Theory Fashoda_Theorem
section ‹Fashoda Meet Theorem›
theory Fashoda_Theorem
imports Brouwer_Fixpoint Path_Connected Cartesian_Euclidean_Space
begin
subsection ‹Bijections between intervals›
definition interval_bij :: "'a × 'a ⇒ 'a × 'a ⇒ 'a ⇒ 'a::euclidean_space"
where "interval_bij =
(λ(a, b) (u, v) x. (∑i∈Basis. (u∙i + (x∙i - a∙i) / (b∙i - a∙i) * (v∙i - u∙i)) *⇩R i))"
lemma interval_bij_affine:
"interval_bij (a,b) (u,v) = (λx. (∑i∈Basis. ((v∙i - u∙i) / (b∙i - a∙i) * (x∙i)) *⇩R i) +
(∑i∈Basis. (u∙i - (v∙i - u∙i) / (b∙i - a∙i) * (a∙i)) *⇩R i))"
by (auto simp add: interval_bij_def sum.distrib [symmetric] scaleR_add_left [symmetric]
fun_eq_iff intro!: sum.cong)
(simp add: algebra_simps diff_divide_distrib [symmetric])
lemma continuous_interval_bij:
fixes a b :: "'a::euclidean_space"
shows "continuous (at x) (interval_bij (a, b) (u, v))"
by (auto simp add: divide_inverse interval_bij_def intro!: continuous_sum continuous_intros)
lemma continuous_on_interval_bij: "continuous_on s (interval_bij (a, b) (u, v))"
apply(rule continuous_at_imp_continuous_on)
apply (rule, rule continuous_interval_bij)
done
lemma in_interval_interval_bij:
fixes a b u v x :: "'a::euclidean_space"
assumes "x ∈ cbox a b"
and "cbox u v ≠ {}"
shows "interval_bij (a, b) (u, v) x ∈ cbox u v"
apply (simp only: interval_bij_def split_conv mem_box inner_sum_left_Basis cong: ball_cong)
apply safe
proof -
fix i :: 'a
assume i: "i ∈ Basis"
have "cbox a b ≠ {}"
using assms by auto
with i have *: "a∙i ≤ b∙i" "u∙i ≤ v∙i"
using assms(2) by (auto simp add: box_eq_empty)
have x: "a∙i≤x∙i" "x∙i≤b∙i"
using assms(1)[unfolded mem_box] using i by auto
have "0 ≤ (x ∙ i - a ∙ i) / (b ∙ i - a ∙ i) * (v ∙ i - u ∙ i)"
using * x by auto
then show "u ∙ i ≤ u ∙ i + (x ∙ i - a ∙ i) / (b ∙ i - a ∙ i) * (v ∙ i - u ∙ i)"
using * by auto
have "((x ∙ i - a ∙ i) / (b ∙ i - a ∙ i)) * (v ∙ i - u ∙ i) ≤ 1 * (v ∙ i - u ∙ i)"
apply (rule mult_right_mono)
unfolding divide_le_eq_1
using * x
apply auto
done
then show "u ∙ i + (x ∙ i - a ∙ i) / (b ∙ i - a ∙ i) * (v ∙ i - u ∙ i) ≤ v ∙ i"
using * by auto
qed
lemma interval_bij_bij:
"∀(i::'a::euclidean_space)∈Basis. a∙i < b∙i ∧ u∙i < v∙i ⟹
interval_bij (a, b) (u, v) (interval_bij (u, v) (a, b) x) = x"
by (auto simp: interval_bij_def euclidean_eq_iff[where 'a='a])
lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "∀i. a$i < b$i ∧ u$i < v$i"
shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
subsection ‹Fashoda meet theorem›
lemma infnorm_2:
fixes x :: "real^2"
shows "infnorm x = max ¦x$1¦ ¦x$2¦"
unfolding infnorm_cart UNIV_2 by (rule cSup_eq) auto
lemma infnorm_eq_1_2:
fixes x :: "real^2"
shows "infnorm x = 1 ⟷
