# Theory Affine

```section "Affine Sets"

theory Affine
imports Linear_Algebra
begin

lemma if_smult: "(if P then x else (y::real)) *⇩R v = (if P then x *⇩R v else y *⇩R v)"
by simp

lemma sum_delta_notmem:
assumes "x ∉ s"
shows "sum (λy. if (y = x) then P x else Q y) s = sum Q s"
and "sum (λy. if (x = y) then P x else Q y) s = sum Q s"
and "sum (λy. if (y = x) then P y else Q y) s = sum Q s"
and "sum (λy. if (x = y) then P y else Q y) s = sum Q s"
by (smt (verit, best) assms sum.cong)+

lemma span_substd_basis:
assumes d: "d ⊆ Basis"
shows "span d = {x. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
(is "_ = ?B")
proof -
have "d ⊆ ?B"
using d by (auto simp: inner_Basis)
moreover have s: "subspace ?B"
using subspace_substandard[of "λi. i ∉ d"] .
ultimately have "span d ⊆ ?B"
using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast
moreover have *: "card d ≤ dim (span d)"
by (simp add: d dim_eq_card_independent independent_substdbasis)
moreover from * have "dim ?B ≤ dim (span d)"
using dim_substandard[OF assms] by auto
ultimately show ?thesis
by (simp add: s subspace_dim_equal)
qed

lemma basis_to_substdbasis_subspace_isomorphism:
fixes B :: "'a::euclidean_space set"
assumes "independent B"
shows "∃f d::'a set. card d = card B ∧ linear f ∧ f ` B = d ∧
f ` span B = {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0} ∧ inj_on f (span B) ∧ d ⊆ Basis"
proof -
have B: "card B = dim B"
using dim_unique[of B B "card B"] assms span_superset[of B] by auto
have "dim B ≤ card (Basis :: 'a set)"
using dim_subset_UNIV[of B] by simp
from obtain_subset_with_card_n[OF this]
obtain d :: "'a set" where d: "d ⊆ Basis" and t: "card d = dim B"
by auto
let ?t = "{x::'a::euclidean_space. ∀i∈Basis. i ∉ d ⟶ x∙i = 0}"
have "∃f. linear f ∧ f ` B = d ∧ f ` span B = ?t ∧ inj_on f (span B)"
proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset)
show "d ⊆ {x. ∀i∈Basis. i ∉ d ⟶ x ∙ i = 0}"
using d inner_not_same_Basis by blast
qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms)
with t ‹card B = dim B› d show ?thesis by auto
qed

subsection ‹Affine set and affine hull›

definition✐‹tag important› affine :: "'a::real_vector set ⇒ bool"
where "affine S ⟷ (∀x∈S. ∀y∈S. ∀u v. u + v = 1 ⟶ u *⇩R x + v *⇩R y ∈ S)"

lemma affine_alt: "affine S ⟷ (∀x∈S. ∀y∈S. ∀u::real. (1 - u) *⇩R x + u *⇩R y ∈ S)"
unfolding affine_def by (metis eq_diff_eq')

lemma affine_empty [iff]: "affine {}"
unfolding affine_def by auto

lemma affine_sing [iff]: "affine {x}"
unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric])

lemma affine_UNIV [iff]: "affine UNIV"
unfolding affine_def by auto

lemma affine_Inter [intro]: "(⋀S. S∈ℱ ⟹ affine S) ⟹ affine (⋂ℱ)"
unfolding affine_def by auto

lemma affine_Int[intro]: "affine S ⟹ affine T ⟹ affine (S ∩ T)"
unfolding affine_def by auto

lemma affine_scaling: "affine S ⟹ affine ((*⇩R) c ` S)"
apply (clarsimp simp: affine_def)
apply (rule_tac x="u *⇩R x + v *⇩R y" in image_eqI)
apply (auto simp: algebra_simps)
done

lemma affine_affine_hull [simp]: "affine(affine hull S)"
unfolding hull_def
using affine_Inter[of "{T. affine T ∧ S ⊆ T}"] by auto

lemma affine_hull_eq[simp]: "(affine hull s = s) ⟷ affine s"
by (metis affine_affine_hull hull_same)

lemma affine_hyperplane: "affine {x. a ∙ x = b}"
by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral)

subsubsection✐‹tag unimportant› ‹Some explicit formulations›

text "Formalized by Lars Schewe."

