^n¦¦ < 1"
using less.prems by (auto simp: field_split_simps)
qed
then have "real i / 2^m = real j / 2^n"
by auto
then show ?thesis
by auto
next
case False
then have "n < m" by auto
obtain k where k: "j = Suc (2*k)"
using ‹odd j› oddE by fastforce
show ?thesis
proof (cases "n > 0")
case False
then have "a (real j / 2^n) = u"
by simp
also have "... ≤ c (real i / 2^m)"
using alec uabv by (blast intro: order_trans)
finally have ac: "a (real j / 2^n) ≤ c (real i / 2^m)" .
have "c (real i / 2^m) ≤ v"
using cleb uabv by (blast intro: order_trans)
also have "... = b (real j / 2^n)"
using False by simp
finally show ?thesis
by (auto simp: ac)
next
case True show ?thesis
proof (cases "i / 2^m" "j / 2^n" rule: linorder_cases)
case less
moreover have "real (4 * k + 1) / 2 ^ Suc n + 1 / (2 ^ Suc n) = real j / 2 ^ n"
using k by (force simp: field_split_simps)
moreover have "¦real i / 2 ^ m - j / 2 ^ n¦ < 2 / (2 ^ Suc n)"
using less.prems by simp
ultimately have closer: "¦real i / 2 ^ m - real (4 * k + 1) / 2 ^ Suc n¦ < 1 / (2 ^ Suc n)"
using less.prems by linarith
have "a (real (4 * k + 1) / 2 ^ Suc n) ≤ c (i / 2 ^ m) ∧
c (real i / 2 ^ m) ≤ b (real (4 * k + 1) / 2 ^ Suc n)"
proof (rule less.IH [OF _ refl])
show "m - Suc n < d"
using ‹n < m› diff_less_mono2 less.prems(1) lessI by presburger
show "¦real i / 2 ^ m - real (4 * k + 1) / 2 ^ Suc n¦ < 1 / 2 ^ Suc n"
using closer ‹n < m› ‹d = m - n› by (auto simp: field_split_simps ‹n < m› diff_less_mono2)
qed auto
then show ?thesis
using LR [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
using alec [of "2*k+1"] cleb [of "2*k+1"] a_ge_0 [of "2*k+1"] b_le_1 [of "2*k+1"]
using k a41 b41 ‹0 < n›
by (simp add: add.commute)
next
case equal then show ?thesis by simp
next
case greater
moreover have "real (4 * k + 3) / 2 ^ Suc n - 1 / (2 ^ Suc n) = real j / 2 ^ n"
using k by (force simp: field_split_simps)
moreover have "¦real i / 2 ^ m - real j / 2 ^ n¦ < 2 * 1 / (2 ^ Suc n)"
using less.prems by simp
ultimately have closer: "¦real i / 2 ^ m - real (4 * k + 3) / 2 ^ Suc n¦ < 1 / (2 ^ Suc n)"
using less.prems by linarith
have "a (real (4 * k + 3) / 2 ^ Suc n) ≤ c (real i / 2 ^ m) ∧
c (real i / 2 ^ m) ≤ b (real (4 * k + 3) / 2 ^ Suc n)"
proof (rule less.IH [OF _ refl])
show "m - Suc n < d"
using ‹n < m› diff_less_mono2 less.prems(1) by blast
show "¦real i / 2 ^ m - real (4 * k + 3) / 2 ^ Suc n¦ < 1 / 2 ^ Suc n"
using closer ‹n < m› ‹d = m - n› by (auto simp: field_split_simps ‹n < m› diff_less_mono2)
qed auto
then show ?thesis
using LR [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
using alec [of "2*k+1"] cleb [of "2*k+1"] a_ge_0 [of "2*k+1"] b_le_1 [of "2*k+1"]
using k a43 b43 ‹0 < n›
by (simp add: add.commute)
qed
qed
qed
qed
then have aj_le_ci: "a (real j / 2 ^ n) ≤ c (real i / 2 ^ m)"
and ci_le_bj: "c (real i / 2 ^ m) ≤ b (real j / 2 ^ n)" if "odd j" "¦real i / 2^m - real j / 2^n¦ < 1/2 ^ n" for i j m n
using that by blast+
have close_ab: "odd m ⟹ ¦a (real m / 2 ^ n) - b (real m / 2 ^ n)¦ ≤ 2 / 2^n" for m n
proof (induction n arbitrary: m)
case 0
with u01 v01 show ?case by auto
next
case (Suc n m)
with oddE obtain k where k: "m = Suc (2*k)" by fastforce
show ?case
proof (cases "n > 0")
case False
with u01 v01 show ?thesis
by (simp add: a_def b_def leftrec_base rightrec_base)
next
case True
show ?thesis
proof (cases "even k")
case True
then obtain j where j: "k = 2*j" by (metis evenE)
have "¦a ((2 * real j + 1) / 2 ^ n) - (b ((2 * real j + 1) / 2 ^ n))¦ ≤ 2/2 ^ n"
proof -
have "odd (Suc k)"
using True by auto
then show ?thesis
by (metis (no_types) Groups.add_ac(2) Suc.IH j of_nat_Suc of_nat_mult of_nat_numeral)
qed
moreover have "a ((2 * real j + 1) / 2 ^ n) ≤
leftcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n))"
using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
by (auto simp: add.commute left_right)
moreover have "leftcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n)) ≤
c ((2 * real j + 1) / 2 ^ n)"
using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
by (auto simp: add.commute left_right_m)
ultimately have "¦a ((2 * real j + 1) / 2 ^ n) -
leftcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n))¦
≤ 2/2 ^ Suc n"
by (simp add: c_def midpoint_def)
with j k ‹n > 0› show ?thesis
by (simp add: add.commute [of 1] a41 b41 del: power_Suc)
next
case False
then obtain j where j: "k = 2*j + 1" by (metis oddE)
have "¦a ((2 * real j + 1) / 2 ^ n) - (b ((2 * real j + 1) / 2 ^ n))¦ ≤ 2/2 ^ n"
using Suc.IH [OF False] j by (auto simp: algebra_simps)
moreover have "c ((2 * real j + 1) / 2 ^ n) ≤
rightcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n))"
using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
by (auto simp: add.commute left_right_m)
moreover have "rightcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n)) ≤
b ((2 * real j + 1) / 2 ^ n)"
using alec [of "2*j+1"] cleb [of "2*j+1"] a_ge_0 [of "2*j+1"] b_le_1 [of "2*j+1"]
by (auto simp: add.commute left_right)
ultimately have "¦rightcut (a ((2 * real j + 1) / 2 ^ n)) (b ((2 * real j + 1) / 2 ^ n)) (c ((2 * real j + 1) / 2 ^ n)) -
b ((2 * real j + 1) / 2 ^ n)¦ ≤ 2/2 ^ Suc n"
by (simp add: c_def midpoint_def)
with j k ‹n > 0› show ?thesis
by (simp add: add.commute [of 3] a43 b43 del: power_Suc)
qed
qed
qed
have m1_to_3: "4 * real k - 1 = real (4 * (k-1)) + 3" if "0 < k" for k
using that by auto
have fb_eq_fa: "⟦0 < j; 2*j < 2 ^ n⟧ ⟹ f(b((2 * real j - 1) / 2^n)) = f(a((2 * real j + 1) / 2^n))" for n j
proof (induction n arbitrary: j)
case 0
then show ?case by auto
next
case (Suc n j) show ?case
proof (cases "n > 0")
case False
with Suc.prems show ?thesis by auto
next
case True
show ?thesis proof (cases "even j")
case True
then obtain k where k: "j = 2*k" by (metis evenE)
with ‹0 < j› have "k > 0" "2 * k < 2 ^ n"
using Suc.prems(2) k by auto
with k ‹0 < n› Suc.IH [of k] show ?thesis
by (simp add: m1_to_3 a41 b43 del: power_Suc) (auto simp: of_nat_diff)
next
case False
then obtain k where k: "j = 2*k + 1" by (metis oddE)
have "f (leftcut (a ((2 * k + 1) / 2^n)) (b ((2 * k + 1) / 2^n)) (c ((2 * k + 1) / 2^n)))
= f (c ((2 * k + 1) / 2^n))"
"f (c ((2 * k + 1) / 2^n))
= f (rightcut (a ((2 * k + 1) / 2^n)) (b ((2 * k + 1) / 2^n)) (c ((2 * k + 1) / 2^n)))"
using alec [of "2*k+1" n] cleb [of "2*k+1" n] a_ge_0 [of "2*k+1" n] b_le_1 [of "2*k+1" n] k
using left_right_m [of "c((2*k + 1) / 2^n)" "a((2*k + 1) / 2^n)" "b((2*k + 1) / 2^n)"]
by (auto simp: add.commute feqm [OF order_refl] feqm [OF _ order_refl, symmetric])
then
show ?thesis