¦x$1¦ ≤ 1 ∧ ¦x$2¦ ≤ 1 ∧ (x$1 = -1 ∨ x$1 = 1 ∨ x$2 = -1 ∨ x$2 = 1)"
unfolding infnorm_2 by auto
lemma infnorm_eq_1_imp:
fixes x :: "real^2"
assumes "infnorm x = 1"
shows "¦x$1¦ ≤ 1" and "¦x$2¦ ≤ 1"
using assms unfolding infnorm_eq_1_2 by auto
proposition fashoda_unit:
fixes f g :: "real ⇒ real^2"
assumes "f ` {-1 .. 1} ⊆ cbox (-1) 1"
and "g ` {-1 .. 1} ⊆ cbox (-1) 1"
and "continuous_on {-1 .. 1} f"
and "continuous_on {-1 .. 1} g"
and "f (- 1)$1 = - 1"
and "f 1$1 = 1" "g (- 1) $2 = -1"
and "g 1 $2 = 1"
shows "∃s∈{-1 .. 1}. ∃t∈{-1 .. 1}. f s = g t"
proof (rule ccontr)
assume "¬ ?thesis"
note as = this[unfolded bex_simps,rule_format]
define sqprojection
where [abs_def]: "sqprojection z = (inverse (infnorm z)) *⇩R z" for z :: "real^2"
define negatex :: "real^2 ⇒ real^2"
where "negatex x = (vector [-(x$1), x$2])" for x
have lem1: "∀z::real^2. infnorm (negatex z) = infnorm z"
unfolding negatex_def infnorm_2 vector_2 by auto
have lem2: "∀z. z ≠ 0 ⟶ infnorm (sqprojection z) = 1"
unfolding sqprojection_def infnorm_mul[unfolded scalar_mult_eq_scaleR]
by (simp add: real_abs_infnorm infnorm_eq_0)
let ?F = "λw::real^2. (f ∘ (λx. x$1)) w - (g ∘ (λx. x$2)) w"
have *: "⋀i. (λx::real^2. x $ i) ` cbox (- 1) 1 = {-1..1}"
proof
show "(λx::real^2. x $ i) ` cbox (- 1) 1 ⊆ {-1..1}" for i
by (auto simp: mem_box_cart)
show "{-1..1} ⊆ (λx::real^2. x $ i) ` cbox (- 1) 1" for i
by (clarsimp simp: image_iff mem_box_cart Bex_def) (metis (no_types, hide_lams) vec_component)
qed
{
fix x
assume "x ∈ (λw. (f ∘ (λx. x $ 1)) w - (g ∘ (λx. x $ 2)) w) ` (cbox (- 1) (1::real^2))"
then obtain w :: "real^2" where w:
"w ∈ cbox (- 1) 1"
"x = (f ∘ (λx. x $ 1)) w - (g ∘ (λx. x $ 2)) w"
unfolding image_iff ..
then have "x ≠ 0"
using as[of "w$1" "w$2"]
unfolding mem_box_cart atLeastAtMost_iff
by auto
} note x0 = this
have 1: "box (- 1) (1::real^2) ≠ {}"
unfolding interval_eq_empty_cart by auto
have "negatex (x + y) $ i = (negatex x + negatex y) $ i ∧ negatex (c *⇩R x) $ i = (c *⇩R negatex x) $ i"
for i x y c
using exhaust_2 [of i] by (auto simp: negatex_def)
then have "bounded_linear negatex"
by (simp add: bounded_linearI' vec_eq_iff)
then have 2: "continuous_on (cbox (- 1) 1) (negatex ∘ sqprojection ∘ ?F)"
apply (intro continuous_intros continuous_on_component)
unfolding * sqprojection_def
apply (intro assms continuous_intros)+
apply (simp_all add: infnorm_eq_0 x0 linear_continuous_on)
done
have 3: "(negatex ∘ sqprojection ∘ ?F) ` cbox (-1) 1 ⊆ cbox (-1) 1"
unfolding subset_eq
proof (rule, goal_cases)
case (1 x)
then obtain y :: "real^2" where y:
"y ∈ cbox (- 1) 1"
"x = (negatex ∘ sqprojection ∘ (λw. (f ∘ (λx. x $ 1)) w - (g ∘ (λx. x $ 2)) w)) y"
unfolding image_iff ..
have "?F y ≠ 0"
by (rule x0) (use y in auto)
then have *: "infnorm (sqprojection (?F y)) = 1"
unfolding y o_def
by - (rule lem2[rule_format])
have inf1: "infnorm x = 1"
unfolding *[symmetric] y o_def
by (rule lem1[rule_format])
show "x ∈ cbox (-1) 1"
unfolding mem_box_cart interval_cbox_cart infnorm_2
proof
fix i
show "(- 1) $ i ≤ x $ i ∧ x $ i ≤ 1 $ i"
using exhaust_2 [of i] inf1 by (auto simp: infnorm_2)
qed
qed
obtain x :: "real^2" where x:
"x ∈ cbox (- 1) 1"
"(negatex ∘ sqprojection ∘ (λw. (f ∘ (λx. x $ 1)) w - (g ∘ (λx. x $ 2)) w)) x = x"
apply (rule brouwer_weak[of "cbox (- 1) (1::real^2)" "negatex ∘ sqprojection ∘ ?F"])
apply (rule compact_cbox convex_box)+
unfolding interior_cbox
apply (rule 1 2 3)+
apply blast
done
have "?F x ≠ 0"
by (rule x0) (use x in auto)
then have *: "infnorm (sqprojection (?F x)) = 1"
unfolding o_def
by (rule lem2[rule_format])
have nx: "infnorm x = 1"
apply (subst x(2)[symmetric])
unfolding *[symmetric] o_def
apply (rule lem1[rule_format])
done
have iff: "0 < sqprojection x$i ⟷ 0 < x$i" "sqprojection x$i < 0 ⟷ x$i < 0" if "x ≠ 0" for x i
proof -
have "inverse (infnorm x) > 0"
by (simp add: infnorm_pos_lt that)
then show "(0 < sqprojection x $ i) = (0 < x $ i)"
and "(sqprojection x $ i < 0) = (x $ i < 0)"
unfolding sqprojection_def vector_component_simps vector_scaleR_component real_scaleR_def
unfolding zero_less_mult_iff mult_less_0_iff
by (auto simp add: field_simps)
qed
have x1: "x $ 1 ∈ {- 1..1::real}" "x $ 2 ∈ {- 1..1::real}"
using x(1) unfolding mem_box_cart by auto
then have nz: "f (x $ 1) - g (x $ 2) ≠ 0"
using as by auto
consider "x $ 1 = -1" | "x $ 1 = 1" | "x $ 2 = -1" | "x $ 2 = 1"
using nx unfolding infnorm_eq_1_2 by auto
then show False
proof cases
case 1
then have *: "f (x $ 1) $ 1 = - 1"
using assms(5) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0"
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
by (auto simp: negatex_def 1)
moreover
from x1 have "g (x $ 2) ∈ cbox (-1) 1"
using assms(2) by blast
ultimately show False
unfolding iff[OF nz] vector_component_simps * mem_box_cart
using not_le by auto
next
case 2
then have *: "f (x $ 1) $ 1 = 1"
using assms(6) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]] 2
by (auto simp: negatex_def)
moreover have "g (x $ 2) ∈ cbox (-1) 1"
using assms(2) x1 by blast
ultimately show False
unfolding iff[OF nz] vector_component_simps * mem_box_cart
using not_le by auto
next
case 3
then have *: "g (x $ 2) $ 2 = - 1"
using assms(7) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]] 3 by (auto simp: negatex_def)
moreover
from x1 have "f (x $ 1) ∈ cbox (-1) 1"
using assms(1) by blast
ultimately show False
unfolding iff[OF nz] vector_component_simps * mem_box_cart
by (erule_tac x=2 in allE) auto
next
case 4
then have *: "g (x $ 2) $ 2 = 1"
using assms(8) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0"
using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]] 4 by (auto simp: negatex_def)
moreover
from x1 have "f (x $ 1) ∈ cbox (-1) 1"
using assms(1) by blast
ultimately show False
unfolding iff[OF nz] vector_component_simps * mem_box_cart
by (erule_tac x=2 in allE) auto
qed
qed
proposition fashoda_unit_path:
fixes f g :: "real ⇒ real^2"
assumes "path f"
and "path g"
and "path_image f ⊆ cbox (-1) 1"
and "path_image g ⊆ cbox (-1) 1"
and "(pathstart f)$1 = -1"
and "(pathfinish f)$1 = 1"
and "(pathstart g)$2 = -1"
and "(pathfinish g)$2 = 1"
obtains z where "z ∈ path_image f" and "z ∈ path_image g"