by (simp add: k add.commute [of 1] add.commute [of 3] a43 b41‹0 < n› del: power_Suc)
qed
qed
qed
have f_eq_fc: "⟦0 < j; j < 2 ^ n⟧
⟹ f(b((2*j - 1) / 2 ^ (Suc n))) = f(c(j / 2^n)) ∧
f(a((2*j + 1) / 2 ^ (Suc n))) = f(c(j / 2^n))" for n and j::nat
proof (induction n arbitrary: j)
case 0
then show ?case by auto
next
case (Suc n)
show ?case
proof (cases "even j")
case True
then obtain k where k: "j = 2*k" by (metis evenE)
then have less2n: "k < 2 ^ n"
using Suc.prems(2) by auto
have "0 < k" using ‹0 < j› k by linarith
then have m1_to_3: "real (4 * k - Suc 0) = real (4 * (k-1)) + 3"
by auto
then show ?thesis
using Suc.IH [of k] k ‹0 < k›
by (simp add: less2n add.commute [of 1] m1_to_3 a41 b43 del: power_Suc) (auto simp: of_nat_diff)
next
case False
then obtain k where k: "j = 2*k + 1" by (metis oddE)
with Suc.prems have "k < 2^n" by auto
show ?thesis
using alec [of "2*k+1" "Suc n"] cleb [of "2*k+1" "Suc n"] a_ge_0 [of "2*k+1" "Suc n"] b_le_1 [of "2*k+1" "Suc n"] k
using left_right_m [of "c((2*k + 1) / 2 ^ Suc n)" "a((2*k + 1) / 2 ^ Suc n)" "b((2*k + 1) / 2 ^ Suc n)"]
apply (simp add: add.commute [of 1] add.commute [of 3] m1_to_3 b41 a43 del: power_Suc)
apply (force intro: feqm)
done
qed
qed
define D01 where "D01 ≡ {0<..<1} ∩ (⋃k m. {real m / 2^k})"
have cloD01 [simp]: "closure D01 = {0..1}"
unfolding D01_def
by (subst closure_dyadic_rationals_in_convex_set_pos_1) auto
have "uniformly_continuous_on D01 (f ∘ c)"
proof (clarsimp simp: uniformly_continuous_on_def)
fix e::real
assume "0 < e"
have ucontf: "uniformly_continuous_on {0..1} f"
by (simp add: compact_uniformly_continuous [OF cont_f])
then obtain d where "0 < d" and d: "⋀x x'. ⟦x ∈ {0..1}; x' ∈ {0..1}; norm (x' - x) < d⟧ ⟹ norm (f x' - f x) < e/2"
unfolding uniformly_continuous_on_def dist_norm
by (metis ‹0 < e› less_divide_eq_numeral1(1) mult_zero_left)
obtain n where n: "1/2^n < min d 1"
by (metis ‹0 < d› divide_less_eq_1 less_numeral_extra(1) min_def one_less_numeral_iff power_one_over real_arch_pow_inv semiring_norm(76) zero_less_numeral)
with gr0I have "n > 0"
by (force simp: field_split_simps)
show "∃d>0. ∀x∈D01. ∀x'∈D01. dist x' x < d ⟶ dist (f (c x')) (f (c x)) < e"
proof (intro exI ballI impI conjI)
show "(0::real) < 1/2^n" by auto
next
have dist_fc_close: "dist (f(c(real i / 2^m))) (f(c(real j / 2^n))) < e/2"
if i: "0 < i" "i < 2 ^ m" and j: "0 < j" "j < 2 ^ n" and clo: "abs(i / 2^m - j / 2^n) < 1/2 ^ n" for i j m
proof -
have abs3: "¦x - a¦ < e ⟹ x = a ∨ ¦x - (a - e/2)¦ < e/2 ∨ ¦x - (a + e/2)¦ < e/2" for x a e::real
by linarith
consider "i / 2 ^ m = j / 2 ^ n"
| "¦i / 2 ^ m - (2 * j - 1) / 2 ^ Suc n¦ < 1/2 ^ Suc n"
| "¦i / 2 ^ m - (2 * j + 1) / 2 ^ Suc n¦ < 1/2 ^ Suc n"
using abs3 [OF clo] j by (auto simp: field_simps of_nat_diff)
then show ?thesis
proof cases
case 1 with ‹0 < e› show ?thesis by auto
next
case 2
have *: "abs(a - b) ≤ 1/2 ^ n ∧ 1/2 ^ n < d ∧ a ≤ c ∧ c ≤ b ⟹ b - c < d" for a b c
by auto
have "norm (c (real i / 2 ^ m) - b (real (2 * j - 1) / 2 ^ Suc n)) < d"
using 2 j n close_ab [of "2*j-1" "Suc n"]
using b_ge_0 [of "2*j-1" "Suc n"] b_le_1 [of "2*j-1" "Suc n"]
using aj_le_ci [of "2*j-1" i m "Suc n"]
using ci_le_bj [of "2*j-1" i m "Suc n"]
apply (simp add: divide_simps of_nat_diff del: power_Suc)
apply (auto simp: divide_simps intro!: *)
done
moreover have "f(c(j / 2^n)) = f(b ((2*j - 1) / 2 ^ (Suc n)))"
using f_eq_fc [OF j] by metis
ultimately show ?thesis
by (metis dist_norm atLeastAtMost_iff b_ge_0 b_le_1 c_ge_0 c_le_1 d)
next
case 3
have *: "abs(a - b) ≤ 1/2 ^ n ∧ 1/2 ^ n < d ∧ a ≤ c ∧ c ≤ b ⟹ c - a < d" for a b c
by auto
have "norm (c (real i / 2 ^ m) - a (real (2 * j + 1) / 2 ^ Suc n)) < d"
using 3 j n close_ab [of "2*j+1" "Suc n"]
using b_ge_0 [of "2*j+1" "Suc n"] b_le_1 [of "2*j+1" "Suc n"]
using aj_le_ci [of "2*j+1" i m "Suc n"]
using ci_le_bj [of "2*j+1" i m "Suc n"]
apply (simp add: divide_simps of_nat_diff del: power_Suc)
apply (auto simp: divide_simps intro!: *)
done
moreover have "f(c(j / 2^n)) = f(a ((2*j + 1) / 2 ^ (Suc n)))"
using f_eq_fc [OF j] by metis
ultimately show ?thesis
by (metis dist_norm a_ge_0 atLeastAtMost_iff a_ge_0 a_le_1 c_ge_0 c_le_1 d)
qed
qed
show "dist (f (c x')) (f (c x)) < e"
if "x ∈ D01" "x' ∈ D01" "dist x' x < 1/2^n" for x x'
using that unfolding D01_def dyadics_in_open_unit_interval
proof clarsimp
fix i k::nat and m p
assume i: "0 < i" "i < 2 ^ m" and k: "0<k" "k < 2 ^ p"
assume clo: "dist (real k / 2 ^ p) (real i / 2 ^ m) < 1/2 ^ n"
obtain j::nat where "0 < j" "j < 2 ^ n"
and clo_ij: "abs(i / 2^m - j / 2^n) < 1/2 ^ n"
and clo_kj: "abs(k / 2^p - j / 2^n) < 1/2 ^ n"
proof -
have "max (2^n * i / 2^m) (2^n * k / 2^p) ≥ 0"
by (auto simp: le_max_iff_disj)
then obtain j where "floor (max (2^n*i / 2^m) (2^n*k / 2^p)) = int j"
using zero_le_floor zero_le_imp_eq_int by blast
then have j_le: "real j ≤ max (2^n * i / 2^m) (2^n * k / 2^p)"
and less_j1: "max (2^n * i / 2^m) (2^n * k / 2^p) < real j + 1"
using floor_correct [of "max (2^n * i / 2^m) (2^n * k / 2^p)"] by linarith+
show thesis
proof (cases "j = 0")
case True
show thesis
proof
show "(1::nat) < 2 ^ n"
by (metis Suc_1 ‹0 < n› lessI one_less_power)
show "¦real i / 2 ^ m - real 1/2 ^ n¦ < 1/2 ^ n"
using i less_j1 by (simp add: dist_norm field_simps True)
show "¦real k / 2 ^ p - real 1/2 ^ n¦ < 1/2 ^ n"
using k less_j1 by (simp add: dist_norm field_simps True)
qed simp
next
case False
have 1: "real j * 2 ^ m < real i * 2 ^ n"
if j: "real j * 2 ^ p ≤ real k * 2 ^ n" and k: "real k * 2 ^ m < real i * 2 ^ p"
for i k m p
proof -
have "real j * 2 ^ p * 2 ^ m ≤ real k * 2 ^ n * 2 ^ m"
using j by simp
moreover have "real k * 2 ^ m * 2 ^ n < real i * 2 ^ p * 2 ^ n"
using k by simp
ultimately have "real j * 2 ^ p * 2 ^ m < real i * 2 ^ p * 2 ^ n"
by (simp only: mult_ac)
then show ?thesis
by simp
qed
have 2: "real j * 2 ^ m < 2 ^ m + real i * 2 ^ n"
if j: "real j * 2 ^ p ≤ real k * 2 ^ n" and k: "real k * (2 ^ m * 2 ^ n) < 2 ^ m * 2 ^ p + real i * (2 ^ n * 2 ^ p)"
for i k m p
proof -
have "real j * 2 ^ p * 2 ^ m ≤ real k * (2 ^ m * 2 ^ n)"
using j by simp
also have "... < 2 ^ m * 2 ^ p + real i * (2 ^ n * 2 ^ p)"
by (rule k)
finally have "(real j * 2 ^ m) * 2 ^ p < (2 ^ m + real i * 2 ^ n) * 2 ^ p"
by (simp add: algebra_simps)
then show ?thesis
by simp
qed
have 3: "real j * 2 ^ p < 2 ^ p + real k * 2 ^ n"
if j: "real j * 2 ^ m ≤ real i * 2 ^ n" and i: "real i * 2 ^ p ≤ real k * 2 ^ m"
proof -
have "real j * 2 ^ m * 2 ^ p ≤ real i * 2 ^ n * 2 ^ p"
using j by simp
moreover have "real i * 2 ^ p * 2 ^ n ≤ real k * 2 ^ m * 2 ^ n"
using i by simp