proof -
note assms=assms[unfolded path_def pathstart_def pathfinish_def path_image_def]
define iscale where [abs_def]: "iscale z = inverse 2 *⇩R (z + 1)" for z :: real
have isc: "iscale ` {- 1..1} ⊆ {0..1}"
unfolding iscale_def by auto
have "∃s∈{- 1..1}. ∃t∈{- 1..1}. (f ∘ iscale) s = (g ∘ iscale) t"
proof (rule fashoda_unit)
show "(f ∘ iscale) ` {- 1..1} ⊆ cbox (- 1) 1" "(g ∘ iscale) ` {- 1..1} ⊆ cbox (- 1) 1"
using isc and assms(3-4) by (auto simp add: image_comp [symmetric])
have *: "continuous_on {- 1..1} iscale"
unfolding iscale_def by (rule continuous_intros)+
show "continuous_on {- 1..1} (f ∘ iscale)" "continuous_on {- 1..1} (g ∘ iscale)"
apply -
apply (rule_tac[!] continuous_on_compose[OF *])
apply (rule_tac[!] continuous_on_subset[OF _ isc])
apply (rule assms)+
done
have *: "(1 / 2) *⇩R (1 + (1::real^1)) = 1"
unfolding vec_eq_iff by auto
show "(f ∘ iscale) (- 1) $ 1 = - 1"
and "(f ∘ iscale) 1 $ 1 = 1"
and "(g ∘ iscale) (- 1) $ 2 = -1"
and "(g ∘ iscale) 1 $ 2 = 1"
unfolding o_def iscale_def
using assms
by (auto simp add: *)
qed
then obtain s t where st:
"s ∈ {- 1..1}"
"t ∈ {- 1..1}"
"(f ∘ iscale) s = (g ∘ iscale) t"
by auto
show thesis
apply (rule_tac z = "f (iscale s)" in that)
using st
unfolding o_def path_image_def image_iff
apply -
apply (rule_tac x="iscale s" in bexI)
prefer 3
apply (rule_tac x="iscale t" in bexI)
using isc[unfolded subset_eq, rule_format]
apply auto
done
qed
theorem fashoda:
fixes b :: "real^2"
assumes "path f"
and "path g"
and "path_image f ⊆ cbox a b"
and "path_image g ⊆ cbox a b"
and "(pathstart f)$1 = a$1"
and "(pathfinish f)$1 = b$1"
and "(pathstart g)$2 = a$2"
and "(pathfinish g)$2 = b$2"
obtains z where "z ∈ path_image f" and "z ∈ path_image g"
proof -
fix P Q S
presume "P ∨ Q ∨ S" "P ⟹ thesis" and "Q ⟹ thesis" and "S ⟹ thesis"
then show thesis
by auto
next
have "cbox a b ≠ {}"
using assms(3) using path_image_nonempty[of f] by auto
then have "a ≤ b"
unfolding interval_eq_empty_cart less_eq_vec_def by (auto simp add: not_less)
then show "a$1 = b$1 ∨ a$2 = b$2 ∨ (a$1 < b$1 ∧ a$2 < b$2)"
unfolding less_eq_vec_def forall_2 by auto
next
assume as: "a$1 = b$1"
have "∃z∈path_image g. z$2 = (pathstart f)$2"
apply (rule connected_ivt_component_cart)
apply (rule connected_path_image assms)+
apply (rule pathstart_in_path_image)
apply (rule pathfinish_in_path_image)
unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"]
unfolding pathstart_def
apply (auto simp add: less_eq_vec_def mem_box_cart)
done
then obtain z :: "real^2" where z: "z ∈ path_image g" "z $ 2 = pathstart f $ 2" ..
have "z ∈ cbox a b"
using z(1) assms(4)
unfolding path_image_def
by blast
then have "z = f 0"
unfolding vec_eq_iff forall_2
unfolding z(2) pathstart_def
using assms(3)[unfolded path_image_def subset_eq mem_box_cart,rule_format,of "f 0" 1]
unfolding mem_box_cart
apply (erule_tac x=1 in allE)
using as
apply auto
done
then show thesis
apply -
apply (rule that[OF _ z(1)])
unfolding path_image_def
apply auto
done
next
assume as: "a$2 = b$2"
have "∃z∈path_image f. z$1 = (pathstart g)$1"
apply (rule connected_ivt_component_cart)
apply (rule connected_path_image assms)+
apply (rule pathstart_in_path_image)
apply (rule pathfinish_in_path_image)
unfolding assms
using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"]
unfolding pathstart_def
apply (auto simp add: less_eq_vec_def mem_box_cart)
done
then obtain z where z: "z ∈ path_image f" "z $ 1 = pathstart g $ 1" ..