ultimately have "real j * 2 ^ m * 2 ^ p ≤ real k * 2 ^ m * 2 ^ n"
by (simp only: mult_ac)
then have "real j * 2 ^ p ≤ real k * 2 ^ n"
by simp
also have "... < 2 ^ p + real k * 2 ^ n"
by auto
finally show ?thesis by simp
qed
show ?thesis
proof
have "2 ^ n * real i / 2 ^ m < 2 ^ n" "2 ^ n * real k / 2 ^ p < 2 ^ n"
using i k by (auto simp: field_simps)
then have "max (2^n * i / 2^m) (2^n * k / 2^p) < 2^n"
by simp
with j_le have "real j < 2 ^ n" by linarith
then show "j < 2 ^ n"
by auto
have "¦real i * 2 ^ n - real j * 2 ^ m¦ < 2 ^ m"
using clo less_j1 j_le
by (auto simp: le_max_iff_disj field_split_simps dist_norm abs_if split: if_split_asm dest: 1 2)
then show "¦real i / 2 ^ m - real j / 2 ^ n¦ < 1/2 ^ n"
by (auto simp: field_split_simps)
have "¦real k * 2 ^ n - real j * 2 ^ p¦ < 2 ^ p"
using clo less_j1 j_le
by (auto simp: le_max_iff_disj field_split_simps dist_norm abs_if split: if_split_asm dest: 3 2)
then show "¦real k / 2 ^ p - real j / 2 ^ n¦ < 1/2 ^ n"
by (auto simp: le_max_iff_disj field_split_simps dist_norm)
qed (use False in simp)
qed
qed
show "dist (f (c (real k / 2 ^ p))) (f (c (real i / 2 ^ m))) < e"
proof (rule dist_triangle_half_l)
show "dist (f (c (real k / 2 ^ p))) (f(c(j / 2^n))) < e/2"
using ‹0 < j› ‹j < 2 ^ n› k clo_kj
by (intro dist_fc_close) auto
show "dist (f (c (real i / 2 ^ m))) (f (c (real j / 2 ^ n))) < e/2"
using ‹0 < j› ‹j < 2 ^ n› i clo_ij
by (intro dist_fc_close) auto
qed
qed
qed
qed
then obtain h where ucont_h: "uniformly_continuous_on {0..1} h"
and fc_eq: "⋀x. x ∈ D01 ⟹ (f ∘ c) x = h x"
proof (rule uniformly_continuous_on_extension_on_closure [of D01 "f ∘ c"])
qed (use closure_subset [of D01] in ‹auto intro!: that›)
then have cont_h: "continuous_on {0..1} h"
using uniformly_continuous_imp_continuous by blast
have h_eq: "h (real k / 2 ^ m) = f (c (real k / 2 ^ m))" if "0 < k" "k < 2^m" for k m
using fc_eq that by (force simp: D01_def)
have "h ` {0..1} = f ` {0..1}"
proof
have "h ` (closure D01) ⊆ f ` {0..1}"
proof (rule image_closure_subset)
show "continuous_on (closure D01) h"
using cont_h by simp
show "closed (f ` {0..1})"
using compact_continuous_image [OF cont_f] compact_imp_closed by blast
show "h ` D01 ⊆ f ` {0..1}"
by (force simp: dyadics_in_open_unit_interval D01_def h_eq)
qed
with cloD01 show "h ` {0..1} ⊆ f ` {0..1}" by simp
have a12 [simp]: "a (1/2) = u"
by (metis a_def leftrec_base numeral_One of_nat_numeral)
have b12 [simp]: "b (1/2) = v"
by (metis b_def rightrec_base numeral_One of_nat_numeral)
have "f ` {0..1} ⊆ closure(h ` D01)"
proof (clarsimp simp: closure_approachable dyadics_in_open_unit_interval D01_def)
fix x e::real
assume "0 ≤ x" "x ≤ 1" "0 < e"
have ucont_f: "uniformly_continuous_on {0..1} f"
using compact_uniformly_continuous cont_f by blast
then obtain δ where "δ > 0"
and δ: "⋀x x'. ⟦x ∈ {0..1}; x' ∈ {0..1}; dist x' x < δ⟧ ⟹ norm (f x' - f x) < e"
using ‹0 < e› by (auto simp: uniformly_continuous_on_def dist_norm)
have *: "∃m::nat. ∃y. odd m ∧ 0 < m ∧ m < 2 ^ n ∧ y ∈ {a(m / 2^n) .. b(m / 2^n)} ∧ f y = f x"
if "n ≠ 0" for n
using that
proof (induction n)
case 0 then show ?case by auto
next
case (Suc n)
show ?case
proof (cases "n=0")
case True
consider "x ∈ {0..u}" | "x ∈ {u..v}" | "x ∈ {v..1}"
using ‹0 ≤ x› ‹x ≤ 1› by force
then have "∃y≥a (real 1/2). y ≤ b (real 1/2) ∧ f y = f x"
proof cases
case 1
then show ?thesis
using uabv [of 1 1] f0u [of u] f0u [of x] by force
next
case 2
then show ?thesis
by (rule_tac x=x in exI) auto
next
case 3
then show ?thesis
using uabv [of 1 1] fv1 [of v] fv1 [of x] by force
qed
with ‹n=0› show ?thesis
by (rule_tac x=1 in exI) auto
next
case False
with Suc obtain m y
where "odd m" "0 < m" and mless: "m < 2 ^ n"
and y: "y ∈ {a (real m / 2 ^ n)..b (real m / 2 ^ n)}" and feq: "f y = f x"
by metis
then obtain j where j: "m = 2*j + 1" by (metis oddE)
have j4: "4 * j + 1 < 2 ^ Suc n"
using mless j by (simp add: algebra_simps)
consider "y ∈ {a((2*j + 1) / 2^n) .. b((4*j + 1) / 2 ^ (Suc n))}"
| "y ∈ {b((4*j + 1) / 2 ^ (Suc n)) .. a((4*j + 3) / 2 ^ (Suc n))}"
| "y ∈ {a((4*j + 3) / 2 ^ (Suc n)) .. b((2*j + 1) / 2^n)}"
using y j by force
then show ?thesis
proof cases
case 1
show ?thesis
proof (intro exI conjI)
show "y ∈ {a (real (4 * j + 1) / 2 ^ Suc n)..b (real (4 * j + 1) / 2 ^ Suc n)}"
using mless j ‹n ≠ 0› 1 by (simp add: a41 b41 add.commute [of 1] del: power_Suc)
qed (use feq j4 in auto)
next
case 2
show ?thesis
proof (intro exI conjI)
show "b (real (4 * j + 1) / 2 ^ Suc n) ∈ {a (real (4 * j + 1) / 2 ^ Suc n)..b (real (4 * j + 1) / 2 ^ Suc n)}"
using ‹n ≠ 0› alec [of "2*j+1" n] cleb [of "2*j+1" n] a_ge_0 [of "2*j+1" n] b_le_1 [of "2*j+1" n]
using left_right [of "c((2*j + 1) / 2^n)" "a((2*j + 1) / 2^n)" "b((2*j + 1) / 2^n)"]
by (simp add: a41 b41 add.commute [of 1] del: power_Suc)
show "f (b (real (4 * j + 1) / 2 ^ Suc n)) = f x"
using ‹n ≠ 0› 2
using alec [of "2*j+1" n] cleb [of "2*j+1" n] a_ge_0 [of "2*j+1" n] b_le_1 [of "2*j+1" n]
by (force simp add: b41 a43 add.commute [of 1] feq [symmetric] simp del: power_Suc intro: f_eqI)
qed (use j4 in auto)
next
case 3
show ?thesis
proof (intro exI conjI)
show "4 * j + 3 < 2 ^ Suc n"
using mless j by simp
show "f y = f x"
by fact
show "y ∈ {a (real (4 * j + 3) / 2 ^ Suc n) .. b (real (4 * j + 3) / 2 ^ Suc n)}"
using 3 False b43 [of n j] by (simp add: add.commute)
qed (use 3 in auto)
qed
qed
qed
obtain n where n: "1/2^n < min (δ / 2) 1"
by (metis ‹0 < δ› divide_less_eq_1 less_numeral_extra(1) min_less_iff_conj one_less_numeral_iff power_one_over real_arch_pow_inv semiring_norm(76) zero_less_divide_iff zero_less_numeral)
with gr0I have "n ≠ 0"
by fastforce
with * obtain m::nat and y
where "odd m" "0 < m" and mless: "m < 2 ^ n"
and y: "a(m / 2^n) ≤ y ∧ y ≤ b(m / 2^n)" and feq: "f x = f y"
by (metis atLeastAtMost_iff)
then have "0 ≤ y" "y ≤ 1"
by (meson a_ge_0 b_le_1 order.trans)+
moreover have "y < δ + c (real m / 2 ^ n)" "c (real m / 2 ^ n) < δ + y"
using y alec [of m n] cleb [of m n] n field_sum_of_halves close_ab [OF ‹odd m›, of n]
by linarith+
moreover note ‹0 < m› mless ‹0 ≤ x› ‹x ≤ 1›
ultimately have "dist (h (real m / 2 ^ n)) (f x) < e"
by (auto simp: dist_norm h_eq feq δ)
then show "∃k. ∃m∈{0<..<2 ^ k}. dist (h (real m / 2 ^ k)) (f x) < e"
using ‹0 < m› greaterThanLessThan_iff mless by blast
qed
also have "... ⊆ h ` {0..1}"
proof (rule closure_minimal)
show "h ` D01 ⊆ h ` {0..1}"
using cloD01 closure_subset by blast
show "closed (h ` {0..1})"
using compact_continuous_image [OF cont_h] compact_imp_closed by auto
qed
finally show "f ` {0..1} ⊆ h ` {0..1}" .