have "z ∈ cbox a b"
using z(1) assms(3)
unfolding path_image_def
by blast
then have "z = g 0"
unfolding vec_eq_iff forall_2
unfolding z(2) pathstart_def
using assms(4)[unfolded path_image_def subset_eq mem_box_cart,rule_format,of "g 0" 2]
unfolding mem_box_cart
apply (erule_tac x=2 in allE)
using as
apply auto
done
then show thesis
apply -
apply (rule that[OF z(1)])
unfolding path_image_def
apply auto
done
next
assume as: "a $ 1 < b $ 1 ∧ a $ 2 < b $ 2"
have int_nem: "cbox (-1) (1::real^2) ≠ {}"
unfolding interval_eq_empty_cart by auto
obtain z :: "real^2" where z:
"z ∈ (interval_bij (a, b) (- 1, 1) ∘ f) ` {0..1}"
"z ∈ (interval_bij (a, b) (- 1, 1) ∘ g) ` {0..1}"
apply (rule fashoda_unit_path[of "interval_bij (a,b) (- 1,1) ∘ f" "interval_bij (a,b) (- 1,1) ∘ g"])
unfolding path_def path_image_def pathstart_def pathfinish_def
apply (rule_tac[1-2] continuous_on_compose)
apply (rule assms[unfolded path_def] continuous_on_interval_bij)+
unfolding subset_eq
apply(rule_tac[1-2] ballI)
proof -
fix x
assume "x ∈ (interval_bij (a, b) (- 1, 1) ∘ f) ` {0..1}"
then obtain y where y:
"y ∈ {0..1}"
"x = (interval_bij (a, b) (- 1, 1) ∘ f) y"
unfolding image_iff ..
show "x ∈ cbox (- 1) 1"
unfolding y o_def
apply (rule in_interval_interval_bij)
using y(1)
using assms(3)[unfolded path_image_def subset_eq] int_nem
apply auto
done
next
fix x
assume "x ∈ (interval_bij (a, b) (- 1, 1) ∘ g) ` {0..1}"
then obtain y where y:
"y ∈ {0..1}"
"x = (interval_bij (a, b) (- 1, 1) ∘ g) y"
unfolding image_iff ..
show "x ∈ cbox (- 1) 1"
unfolding y o_def
apply (rule in_interval_interval_bij)
using y(1)
using assms(4)[unfolded path_image_def subset_eq] int_nem
apply auto
done
next
show "(interval_bij (a, b) (- 1, 1) ∘ f) 0 $ 1 = -1"
and "(interval_bij (a, b) (- 1, 1) ∘ f) 1 $ 1 = 1"
and "(interval_bij (a, b) (- 1, 1) ∘ g) 0 $ 2 = -1"
and "(interval_bij (a, b) (- 1, 1) ∘ g) 1 $ 2 = 1"
using assms as
by (simp_all add: cart_eq_inner_axis pathstart_def pathfinish_def interval_bij_def)
(simp_all add: inner_axis)
qed
from z(1) obtain zf where zf:
"zf ∈ {0..1}"
"z = (interval_bij (a, b) (- 1, 1) ∘ f) zf"
unfolding image_iff ..
from z(2) obtain zg where zg:
"zg ∈ {0..1}"
"z = (interval_bij (a, b) (- 1, 1) ∘ g) zg"
unfolding image_iff ..
have *: "∀i. (- 1) $ i < (1::real^2) $ i ∧ a $ i < b $ i"
unfolding forall_2
using as
by auto
show thesis
proof (rule_tac z="interval_bij (- 1,1) (a,b) z" in that)
show "interval_bij (- 1, 1) (a, b) z ∈ path_image f"
using zf by (simp add: interval_bij_bij_cart[OF *] path_image_def)
show "interval_bij (- 1, 1) (a, b) z ∈ path_image g"
using zg by (simp add: interval_bij_bij_cart[OF *] path_image_def)
qed
qed
subsection ‹Some slightly ad hoc lemmas I use below›
lemma segment_vertical:
fixes a :: "real^2"
assumes "a$1 = b$1"
shows "x ∈ closed_segment a b ⟷
x$1 = a$1 ∧ x$1 = b$1 ∧ (a$2 ≤ x$2 ∧ x$2 ≤ b$2 ∨ b$2 ≤ x$2 ∧ x$2 ≤ a$2)"
(is "_ = ?R")
proof -
let ?L = "∃u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 ∧ x $ 2 = (1 - u) * a $ 2 + u * b $ 2) ∧ 0 ≤ u ∧ u ≤ 1"
{
presume "?L ⟹ ?R" and "?R ⟹ ?L"
then show ?thesis
unfolding closed_segment_def mem_Collect_eq
unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[symmetric] vector_component_simps
by blast
}
{
assume ?L
then obtain u where u:
"x $ 1 = (1 - u) * a $ 1 + u * b $ 1"
"x $ 2 = (1 - u) * a $ 2 + u * b $ 2"
"0 ≤ u"
"u ≤ 1"
by blast
{ fix b a
assume "b + u * a > a + u * b"
then have "(1 - u) * b > (1 - u) * a"
by (auto simp add:field_simps)
then have "b ≥ a"
apply (drule_tac mult_left_less_imp_less)
using u
apply auto
done
then have "u * a ≤ u * b"
apply -
apply (rule mult_left_mono[OF _ u(3)])
using u(3-4)
apply (auto simp add: field_simps)
done
} note * = this
{
fix a b
assume "u * b > u * a"
then have "(1 - u) * a ≤ (1 - u) * b"
apply -
apply (rule mult_left_mono)
apply (drule mult_left_less_imp_less)
using u
apply auto
done
then have "a + u * b ≤ b + u * a"
by (auto simp add: field_simps)
} note ** = this
then show ?R
unfolding u assms
using u
by (auto simp add:field_simps not_le intro: * **)
}
{
assume ?R
then show ?L
proof (cases "x$2 = b$2")
case True
then show ?L
apply (rule_tac x="(x$2 - a$2) / (b$2 - a$2)" in exI)
unfolding assms True using ‹?R› apply (auto simp add: field_simps)
done
next
case False
then show ?L
apply (rule_tac x="1 - (x$2 - b$2) / (a$2 - b$2)" in exI)
unfolding assms using ‹?R› apply (auto simp add: field_simps)
done
qed
}
qed
lemma segment_horizontal:
fixes a :: "real^2"
assumes "a$2 = b$2"
shows "x ∈ closed_segment a b ⟷
x$2 = a$2 ∧ x$2 = b$2 ∧ (a$1 ≤ x$1 ∧ x$1 ≤ b$1 ∨ b$1 ≤ x$1 ∧ x$1 ≤ a$1)"
(is "_ = ?R")
proof -
let ?L = "∃u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 ∧ x $ 2 = (1 - u) * a $ 2 + u * b $ 2) ∧ 0 ≤ u ∧ u ≤ 1"
{
presume "?L ⟹ ?R" and "?R ⟹ ?L"
then show ?thesis
unfolding closed_segment_def mem_Collect_eq
unfolding vec_eq_iff forall_2 scalar_mult_eq_scaleR[symmetric] vector_component_simps
by blast
}
{
assume ?L
then obtain u where u:
"x $ 1 = (1 - u) * a $ 1 + u * b $ 1"
"x $ 2 = (1 - u) * a $ 2 + u * b $ 2"
"0 ≤ u"
"u ≤ 1"
by blast
{
fix b a
assume "b + u * a > a + u * b"
then have "(1 - u) * b > (1 - u) * a"
by (auto simp add: field_simps)
then have "b ≥ a"
apply (drule_tac mult_left_less_imp_less)
using u
apply auto
done
then have "u * a ≤ u * b"
apply -
apply (rule mult_left_mono[OF _ u(3)])
using u(3-4)
apply (auto simp add: field_simps)
done
} note * = this
{
fix a b
assume "u * b > u * a"
then have "(1 - u) * a ≤ (1 - u) * b"
apply -
apply (rule mult_left_mono)
apply (drule mult_left_less_imp_less)
using u
apply auto
done
then have "a + u * b ≤ b + u * a"
by (auto simp add: field_simps)