qed
moreover have "inj_on h {0..1}"
proof -
have "u < v"
by (metis atLeastAtMost_iff f0u f_1not0 fv1 order.not_eq_order_implies_strict u01(1) u01(2) v01(1))
have f_not_fu: "⋀x. ⟦u < x; x ≤ v⟧ ⟹ f x ≠ f u"
by (metis atLeastAtMost_iff f0u fu1 greaterThanAtMost_iff order_refl order_trans u01(1) v01(2))
have f_not_fv: "⋀x. ⟦u ≤ x; x < v⟧ ⟹ f x ≠ f v"
by (metis atLeastAtMost_iff order_refl order_trans v01(2) atLeastLessThan_iff fuv fv1)
have a_less_b:
"a(j / 2^n) < b(j / 2^n) ∧
(∀x. a(j / 2^n) < x ⟶ x ≤ b(j / 2^n) ⟶ f x ≠ f(a(j / 2^n))) ∧
(∀x. a(j / 2^n) ≤ x ⟶ x < b(j / 2^n) ⟶ f x ≠ f(b(j / 2^n)))" for n and j::nat
proof (induction n arbitrary: j)
case 0 then show ?case
by (simp add: ‹u < v› f_not_fu f_not_fv)
next
case (Suc n j) show ?case
proof (cases "n > 0")
case False then show ?thesis
by (auto simp: a_def b_def leftrec_base rightrec_base ‹u < v› f_not_fu f_not_fv)
next
case True show ?thesis
proof (cases "even j")
case True
with ‹0 < n› Suc.IH show ?thesis
by (auto elim!: evenE)
next
case False
then obtain k where k: "j = 2*k + 1" by (metis oddE)
then show ?thesis
proof (cases "even k")
case True
then obtain m where m: "k = 2*m" by (metis evenE)
have fleft: "f (leftcut (a ((2*m + 1) / 2^n)) (b ((2*m + 1) / 2^n)) (c ((2*m + 1) / 2^n))) =
f (c((2*m + 1) / 2^n))"
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
by (auto intro: f_eqI)
show ?thesis
proof (intro conjI impI notI allI)
have False if "b (real j / 2 ^ Suc n) ≤ a (real j / 2 ^ Suc n)"
proof -
have "f (c ((1 + real m * 2) / 2 ^ n)) = f (a ((1 + real m * 2) / 2 ^ n))"
using k m ‹0 < n› fleft that a41 [of n m] b41 [of n m]
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
using left_right [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
by (auto simp: algebra_simps)
moreover have "a (real (1 + m * 2) / 2 ^ n) < c (real (1 + m * 2) / 2 ^ n)"
using Suc.IH [of "1 + m * 2"] by (simp add: c_def midpoint_def)
moreover have "c (real (1 + m * 2) / 2 ^ n) ≤ b (real (1 + m * 2) / 2 ^ n)"
using cleb by blast
ultimately show ?thesis
using Suc.IH [of "1 + m * 2"] by force
qed
then show "a (real j / 2 ^ Suc n) < b (real j / 2 ^ Suc n)" by force
next
fix x
assume "a (real j / 2 ^ Suc n) < x" "x ≤ b (real j / 2 ^ Suc n)" "f x = f (a (real j / 2 ^ Suc n))"
then show False
using Suc.IH [of "1 + m * 2", THEN conjunct2, THEN conjunct1]
using k m ‹0 < n› a41 [of n m] b41 [of n m]
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
by (auto simp: algebra_simps)
next
fix x
assume "a (real j / 2 ^ Suc n) ≤ x" "x < b (real j / 2 ^ Suc n)" "f x = f (b (real j / 2 ^ Suc n))"
then show False
using k m ‹0 < n› a41 [of n m] b41 [of n m] fleft left_neq
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
by (auto simp: algebra_simps)
qed
next
case False
with oddE obtain m where m: "k = Suc (2*m)" by fastforce
have fright: "f (rightcut (a ((2*m + 1) / 2^n)) (b ((2*m + 1) / 2^n)) (c ((2*m + 1) / 2^n))) = f (c((2*m + 1) / 2^n))"
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
by (auto intro: f_eqI [OF _ order_refl])
show ?thesis
proof (intro conjI impI notI allI)
have False if "b (real j / 2 ^ Suc n) ≤ a (real j / 2 ^ Suc n)"
proof -
have "f (c ((1 + real m * 2) / 2 ^ n)) = f (b ((1 + real m * 2) / 2 ^ n))"
using k m ‹0 < n› fright that a43 [of n m] b43 [of n m]
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
using left_right [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
by (auto simp: algebra_simps)
moreover have "a (real (1 + m * 2) / 2 ^ n) ≤ c (real (1 + m * 2) / 2 ^ n)"
using alec by blast
moreover have "c (real (1 + m * 2) / 2 ^ n) < b (real (1 + m * 2) / 2 ^ n)"
using Suc.IH [of "1 + m * 2"] by (simp add: c_def midpoint_def)
ultimately show ?thesis
using Suc.IH [of "1 + m * 2"] by force
qed
then show "a (real j / 2 ^ Suc n) < b (real j / 2 ^ Suc n)" by force
next
fix x
assume "a (real j / 2 ^ Suc n) < x" "x ≤ b (real j / 2 ^ Suc n)" "f x = f (a (real j / 2 ^ Suc n))"
then show False
using k m ‹0 < n› a43 [of n m] b43 [of n m] fright right_neq
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
by (auto simp: algebra_simps)
next
fix x
assume "a (real j / 2 ^ Suc n) ≤ x" "x < b (real j / 2 ^ Suc n)" "f x = f (b (real j / 2 ^ Suc n))"
then show False
using Suc.IH [of "1 + m * 2", THEN conjunct2, THEN conjunct2]
using k m ‹0 < n› a43 [of n m] b43 [of n m]
using alec [of "2*m+1" n] cleb [of "2*m+1" n] a_ge_0 [of "2*m+1" n] b_le_1 [of "2*m+1" n]
using left_right_m [of "c((2*m + 1) / 2^n)" "a((2*m + 1) / 2^n)" "b((2*m + 1) / 2^n)"]
by (auto simp: algebra_simps fright simp del: power_Suc)
qed
qed
qed
qed
qed
have c_gt_0 [simp]: "0 < c(m / 2^n)" and c_less_1 [simp]: "c(m / 2^n) < 1" for m::nat and n
using a_less_b [of m n] apply (simp_all add: c_def midpoint_def)
using a_ge_0 [of m n] b_le_1 [of m n] by linarith+
have approx: "∃j n. odd j ∧ n ≠ 0 ∧
real i / 2^m ≤ real j / 2^n ∧
real j / 2^n ≤ real k / 2^p ∧
¦real i / 2 ^ m - real j / 2 ^ n¦ < 1/2^n ∧
¦real k / 2 ^ p - real j / 2 ^ n¦ < 1/2^n"
if "0 < i" "i < 2 ^ m" "0 < k" "k < 2 ^ p" "i / 2^m < k / 2^p" "m + p = N" for N m p i k
using that
proof (induction N arbitrary: m p i k rule: less_induct)
case (less N)
then consider "i / 2^m ≤ 1/2" "1/2 ≤ k / 2^p" | "k / 2^p < 1/2" | "k / 2^p ≥ 1/2" "1/2 < i / 2^m"
by linarith
then show ?case
proof cases
case 1
with less.prems show ?thesis
by (rule_tac x=1 in exI)+ (fastforce simp: field_split_simps)
next
case 2 show ?thesis
proof (cases m)
case 0 with less.prems show ?thesis
by auto
next
case (Suc m') show ?thesis
proof (cases p)
case 0 with less.prems show ?thesis by auto
next
case (Suc p')
have §: False if "real i * 2 ^ p' < real k * 2 ^ m'" "k < 2 ^ p'" "2 ^ m' ≤ i"
proof -
have "real k * 2 ^ m' < 2 ^ p' * 2 ^ m'"
using that by simp
then have "real i * 2 ^ p' < 2 ^ p' * 2 ^ m'"
using that by linarith
with that show ?thesis by simp
qed
moreover have *: "real i / 2 ^ m' < real k / 2^p'" "k < 2 ^ p'"
using less.prems ‹m = Suc m'› 2 Suc by (force simp: field_split_simps)+
moreover have "i < 2 ^ m' "
using § * by (clarsimp simp: divide_simps linorder_not_le) (meson linorder_not_le)
ultimately show ?thesis
using less.IH [of "m'+p'" i m' k p'] less.prems ‹m = Suc m'› 2 Suc
by (force simp: field_split_simps)
qed
qed
next
case 3 show ?thesis
proof (cases m)
case 0 with less.prems show ?thesis
by auto
next
case (Suc m') show ?thesis
proof (cases p)
case 0 with less.prems show ?thesis by auto
next
case (Suc p')
have "real (i - 2 ^ m') / 2 ^ m' < real (k - 2 ^ p') / 2 ^ p'"
using less.prems ‹m = Suc m'› Suc 3 by (auto simp: field_simps of_nat_diff)
moreover have "k - 2 ^ p' < 2 ^ p'" "i - 2 ^ m' < 2 ^ m'"
using less.prems Suc ‹m = Suc m'› by auto
moreover
have "2 ^ p' ≤ k" "2 ^ p' ≠ k"
using less.prems ‹m = Suc m'› Suc 3 by auto
then have "2 ^ p' < k"
by linarith
ultimately show ?thesis
using less.IH [of "m'+p'" "i - 2^m'" m' "k - 2 ^ p'" p'] less.prems ‹m = Suc m'› Suc 3
apply (clarsimp simp: field_simps of_nat_diff)
apply (rule_tac x="2 ^ n + j" in exI, simp)
apply (rule_tac x="Suc n" in exI)
apply (auto simp: field_simps)
done
qed
qed
qed
qed
have clec: "c(real i / 2^m) ≤ c(real j / 2^n)"
if i: "0 < i" "i < 2 ^ m" and j: "0 < j" "j < 2 ^ n" and ij: "i / 2^m < j / 2^n" for m i n j
proof -
obtain j' n' where "odd j'" "n' ≠ 0"
and i_le_j: "real i / 2 ^ m ≤ real j' / 2 ^ n'"
and j_le_j: "real j' / 2 ^ n' ≤ real j / 2 ^ n"
and clo_ij: "¦real i / 2 ^ m - real j' / 2 ^ n'¦ < 1/2 ^ n'"
and clo_jj: "¦real j / 2 ^ n - real j' / 2 ^ n'¦ < 1/2 ^ n'"
using approx [of i m j n "m+n"] that i j ij by auto
with oddE obtain q where q: "j' = Suc (2*q)" by fastforce
have "c (real i / 2 ^ m) ≤ c((2*q + 1) / 2^n')"
proof (cases "i / 2^m = (2*q + 1) / 2^n'")
case True then show ?thesis by simp
next
case False
with i_le_j clo_ij q have "¦real i / 2 ^ m - real (4 * q + 1) / 2 ^ Suc n'¦ < 1 / 2 ^ Suc n'"
by (auto simp: field_split_simps)
then have "c(i / 2^m) ≤ b(real(4 * q + 1) / 2 ^ (Suc n'))"
by (meson ci_le_bj even_mult_iff even_numeral even_plus_one_iff)
then show ?thesis
using alec [of "2*q+1" n'] cleb [of "2*q+1" n'] a_ge_0 [of "2*q+1" n'] b_le_1 [of "2*q+1" n'] b41 [of n' q] ‹n' ≠ 0›
using left_right_m [of "c((2*q + 1) / 2^n')" "a((2*q + 1) / 2^n')" "b((2*q + 1) / 2^n')"]
by (auto simp: algebra_simps)
qed
also have "... ≤ c(real j / 2^n)"
proof (cases "j / 2^n = (2*q + 1) / 2^n'")
case True
then show ?thesis by simp
next
case False
with j_le_j q have less: "(2*q + 1) / 2^n' < j / 2^n"
by auto
have *: "⟦q < i; abs(i - q) < s*2; r = q + s⟧ ⟹ abs(i - r) < s" for i q s r::real
by auto
have "¦real j / 2 ^ n - real (4 * q + 3) / 2 ^ Suc n'¦ < 1 / 2 ^ Suc n'"
by (rule * [OF less]) (use j_le_j clo_jj q in ‹auto simp: field_split_simps›)
then have "a(real(4*q + 3) / 2 ^ (Suc n')) ≤ c(j / 2^n)"
by (metis Suc3_eq_add_3 add.commute aj_le_ci even_Suc even_mult_iff even_numeral)
then show ?thesis
using alec [of "2*q+1" n'] cleb [of "2*q+1" n'] a_ge_0 [of "2*q+1" n'] b_le_1 [of "2*q+1" n'] a43 [of n' q] ‹n' ≠ 0›
using left_right_m [of "c((2*q + 1) / 2^n')" "a((2*q + 1) / 2^n')" "b((2*q + 1) / 2^n')"]
by (auto simp: algebra_simps)
qed
finally show ?thesis .