} note ** = this
then show ?R
unfolding u assms
using u
by (auto simp add: field_simps not_le intro: * **)
}
{
assume ?R
then show ?L
proof (cases "x$1 = b$1")
case True
then show ?L
apply (rule_tac x="(x$1 - a$1) / (b$1 - a$1)" in exI)
unfolding assms True
using ‹?R›
apply (auto simp add: field_simps)
done
next
case False
then show ?L
apply (rule_tac x="1 - (x$1 - b$1) / (a$1 - b$1)" in exI)
unfolding assms
using ‹?R›
apply (auto simp add: field_simps)
done
qed
}
qed
subsection ‹Useful Fashoda corollary pointed out to me by Tom Hales›
corollary fashoda_interlace:
fixes a :: "real^2"
assumes "path f"
and "path g"
and paf: "path_image f ⊆ cbox a b"
and pag: "path_image g ⊆ cbox a b"
and "(pathstart f)$2 = a$2"
and "(pathfinish f)$2 = a$2"
and "(pathstart g)$2 = a$2"
and "(pathfinish g)$2 = a$2"
and "(pathstart f)$1 < (pathstart g)$1"
and "(pathstart g)$1 < (pathfinish f)$1"
and "(pathfinish f)$1 < (pathfinish g)$1"
obtains z where "z ∈ path_image f" and "z ∈ path_image g"
proof -
have "cbox a b ≠ {}"
using path_image_nonempty[of f] using assms(3) by auto
note ab=this[unfolded interval_eq_empty_cart not_ex forall_2 not_less]
have "pathstart f ∈ cbox a b"
and "pathfinish f ∈ cbox a b"
and "pathstart g ∈ cbox a b"
and "pathfinish g ∈ cbox a b"
using pathstart_in_path_image pathfinish_in_path_image
using assms(3-4)
by auto
note startfin = this[unfolded mem_box_cart forall_2]
let ?P1 = "linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2]) +++
linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f) +++ f +++
linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2]) +++
linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2])"
let ?P2 = "linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g) +++ g +++
linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1]) +++
linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1]) +++
linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3])"
let ?a = "vector[a$1 - 2, a$2 - 3]"
let ?b = "vector[b$1 + 2, b$2 + 3]"
have P1P2: "path_image ?P1 = path_image (linepath (vector[a$1 - 2, a$2 - 2]) (vector[(pathstart f)$1,a$2 - 2])) ∪
path_image (linepath(vector[(pathstart f)$1,a$2 - 2])(pathstart f)) ∪ path_image f ∪
path_image (linepath(pathfinish f)(vector[(pathfinish f)$1,a$2 - 2])) ∪
path_image (linepath(vector[(pathfinish f)$1,a$2 - 2])(vector[b$1 + 2,a$2 - 2]))"
"path_image ?P2 = path_image(linepath(vector[(pathstart g)$1, (pathstart g)$2 - 3])(pathstart g)) ∪ path_image g ∪
path_image(linepath(pathfinish g)(vector[(pathfinish g)$1,a$2 - 1])) ∪
path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) ∪
path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
by(auto simp add: path_image_join)
have abab: "cbox a b ⊆ cbox ?a ?b"
unfolding interval_cbox_cart[symmetric]
by (auto simp add:less_eq_vec_def forall_2)
obtain z where
"z ∈ path_image
(linepath (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) +++
linepath (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f) +++
f +++
linepath (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) +++
linepath (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]))"
"z ∈ path_image
(linepath (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g) +++
g +++
linepath (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1]) +++
linepath (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1]) +++