qed
have "x = y" if "0 ≤ x" "x ≤ 1" "0 ≤ y" "y ≤ 1" "h x = h y" for x y
using that
proof (induction x y rule: linorder_class.linorder_less_wlog)
case (less x1 x2)
obtain m n where m: "0 < m" "m < 2 ^ n"
and x12: "x1 < m / 2^n" "m / 2^n < x2"
and neq: "h x1 ≠ h (real m / 2^n)"
proof -
have "(x1 + x2) / 2 ∈ closure D01"
using cloD01 less.hyps less.prems by auto
with less obtain y where "y ∈ D01" and dist_y: "dist y ((x1 + x2) / 2) < (x2 - x1) / 64"
unfolding closure_approachable
by (metis diff_gt_0_iff_gt less_divide_eq_numeral1(1) mult_zero_left)
obtain m n where m: "0 < m" "m < 2 ^ n"
and clo: "¦real m / 2 ^ n - (x1 + x2) / 2¦ < (x2 - x1) / 64"
and n: "1/2^n < (x2 - x1) / 128"
proof -
have "min 1 ((x2 - x1) / 128) > 0" "1/2 < (1::real)"
using less by auto
then obtain N where N: "1/2^N < min 1 ((x2 - x1) / 128)"
by (metis power_one_over real_arch_pow_inv)
then have "N > 0"
using less_divide_eq_1 by force
obtain p q where p: "p < 2 ^ q" "p ≠ 0" and yeq: "y = real p / 2 ^ q"
using ‹y ∈ D01› by (auto simp: zero_less_divide_iff D01_def)
show ?thesis
proof
show "0 < 2^N * p"
using p by auto
show "2 ^ N * p < 2 ^ (N+q)"
by (simp add: p power_add)
have "¦real (2 ^ N * p) / 2 ^ (N + q) - (x1 + x2) / 2¦ = ¦real p / 2 ^ q - (x1 + x2) / 2¦"
by (simp add: power_add)
also have "... = ¦y - (x1 + x2) / 2¦"
by (simp add: yeq)
also have "... < (x2 - x1) / 64"
using dist_y by (simp add: dist_norm)
finally show "¦real (2 ^ N * p) / 2 ^ (N + q) - (x1 + x2) / 2¦ < (x2 - x1) / 64" .
have "(1::real) / 2 ^ (N + q) ≤ 1/2^N"
by (simp add: field_simps)
also have "... < (x2 - x1) / 128"
using N by force
finally show "1/2 ^ (N + q) < (x2 - x1) / 128" .
qed
qed
obtain m' n' m'' n'' where "0 < m'" "m' < 2 ^ n'" "x1 < m' / 2^n'" "m' / 2^n' < x2"
and "0 < m''" "m'' < 2 ^ n''" "x1 < m'' / 2^n''" "m'' / 2^n'' < x2"
and neq: "h (real m'' / 2^n'') ≠ h (real m' / 2^n')"
proof
show "0 < Suc (2*m)"
by simp
show m21: "Suc (2*m) < 2 ^ Suc n"
using m by auto
show "x1 < real (Suc (2 * m)) / 2 ^ Suc n"
using clo by (simp add: field_simps abs_if split: if_split_asm)
show "real (Suc (2 * m)) / 2 ^ Suc n < x2"
using n clo by (simp add: field_simps abs_if split: if_split_asm)
show "0 < 4*m + 3"
by simp
have "m+1 ≤ 2 ^ n"
using m by simp
then have "4 * (m+1) ≤ 4 * (2 ^ n)"
by simp
then show m43: "4*m + 3 < 2 ^ (n+2)"
by (simp add: algebra_simps)
show "x1 < real (4 * m + 3) / 2 ^ (n + 2)"
using clo by (simp add: field_simps abs_if split: if_split_asm)
show "real (4 * m + 3) / 2 ^ (n + 2) < x2"
using n clo by (simp add: field_simps abs_if split: if_split_asm)
have c_fold: "midpoint (a ((2 * real m + 1) / 2 ^ Suc n)) (b ((2 * real m + 1) / 2 ^ Suc n)) = c ((2 * real m + 1) / 2 ^ Suc n)"
by (simp add: c_def)
define R where "R ≡ rightcut (a ((2 * real m + 1) / 2 ^ Suc n)) (b ((2 * real m + 1) / 2 ^ Suc n)) (c ((2 * real m + 1) / 2 ^ Suc n))"
have "R < b ((2 * real m + 1) / 2 ^ Suc n)"
unfolding R_def using a_less_b [of "4*m + 3" "n+2"] a43 [of "Suc n" m] b43 [of "Suc n" m]
by simp
then have Rless: "R < midpoint R (b ((2 * real m + 1) / 2 ^ Suc n))"
by (simp add: midpoint_def)
have midR_le: "midpoint R (b ((2 * real m + 1) / 2 ^ Suc n)) ≤ b ((2 * real m + 1) / (2 * 2 ^ n))"
using ‹R < b ((2 * real m + 1) / 2 ^ Suc n)›
by (simp add: midpoint_def)
have "(real (Suc (2 * m)) / 2 ^ Suc n) ∈ D01" "real (4 * m + 3) / 2 ^ (n + 2) ∈ D01"
by (simp_all add: D01_def m21 m43 del: power_Suc of_nat_Suc of_nat_add add_2_eq_Suc') blast+
then show "h (real (4 * m + 3) / 2 ^ (n + 2)) ≠ h (real (Suc (2 * m)) / 2 ^ Suc n)"
using a_less_b [of "4*m + 3" "n+2", THEN conjunct1]
using a43 [of "Suc n" m] b43 [of "Suc n" m]
using alec [of "2*m+1" "Suc n"] cleb [of "2*m+1" "Suc n"] a_ge_0 [of "2*m+1" "Suc n"] b_le_1 [of "2*m+1" "Suc n"]
apply (simp add: fc_eq [symmetric] c_def del: power_Suc)
apply (simp only: add.commute [of 1] c_fold R_def [symmetric])
apply (rule right_neq)
using Rless apply (simp add: R_def)
apply (rule midR_le, auto)
done
qed
then show ?thesis by (metis that)
qed
have m_div: "0 < m / 2^n" "m / 2^n < 1"
using m by (auto simp: field_split_simps)
have closure0m: "{0..m / 2^n} = closure ({0<..< m / 2^n} ∩ (⋃k m. {real m / 2 ^ k}))"
by (subst closure_dyadic_rationals_in_convex_set_pos_1, simp_all add: not_le m)
have "2^n > m"
by (simp add: m(2) not_le)
then have closurem1: "{m / 2^n .. 1} = closure ({m / 2^n <..< 1} ∩ (⋃k m. {real m / 2 ^ k}))"
using closure_dyadic_rationals_in_convex_set_pos_1 m_div(1) by fastforce
have cont_h': "continuous_on (closure ({u<..<v} ∩ (⋃k m. {real m / 2 ^ k}))) h"
if "0 ≤ u" "v ≤ 1" for u v
using that by (intro continuous_on_subset [OF cont_h] closure_minimal [OF subsetI]) auto
have closed_f': "closed (f ` {u..v})" if "0 ≤ u" "v ≤ 1" for u v
by (metis compact_continuous_image cont_f compact_interval atLeastatMost_subset_iff
compact_imp_closed continuous_on_subset that)
have less_2I: "⋀k i. real i / 2 ^ k < 1 ⟹ i < 2 ^ k"
by simp
have "h ` ({0<..<m / 2 ^ n} ∩ (⋃q p. {real p / 2 ^ q})) ⊆ f ` {0..c (m / 2 ^ n)}"
proof clarsimp
fix p q
assume p: "0 < real p / 2 ^ q" "real p / 2 ^ q < real m / 2 ^ n"
then have [simp]: "0 < p"
by (simp add: field_split_simps)
have [simp]: "p < 2 ^ q"
by (blast intro: p less_2I m_div less_trans)
have "f (c (real p / 2 ^ q)) ∈ f ` {0..c (real m / 2 ^ n)}"
by (auto simp: clec p m)
then show "h (real p / 2 ^ q) ∈ f ` {0..c (real m / 2 ^ n)}"
by (simp add: h_eq)
qed
with m_div have "h ` {0 .. m / 2^n} ⊆ f ` {0 .. c(m / 2^n)}"
apply (subst closure0m)
by (rule image_closure_subset [OF cont_h' closed_f']) auto
then have hx1: "h x1 ∈ f ` {0 .. c(m / 2^n)}"
using x12 less.prems(1) by auto
then obtain t1 where t1: "h x1 = f t1" "0 ≤ t1" "t1 ≤ c (m / 2 ^ n)"
by auto
have "h ` ({m / 2 ^ n<..<1} ∩ (⋃q p. {real p / 2 ^ q})) ⊆ f ` {c (m / 2 ^ n)..1}"
proof clarsimp
fix p q
assume p: "real m / 2 ^ n < real p / 2 ^ q" and [simp]: "p < 2 ^ q"
then have [simp]: "0 < p"
using gr_zeroI m_div by fastforce
have "f (c (real p / 2 ^ q)) ∈ f ` {c (m / 2 ^ n)..1}"
by (auto simp: clec p m)
then show "h (real p / 2 ^ q) ∈ f ` {c (real m / 2 ^ n)..1}"
by (simp add: h_eq)
qed
with m have "h ` {m / 2^n .. 1} ⊆ f ` {c(m / 2^n) .. 1}"
apply (subst closurem1)
by (rule image_closure_subset [OF cont_h' closed_f']) auto
then have hx2: "h x2 ∈ f ` {c(m / 2^n)..1}"
using x12 less.prems by auto
then obtain t2 where t2: "h x2 = f t2" "c (m / 2 ^ n) ≤ t2" "t2 ≤ 1"
by auto
with t1 less neq have False
using conn [of "h x2", unfolded is_interval_connected_1 [symmetric] is_interval_1, rule_format, of t1 t2 "c(m / 2^n)"]
by (simp add: h_eq m)
then show ?case by blast
qed auto
then show ?thesis
by (auto simp: inj_on_def)
qed
ultimately have "{0..1::real} homeomorphic f ` {0..1}"
using homeomorphic_compact [OF _ cont_h] by blast
then show ?thesis
using homeomorphic_sym by blast
qed
theorem path_contains_arc:
fixes p :: "real ⇒ 'a::{complete_space,real_normed_vector}"
assumes "path p" and a: "pathstart p = a" and b: "pathfinish p = b" and "a ≠ b"
obtains q where "arc q" "path_image q ⊆ path_image p" "pathstart q = a" "pathfinish q = b"
proof -
have ucont_p: "uniformly_continuous_on {0..1} p"
using ‹path p› unfolding path_def
by (metis compact_Icc compact_uniformly_continuous)
define φ where "φ ≡ λS. S ⊆ {0..1} ∧ 0 ∈ S ∧ 1 ∈ S ∧