linepath (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]))"
apply (rule fashoda[of ?P1 ?P2 ?a ?b])
unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2
proof -
show "path ?P1" and "path ?P2"
using assms by auto
show "path_image ?P1 ⊆ cbox ?a ?b" "path_image ?P2 ⊆ cbox ?a ?b"
unfolding P1P2 path_image_linepath using startfin paf pag
by (auto simp: mem_box_cart segment_horizontal segment_vertical forall_2)
show "a $ 1 - 2 = a $ 1 - 2"
and "b $ 1 + 2 = b $ 1 + 2"
and "pathstart g $ 2 - 3 = a $ 2 - 3"
and "b $ 2 + 3 = b $ 2 + 3"
by (auto simp add: assms)
qed
note z=this[unfolded P1P2 path_image_linepath]
show thesis
proof (rule that[of z])
have "(z ∈ closed_segment (vector [a $ 1 - 2, a $ 2 - 2]) (vector [pathstart f $ 1, a $ 2 - 2]) ∨
z ∈ closed_segment (vector [pathstart f $ 1, a $ 2 - 2]) (pathstart f)) ∨
z ∈ closed_segment (pathfinish f) (vector [pathfinish f $ 1, a $ 2 - 2]) ∨
z ∈ closed_segment (vector [pathfinish f $ 1, a $ 2 - 2]) (vector [b $ 1 + 2, a $ 2 - 2]) ⟹
(((z ∈ closed_segment (vector [pathstart g $ 1, pathstart g $ 2 - 3]) (pathstart g)) ∨
z ∈ closed_segment (pathfinish g) (vector [pathfinish g $ 1, a $ 2 - 1])) ∨
z ∈ closed_segment (vector [pathfinish g $ 1, a $ 2 - 1]) (vector [b $ 1 + 1, a $ 2 - 1])) ∨
z ∈ closed_segment (vector [b $ 1 + 1, a $ 2 - 1]) (vector [b $ 1 + 1, b $ 2 + 3]) ⟹ False"
proof (simp only: segment_vertical segment_horizontal vector_2, goal_cases)
case prems: 1
have "pathfinish f ∈ cbox a b"
using assms(3) pathfinish_in_path_image[of f] by auto
then have "1 + b $ 1 ≤ pathfinish f $ 1 ⟹ False"
unfolding mem_box_cart forall_2 by auto
then have "z$1 ≠ pathfinish f$1"
using prems(2)
using assms ab
by (auto simp add: field_simps)
moreover have "pathstart f ∈ cbox a b"
using assms(3) pathstart_in_path_image[of f]
by auto
then have "1 + b $ 1 ≤ pathstart f $ 1 ⟹ False"
unfolding mem_box_cart forall_2
by auto
then have "z$1 ≠ pathstart f$1"
using prems(2) using assms ab
by (auto simp add: field_simps)
ultimately have *: "z$2 = a$2 - 2"
using prems(1) by auto
have "z$1 ≠ pathfinish g$1"
using prems(2) assms ab
by (auto simp add: field_simps *)
moreover have "pathstart g ∈ cbox a b"
using assms(4) pathstart_in_path_image[of g]
by auto
note this[unfolded mem_box_cart forall_2]
then have "z$1 ≠ pathstart g$1"
using prems(1) assms ab
by (auto simp add: field_simps *)
ultimately have "a $ 2 - 1 ≤ z $ 2 ∧ z $ 2 ≤ b $ 2 + 3 ∨ b $ 2 + 3 ≤ z $ 2 ∧ z $ 2 ≤ a $ 2 - 1"
using prems(2) unfolding * assms by (auto simp add: field_simps)
then show False
unfolding * using ab by auto
qed
then have "z ∈ path_image f ∨ z ∈ path_image g"
using z unfolding Un_iff by blast
then have z': "z ∈ cbox a b"
using assms(3-4) by auto
have "a $ 2 = z $ 2 ⟹ (z $ 1 = pathstart f $ 1 ∨ z $ 1 = pathfinish f $ 1) ⟹
z = pathstart f ∨ z = pathfinish f"
unfolding vec_eq_iff forall_2 assms
by auto
with z' show "z ∈ path_image f"
using z(1)
unfolding Un_iff mem_box_cart forall_2
by (simp only: segment_vertical segment_horizontal vector_2) (auto simp: assms)
have "a $ 2 = z $ 2 ⟹ (z $ 1 = pathstart g $ 1 ∨ z $ 1 = pathfinish g $ 1) ⟹
z = pathstart g ∨ z = pathfinish g"
unfolding vec_eq_iff forall_2 assms
by auto
with z' show "z ∈ path_image g"
using z(2)
unfolding Un_iff mem_box_cart forall_2
by (simp only: segment_vertical segment_horizontal vector_2) (auto simp: assms)
qed
qed
end