(∀x ∈ S. ∀y ∈ S. open_segment x y ∩ S = {} ⟶ p x = p y)"
obtain T where "closed T" "φ T" and T: "⋀U. ⟦closed U; φ U⟧ ⟹ ¬ (U ⊂ T)"
proof (rule Brouwer_reduction_theorem_gen [of "{0..1}" φ])
have *: "{x<..<y} ∩ {0..1} = {x<..<y}" if "0 ≤ x" "y ≤ 1" "x ≤ y" for x y::real
using that by auto
show "φ {0..1}"
by (auto simp: φ_def open_segment_eq_real_ivl *)
show "φ (⋂(F ` UNIV))"
if "⋀n. closed (F n)" and φ: "⋀n. φ (F n)" and Fsub: "⋀n. F (Suc n) ⊆ F n" for F
proof -
have F01: "⋀n. F n ⊆ {0..1} ∧ 0 ∈ F n ∧ 1 ∈ F n"
and peq: "⋀n x y. ⟦x ∈ F n; y ∈ F n; open_segment x y ∩ F n = {}⟧ ⟹ p x = p y"
by (metis φ φ_def)+
have pqF: False if "∀u. x ∈ F u" "∀x. y ∈ F x" "open_segment x y ∩ (⋂x. F x) = {}" and neg: "p x ≠ p y"
for x y
using that
proof (induction x y rule: linorder_class.linorder_less_wlog)
case (less x y)
have xy: "x ∈ {0..1}" "y ∈ {0..1}"
by (metis less.prems subsetCE F01)+
have "norm(p x - p y) / 2 > 0"
using less by auto
then obtain e where "e > 0"
and e: "⋀u v. ⟦u ∈ {0..1}; v ∈ {0..1}; dist v u < e⟧ ⟹ dist (p v) (p u) < norm(p x - p y) / 2"
by (metis uniformly_continuous_onE [OF ucont_p])
have minxy: "min e (y - x) < (y - x) * (3 / 2)"
by (subst min_less_iff_disj) (simp add: less)
define w where "w ≡ x + (min e (y - x) / 3)"
define z where "z ≡y - (min e (y - x) / 3)"
have "w < z" and w: "w ∈ {x<..<y}" and z: "z ∈ {x<..<y}"
and wxe: "norm(w - x) < e" and zye: "norm(z - y) < e"
using minxy ‹0 < e› less unfolding w_def z_def by auto
have Fclo: "⋀T. T ∈ range F ⟹ closed T"
by (metis ‹⋀n. closed (F n)› image_iff)
have eq: "{w..z} ∩ ⋂(F ` UNIV) = {}"
using less w z by (simp add: open_segment_eq_real_ivl disjoint_iff)
then obtain K where "finite K" and K: "{w..z} ∩ (⋂ (F ` K)) = {}"
by (metis finite_subset_image compact_imp_fip [OF compact_interval Fclo])
then have "K ≠ {}"
using ‹w < z› ‹{w..z} ∩ ⋂(F ` K) = {}› by auto
define n where "n ≡ Max K"
have "n ∈ K" unfolding n_def by (metis ‹K ≠ {}› ‹finite K› Max_in)
have "F n ⊆ ⋂ (F ` K)"
unfolding n_def by (metis Fsub Max_ge ‹K ≠ {}› ‹finite K› cINF_greatest lift_Suc_antimono_le)
with K have wzF_null: "{w..z} ∩ F n = {}"
by (metis disjoint_iff_not_equal subset_eq)
obtain u where u: "u ∈ F n" "u ∈ {x..w}" "({u..w} - {u}) ∩ F n = {}"
proof (cases "w ∈ F n")
case True
then show ?thesis
by (metis wzF_null ‹w < z› atLeastAtMost_iff disjoint_iff_not_equal less_eq_real_def)
next
case False
obtain u where "u ∈ F n" "u ∈ {x..w}" "{u<..<w} ∩ F n = {}"
proof (rule segment_to_point_exists [of "F n ∩ {x..w}" w])
show "closed (F n ∩ {x..w})"
by (metis ‹⋀n. closed (F n)› closed_Int closed_real_atLeastAtMost)
show "F n ∩ {x..w} ≠ {}"
by (metis atLeastAtMost_iff disjoint_iff_not_equal greaterThanLessThan_iff less.prems(1) less_eq_real_def w)
qed (auto simp: open_segment_eq_real_ivl intro!: that)
with False show thesis
by (auto simp add: disjoint_iff less_eq_real_def intro!: that)
qed
obtain v where v: "v ∈ F n" "v ∈ {z..y}" "({z..v} - {v}) ∩ F n = {}"
proof (cases "z ∈ F n")
case True
have "z ∈ {w..z}"
using ‹w < z› by auto
then show ?thesis
by (metis wzF_null Int_iff True empty_iff)
next
case False
show ?thesis
proof (rule segment_to_point_exists [of "F n ∩ {z..y}" z])
show "closed (F n ∩ {z..y})"
by (metis ‹⋀n. closed (F n)› closed_Int closed_atLeastAtMost)
show "F n ∩ {z..y} ≠ {}"
by (metis atLeastAtMost_iff disjoint_iff_not_equal greaterThanLessThan_iff less.prems(2) less_eq_real_def z)
show "⋀b. ⟦b ∈ F n ∩ {z..y}; open_segment z b ∩ (F n ∩ {z..y}) = {}⟧ ⟹ thesis"
proof
show "⋀b. ⟦b ∈ F n ∩ {z..y}; open_segment z b ∩ (F n ∩ {z..y}) = {}⟧ ⟹ ({z..b} - {b}) ∩ F n = {}"
using False by (auto simp: open_segment_eq_real_ivl less_eq_real_def)
qed auto
qed
qed
obtain u v where "u ∈ {0..1}" "v ∈ {0..1}" "norm(u - x) < e" "norm(v - y) < e" "p u = p v"
proof
show "u ∈ {0..1}" "v ∈ {0..1}"
by (metis F01 ‹u ∈ F n› ‹v ∈ F n› subsetD)+
show "norm(u - x) < e" "norm (v - y) < e"
using ‹u ∈ {x..w}› ‹v ∈ {z..y}› atLeastAtMost_iff real_norm_def wxe zye by auto
show "p u = p v"
proof (rule peq)
show "u ∈ F n" "v ∈ F n"
by (auto simp: u v)
have "False" if "ξ ∈ F n" "u < ξ" "ξ < v" for ξ
proof -
have "ξ ∉ {z..v}"
by (metis DiffI disjoint_iff_not_equal less_irrefl singletonD that(1,3) v(3))
moreover have "ξ ∉ {w..z} ∩ F n"
by (metis equals0D wzF_null)
ultimately have "ξ ∈ {u..w}"
using that by auto
then show ?thesis
by (metis DiffI disjoint_iff_not_equal less_eq_real_def not_le singletonD that(1,2) u(3))
qed
moreover
have "⟦ξ ∈ F n; v < ξ; ξ < u⟧ ⟹ False" for ξ
using ‹u ∈ {x..w}› ‹v ∈ {z..y}› ‹w < z› by simp
ultimately
show "open_segment u v ∩ F n = {}"
by (force simp: open_segment_eq_real_ivl)
qed
qed
then show ?case
using e [of x u] e [of y v] xy
by (metis dist_norm dist_triangle_half_r order_less_irrefl)
qed (auto simp: open_segment_commute)
show ?thesis
unfolding φ_def by (metis (no_types, opaque_lifting) INT_I Inf_lower2 rangeI that(3) F01 subsetCE pqF)
qed
show "closed {0..1::real}" by auto
qed (meson φ_def)
then have "T ⊆ {0..1}" "0 ∈ T" "1 ∈ T"
and peq: "⋀x y. ⟦x ∈ T; y ∈ T; open_segment x y ∩ T = {}⟧ ⟹ p x = p y"
unfolding φ_def by metis+
then have "T ≠ {}" by auto
define h where "h ≡ λx. p(SOME y. y ∈ T ∧ open_segment x y ∩ T = {})"
have "p y = p z" if "y ∈ T" "z ∈ T" and xyT: "open_segment x y ∩ T = {}" and xzT: "open_segment x z ∩ T = {}"
for x y z
proof (cases "x ∈ T")
case True
with that show ?thesis by (metis ‹φ T› φ_def)
next
case False
have "insert x (open_segment x y ∪ open_segment x z) ∩ T = {}"
by (metis False Int_Un_distrib2 Int_insert_left Un_empty_right xyT xzT)
moreover have "open_segment y z ∩ T ⊆ insert x (open_segment x y ∪ open_segment x z) ∩ T"
by (auto simp: open_segment_eq_real_ivl)
ultimately have "open_segment y z ∩ T = {}"
by blast
with that peq show ?thesis by metis
qed
then have h_eq_p_gen: "h x = p y" if "y ∈ T" "open_segment x y ∩ T = {}" for x y
using that unfolding h_def
by (metis (mono_tags, lifting) some_eq_ex)
then have h_eq_p: "⋀x. x ∈ T ⟹ h x = p x"
by simp
have disjoint: "⋀x. ∃y. y ∈ T ∧ open_segment x y ∩ T = {}"
by (meson ‹T ≠ {}› ‹closed T› segment_to_point_exists)
have heq: "h x = h x'" if "open_segment x x' ∩ T = {}" for x x'
proof (cases "x ∈ T ∨ x' ∈ T")
case True
then show ?thesis
by (metis h_eq_p h_eq_p_gen open_segment_commute that)
next
case False
obtain y y' where "y ∈ T" "open_segment x y ∩ T = {}" "h x = p y"
"y' ∈ T" "open_segment x' y' ∩ T = {}" "h x' = p y'"
by (meson disjoint h_eq_p_gen)
moreover have "open_segment y y' ⊆ (insert x (insert x' (open_segment x y ∪ open_segment x' y' ∪ open_segment x x')))"
by (auto simp: open_segment_eq_real_ivl)
ultimately show ?thesis
using False that by (fastforce simp add: h_eq_p intro!: peq)
qed
have "h ` {0..1} homeomorphic {0..1::real}"
proof (rule homeomorphic_monotone_image_interval)
show "continuous_on {0..1} h"
proof (clarsimp simp add: continuous_on_iff)
fix u ε::real
assume "0 < ε" "0 ≤ u" "u ≤ 1"
then obtain δ where "δ > 0" and δ: "⋀v. v ∈ {0..1} ⟹ dist v u < δ ⟶ dist (p v) (p u) < ε / 2"
using ucont_p [unfolded uniformly_continuous_on_def]
by (metis atLeastAtMost_iff half_gt_zero_iff)
then have "dist (h v) (h u) < ε" if "v ∈ {0..1}" "dist v u < δ" for v
proof (cases "open_segment u v ∩ T = {}")
case True
then show ?thesis
using ‹0 < ε› heq by auto
next
case False
have uvT: "closed (closed_segment u v ∩ T)" "closed_segment u v ∩ T ≠ {}"
using False open_closed_segment by (auto simp: ‹closed T› closed_Int)
obtain w where "w ∈ T" and w: "w ∈ closed_segment u v" "open_segment u w ∩ T = {}"
proof (rule segment_to_point_exists [OF uvT])
fix b
assume "b ∈ closed_segment u v ∩ T" "open_segment u b ∩ (closed_segment u v ∩ T) = {}"
then show thesis
by (metis IntD1 IntD2 ends_in_segment(1) inf.orderE inf_assoc subset_oc_segment that)
qed
then have puw: "dist (p u) (p w) < ε / 2"
by (metis (no_types) ‹T ⊆ {0..1}› ‹dist v u < δ› δ dist_commute dist_in_closed_segment le_less_trans subsetCE)
obtain z where "z ∈ T" and z: "z ∈ closed_segment u v" "open_segment v z ∩ T = {}"
proof (rule segment_to_point_exists [OF uvT])
fix b
assume "b ∈ closed_segment u v ∩ T" "open_segment v b ∩ (closed_segment u v ∩ T) = {}"
then show thesis
by (metis IntD1 IntD2 ends_in_segment(2) inf.orderE inf_assoc subset_oc_segment that)
qed
then have "dist (p u) (p z) < ε / 2"
by (metis ‹T ⊆ {0..1}› ‹dist v u < δ› δ dist_commute dist_in_closed_segment le_less_trans subsetCE)
then show ?thesis
using puw by (metis (no_types) ‹w ∈ T› ‹z ∈ T› dist_commute dist_triangle_half_l h_eq_p_gen w(2) z(2))
qed
with ‹0 < δ› show "∃δ>0. ∀v∈{0..1}. dist v u < δ ⟶ dist (h v) (h u) < ε" by blast
qed
show "connected ({0..1} ∩ h -` {z})" for z
proof (clarsimp simp add: connected_iff_connected_component)
fix u v
assume huv_eq: "h v = h u" and uv: "0 ≤ u" "u ≤ 1" "0 ≤ v" "v ≤ 1"
have "∃T. connected T ∧ T ⊆ {0..1} ∧ T ⊆ h -` {h u} ∧ u ∈ T ∧ v ∈ T"
proof (intro exI conjI)
show "connected (closed_segment u v)"
by simp
show "closed_segment u v ⊆ {0..1}"
by (simp add: uv closed_segment_eq_real_ivl)
have pxy: "p x = p y"
if "T ⊆ {0..1}" "0 ∈ T" "1 ∈ T" "x ∈ T" "y ∈ T"
and disjT: "open_segment x y ∩ (T - open_segment u v) = {}"
and xynot: "x ∉ open_segment u v" "y ∉ open_segment u v"
for x y
proof (cases "open_segment x y ∩ open_segment u v = {}")
case True
then show ?thesis
by (metis Diff_Int_distrib Diff_empty peq disjT ‹x ∈ T› ‹y ∈ T›)
next
case False
then have "open_segment x u ∪ open_segment y v ⊆ open_segment x y - open_segment u v ∨
open_segment y u ∪ open_segment x v ⊆ open_segment x y - open_segment u v" (is "?xuyv ∨ ?yuxv")
using xynot by (fastforce simp add: open_segment_eq_real_ivl not_le not_less split: if_split_asm)
then show "p x = p y"
proof
assume "?xuyv"
then have "open_segment x u ∩ T = {}" "open_segment y v ∩ T = {}"
using disjT by auto
then have "h x = h y"
using heq huv_eq by auto
then show ?thesis
using h_eq_p ‹x ∈ T› ‹y ∈ T› by auto
next
assume "?yuxv"
then have "open_segment y u ∩ T = {}" "open_segment x v ∩ T = {}"
using disjT by auto
then have "h x = h y"
using heq [of y u] heq [of x v] huv_eq by auto
then show ?thesis
using h_eq_p ‹x ∈ T› ‹y ∈ T› by auto
qed
qed
have "¬ T - open_segment u v ⊂ T"
proof (rule T)
show "closed (T - open_segment u v)"
by (simp add: closed_Diff [OF ‹closed T›] open_segment_eq_real_ivl)
have "0 ∉ open_segment u v" "1 ∉ open_segment u v"
using open_segment_eq_real_ivl uv by auto
then show "φ (T - open_segment u v)"
using ‹T ⊆ {0..1}› ‹0 ∈ T› ‹1 ∈ T›
by (auto simp: φ_def) (meson peq pxy)
qed
then have "open_segment u v ∩ T = {}"
by blast
then show "closed_segment u v ⊆ h -` {h u}"
by (force intro: heq simp: open_segment_eq_real_ivl closed_segment_eq_real_ivl split: if_split_asm)+
qed auto
then show "connected_component ({0..1} ∩ h -` {h u}) u v"
by (simp add: connected_component_def)
qed
show "h 1 ≠ h 0"
by (metis ‹φ T› φ_def a ‹a ≠ b› b h_eq_p pathfinish_def pathstart_def)
qed
then obtain f and g :: "real ⇒ 'a"
where gfeq: "(∀x∈h ` {0..1}. (g(f x) = x))" and fhim: "f ` h ` {0..1} = {0..1}" and contf: "continuous_on (h ` {0..1}) f"
and fgeq: "(∀y∈{0..1}. (f(g y) = y))" and pag: "path_image g = h ` {0..1}" and contg: "continuous_on {0..1} g"
by (auto simp: homeomorphic_def homeomorphism_def path_image_def)
then have "arc g"
by (metis arc_def path_def inj_on_def)
obtain u v where "u ∈ {0..1}" "a = g u" "v ∈ {0..1}" "b = g v"
by (metis (mono_tags, opaque_lifting) ‹φ T› φ_def a b fhim gfeq h_eq_p imageI path_image_def pathfinish_def pathfinish_in_path_image pathstart_def pathstart_in_path_image)
then have "a ∈ path_image g" "b ∈ path_image g"
using path_image_def by blast+
have ph: "path_image h ⊆ path_image p"
by (metis image_mono image_subset_iff path_image_def disjoint h_eq_p_gen ‹T ⊆ {0..1}›)
show ?thesis
proof
show "pathstart (subpath u v g) = a" "pathfinish (subpath u v g) = b"
by (simp_all add: ‹a = g u› ‹b = g v›)
show "path_image (subpath u v g) ⊆ path_image p"
by (metis ‹u ∈ {0..1}› ‹v ∈ {0..1}› order_trans pag path_image_def path_image_subpath_subset ph)
show "arc (subpath u v g)"
using ‹arc g› ‹a = g u› ‹b = g v› ‹u ∈ {0..1}› ‹v ∈ {0..1}› arc_subpath_arc ‹a ≠ b› by blast
qed
qed
corollary path_connected_arcwise:
fixes S :: "'a::{complete_space,real_normed_vector} set"
shows "path_connected S ⟷
(∀x ∈ S. ∀y ∈ S. x ≠ y ⟶ (∃g. arc g ∧ path_image g ⊆ S ∧ pathstart g = x ∧ pathfinish g = y))"
(is "?lhs = ?rhs")
proof (intro iffI impI ballI)
fix x y
assume "path_connected S" "x ∈ S" "y ∈ S" "x ≠ y"
then obtain p where p: "path p" "path_image p ⊆ S" "pathstart p = x" "pathfinish p = y"
by (force simp: path_connected_def)
then show "∃g. arc g ∧ path_image g ⊆ S ∧ pathstart g = x ∧ pathfinish g = y"
by (metis ‹x ≠ y› order_trans path_contains_arc)
next
assume R [rule_format]: ?rhs
show ?lhs
unfolding path_connected_def
proof (intro ballI)
fix x y
assume "x ∈ S" "y ∈ S"
show "∃g. path g ∧ path_image g ⊆ S ∧ pathstart g = x ∧ pathfinish g = y"
proof (cases "x = y")
case True with ‹x ∈ S› path_component_def path_component_refl show ?thesis
by blast
next
case False with R [OF ‹x ∈ S› ‹y ∈ S›] show ?thesis
by (auto intro: arc_imp_path)
qed
qed
qed
corollary arc_connected_trans:
fixes g :: "real ⇒ 'a::{complete_space,real_normed_vector}"
assumes "arc g" "arc h" "pathfinish g = pathstart h" "pathstart g ≠ pathfinish h"
obtains i where "arc i" "path_image i ⊆ path_image g ∪ path_image h"
"pathstart i = pathstart g" "pathfinish i = pathfinish h"
by (metis (no_types, opaque_lifting) arc_imp_path assms path_contains_arc path_image_join path_join pathfinish_join pathstart_join)
subsection‹Accessibility of frontier points›
lemma dense_accessible_frontier_points:
fixes S :: "'a::{complete_space,real_normed_vector} set"
assumes "open S" and opeSV: "openin (top_of_set (frontier S)) V" and "V ≠ {}"
obtains g where "arc g" "g ` {0..<1} ⊆ S" "pathstart g ∈ S" "pathfinish g ∈ V"
proof -
obtain z where "z ∈ V"
using ‹V ≠ {}› by auto
then obtain r where "r > 0" and r: "ball z r ∩ frontier S ⊆ V"
by (metis openin_contains_ball opeSV)
then have "z ∈ frontier S"
using ‹z ∈ V› opeSV openin_contains_ball by blast
then have "z ∈ closure S" "z ∉ S"
by (simp_all add: frontier_def assms interior_open)
with ‹r > 0› have "infinite (S ∩ ball z r)"
by (auto simp: closure_def islimpt_eq_infinite_ball)
then obtain y where "y ∈ S" and y: "y ∈ ball z r"
using infinite_imp_nonempty by force
then have "y ∉ frontier S"
by (meson ‹open S› disjoint_iff_not_equal frontier_disjoint_eq)
have "y ≠ z"
using ‹y ∈ S› ‹z ∉ S› by blast
have "path_connected(ball z r)"
by (simp add: convex_imp_path_connected)
with y ‹r > 0› obtain g where "arc g" and pig: "path_image g ⊆ ball z r"
and g: "pathstart g = y" "pathfinish g = z"
using ‹y ≠ z› by (force simp: path_connected_arcwise)
have "continuous_on {0..1} g"
using ‹arc g› arc_imp_path path_def by blast
then have "compact (g -` frontier S ∩ {0..1})"
by (simp add: bounded_Int closed_Diff closed_vimage_Int compact_eq_bounded_closed)
moreover have "g -` frontier S ∩ {0..1} ≠ {}"
proof -
have "∃r. r ∈ g -` frontier S ∧ r ∈ {0..1}"
by (metis ‹z ∈ frontier S› g(2) imageE path_image_def pathfinish_in_path_image vimageI2)
then show ?thesis
by blast
qed
ultimately obtain t where gt: "g t ∈ frontier S" and "0 ≤ t" "t ≤ 1"
and t: "⋀u. ⟦g u ∈ frontier S; 0 ≤ u; u ≤ 1⟧ ⟹ t ≤ u"
by (force simp: dest!: compact_attains_inf)
moreover have "t ≠ 0"
by (metis ‹y ∉ frontier S› g(1) gt pathstart_def)
ultimately have t01: "0 < t" "t ≤ 1"
by auto
have "V ⊆ frontier S"
using opeSV openin_contains_ball by blast
show ?thesis
proof
show "arc (subpath 0 t g)"
by (simp add: ‹0 ≤ t› ‹t ≤ 1› ‹arc g› ‹t ≠ 0› arc_subpath_arc)
have "g 0 ∈ S"
by (metis ‹y ∈ S› g(1) pathstart_def)
then show "pathstart (subpath 0 t g) ∈ S"
by auto
have "g t ∈ V"
by (metis IntI atLeastAtMost_iff gt image_eqI path_image_def pig r subsetCE ‹0 ≤ t› ‹t ≤ 1›)
then show "pathfinish (subpath 0 t g) ∈ V"
by auto
then have "inj_on (subpath 0 t g) {0..1}"
using t01 ‹arc (subpath 0 t g)› arc_imp_inj_on by blast
then have "subpath 0 t g ` {0..<1} ⊆ subpath 0 t g ` {0..1} - {subpath 0 t g 1}"
by (force simp: dest: inj_onD)
moreover have False if "subpath 0 t g ` ({0..<1}) - S ≠ {}"
proof -
have contg: "continuous_on {0..1} g"
using ‹arc g› by (auto simp: arc_def path_def)
have "subpath 0 t g ` {0..<1} ∩ frontier S ≠ {}"
proof (rule connected_Int_frontier [OF _ _ that])
show "connected (subpath 0 t g ` {0..<1})"
proof (rule connected_continuous_image)
show "continuous_on {0..<1} (subpath 0 t g)"
by (meson ‹arc (subpath 0 t g)› arc_def atLeastLessThan_subseteq_atLeastAtMost_iff continuous_on_subset order_refl path_def)
qed auto
show "subpath 0 t g ` {0..<1} ∩ S ≠ {}"
using ‹y ∈ S› g(1) by (force simp: subpath_def image_def pathstart_def)
qed
then obtain x where "x ∈ subpath 0 t g ` {0..<1}" "x ∈ frontier S"
by blast
with t01 ‹0 ≤ t› mult_le_one t show False
by (fastforce simp: subpath_def)
qed
then have "subpath 0 t g ` {0..1} - {subpath 0 t g 1} ⊆ S"
using subsetD by fastforce
ultimately show "subpath 0 t g ` {0..<1} ⊆ S"
by auto
qed
qed
lemma dense_accessible_frontier_points_connected:
fixes S :: "'a::{complete_space,real_normed_vector} set"
assumes "open S" "connected S" "x ∈ S" "V ≠ {}"
and ope: "openin (top_of_set (frontier S)) V"
obtains g where "arc g" "g ` {0..<1} ⊆ S" "pathstart g = x" "pathfinish g ∈ V"
proof -
have "V ⊆ frontier S"
using ope openin_imp_subset by blast
with ‹open S› ‹x ∈ S› have "x ∉ V"
using interior_open by (auto simp: frontier_def)
obtain g where "arc g" and g: "g ` {0..<1} ⊆ S" "pathstart g ∈ S" "pathfinish g ∈ V"
by (metis dense_accessible_frontier_points [OF ‹open S› ope ‹V ≠ {}›])
then have "path_connected S"
by (simp add: assms connected_open_path_connected)
with ‹pathstart g ∈ S› ‹x ∈ S› have "path_component S x (pathstart g)"
by (simp add: path_connected_component)
then obtain f where "path f" and f: "path_image f ⊆ S" "pathstart f = x" "pathfinish f = pathstart g"
by (auto simp: path_component_def)
then have "path (f +++ g)"
by (simp add: ‹arc g› arc_imp_path)
then obtain h where "arc h"
and h: "path_image h ⊆ path_image (f +++ g)" "pathstart h = x" "pathfinish h = pathfinish g"
using path_contains_arc [of "f +++ g" x "pathfinish g"] ‹x ∉ V› ‹pathfinish g ∈ V› f
by (metis pathfinish_join pathstart_join)
have "path_image h ⊆ path_image f ∪ path_image g"
using h(1) path_image_join_subset by auto
then have "h ` {0..1} - {h 1} ⊆ S"
using f g h
apply (simp add: path_image_def pathfinish_def subset_iff image_def Bex_def)
by (metis le_less)
then have "h ` {0..<1} ⊆ S"
using ‹arc h› by (force simp: arc_def dest: inj_onD)
then show thesis
using ‹arc h› g(3) h that by presburger
qed
lemma dense_access_fp_aux:
fixes S :: "'a::{complete_space,real_normed_vector} set"
assumes S: "open S" "connected S"
and opeSU: "openin (top_of_set (frontier S)) U"
and opeSV: "openin (top_of_set (frontier S)) V"
and "V ≠ {}" "¬ U ⊆ V"
obtains g where "arc g" "pathstart g ∈ U" "pathfinish g ∈ V" "g ` {0<..<1} ⊆ S"
proof -
have "S ≠ {}"
using opeSV ‹V ≠ {}› by (metis frontier_empty openin_subtopology_empty)
then obtain x where "x ∈ S" by auto
obtain g where "arc g" and g: "g ` {0..<1} ⊆ S" "pathstart g = x" "pathfinish g ∈ V"
using dense_accessible_frontier_points_connected [OF S ‹x ∈ S› ‹V ≠ {}› opeSV] by blast
obtain h where "arc h" and h: "h ` {0..<1} ⊆ S" "pathstart h = x" "pathfinish h ∈ U - {pathfinish g}"
proof (rule dense_accessible_frontier_points_connected [OF S ‹x ∈ S›])
show "U - {pathfinish g} ≠ {}"
using ‹pathfinish g ∈ V› ‹¬ U ⊆ V› by blast
show "openin (top_of_set (frontier S)) (U - {pathfinish g})"
by (simp add: opeSU openin_delete)
qed auto
obtain γ where "arc γ"
and γ: "path_image γ ⊆ path_image (reversepath h +++ g)"
"pathstart γ = pathfinish h" "pathfinish γ = pathfinish g"
proof (rule path_contains_arc [of "(reversepath h +++ g)" "pathfinish h" "pathfinish g"])
show "path (reversepath h +++ g)"
by (simp add: ‹arc g› ‹arc h› ‹pathstart g = x› ‹pathstart h = x› arc_imp_path)
show "pathstart (reversepath h +++ g) = pathfinish h"
"pathfinish (reversepath h +++ g) = pathfinish g"
by auto
show "pathfinish h ≠ pathfinish g"
using ‹pathfinish h ∈ U - {pathfinish g}› by auto
qed auto
show ?thesis
proof
show "arc γ" "pathstart γ ∈ U" "pathfinish γ ∈ V"
using γ ‹arc γ› ‹pathfinish h ∈ U - {pathfinish g}› ‹pathfinish g ∈ V› by auto
have "path_image γ ⊆ path_image h ∪ path_image g"
by (metis γ(1) g(2) h(2) path_image_join path_image_reversepath pathfinish_reversepath)
then have "γ ` {0..1} - {γ 0, γ 1} ⊆ S"
using γ g h
apply (simp add: path_image_def pathstart_def pathfinish_def subset_iff image_def Bex_def)
by (metis linorder_neqE_linordered_idom not_less)
then show "γ ` {0<..<1} ⊆ S"
using ‹arc h› ‹arc γ›
by (metis arc_imp_simple_path path_image_def pathfinish_def pathstart_def simple_path_endless)
qed
qed
lemma dense_accessible_frontier_point_pairs:
fixes S :: "'a::{complete_space,real_normed_vector} set"
assumes S: "open S" "connected S"
and opeSU: "openin (top_of_set (frontier S)) U"
and opeSV: "openin (top_of_set (frontier S)) V"
and "U ≠ {}" "V ≠ {}" "U ≠ V"
obtains g where "arc g" "pathstart g ∈ U" "pathfinish g ∈ V" "g ` {0<..<1} ⊆ S"
proof -
consider "¬ U ⊆ V" | "¬ V ⊆ U"
using ‹U ≠ V› by blast
then show ?thesis
proof cases
case 1 then show ?thesis
using assms dense_access_fp_aux [OF S opeSU opeSV] that by blast
next
case 2
obtain g where "arc g" and g: "pathstart g ∈ V" "pathfinish g ∈ U" "g ` {0<..<1} ⊆ S"
using assms dense_access_fp_aux [OF S opeSV opeSU] "2" by blast
show ?thesis
proof
show "arc (reversepath g)"
by (simp add: ‹arc g› arc_reversepath)
show "pathstart (reversepath g) ∈ U" "pathfinish (reversepath g) ∈ V"
using g by auto
show "reversepath g ` {0<..<1} ⊆ S"
using g by (auto simp: reversepath_def)
qed
qed
qed
end