lemma affine:
fixes V::"'a::real_vector set"
shows "affine V ⟷
(∀S u. finite S ∧ S ≠ {} ∧ S ⊆ V ∧ sum u S = 1 ⟶ (∑x∈S. u x *⇩R x) ∈ V)"
proof -
have "u *⇩R x + v *⇩R y ∈ V" if "x ∈ V" "y ∈ V" "u + v = (1::real)"
and *: "⋀S u. ⟦finite S; S ≠ {}; S ⊆ V; sum u S = 1⟧ ⟹ (∑x∈S. u x *⇩R x) ∈ V" for x y u v
proof (cases "x = y")
case True
then show ?thesis
using that by (metis scaleR_add_left scaleR_one)
next
case False
then show ?thesis
using that *[of "{x,y}" "λw. if w = x then u else v"] by auto
qed
moreover have "(∑x∈S. u x *⇩R x) ∈ V"
if *: "⋀x y u v. ⟦x∈V; y∈V; u + v = 1⟧ ⟹ u *⇩R x + v *⇩R y ∈ V"
and "finite S" "S ≠ {}" "S ⊆ V" "sum u S = 1" for S u
proof -
define n where "n = card S"
consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith
then show "(∑x∈S. u x *⇩R x) ∈ V"
proof cases
assume "card S = 1"
then obtain a where "S={a}"
by (auto simp: card_Suc_eq)
then show ?thesis
using that by simp
next
assume "card S = 2"
then obtain a b where "S = {a, b}"
by (metis Suc_1 card_1_singletonE card_Suc_eq)
then show ?thesis
using *[of a b] that
by (auto simp: sum_clauses(2))
next
assume "card S > 2"
then show ?thesis using that n_def
proof (induct n arbitrary: u S)
case 0
then show ?case by auto
next
case (Suc n u S)
have "sum u S = card S" if "¬ (∃x∈S. u x ≠ 1)"
using that unfolding card_eq_sum by auto
with Suc.prems obtain x where "x ∈ S" and x: "u x ≠ 1" by force
have c: "card (S - {x}) = card S - 1"
by (simp add: Suc.prems(3) ‹x ∈ S›)
have "sum u (S - {x}) = 1 - u x"
by (simp add: Suc.prems sum_diff1 ‹x ∈ S›)
with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1"
by auto
have inV: "(∑y∈S - {x}. (inverse (1 - u x) * u y) *⇩R y) ∈ V"
proof (cases "card (S - {x}) > 2")
case True
then have S: "S - {x} ≠ {}" "card (S - {x}) = n"
using Suc.prems c by force+
show ?thesis
proof (rule Suc.hyps)
show "(∑a∈S - {x}. inverse (1 - u x) * u a) = 1"
by (auto simp: eq1 sum_distrib_left[symmetric])
qed (use S Suc.prems True in auto)
next
case False
then have "card (S - {x}) = Suc (Suc 0)"
using Suc.prems c by auto
then obtain a b where ab: "(S - {x}) = {a, b}" "a≠b"
unfolding card_Suc_eq by auto
then show ?thesis
using eq1 ‹S ⊆ V›
by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b])
qed
have "u x + (1 - u x) = 1 ⟹
u x *⇩R x + (1 - u x) *⇩R ((∑y∈S - {x}. u y *⇩R y) /⇩R (1 - u x)) ∈ V"
by (rule Suc.prems) (use ‹x ∈ S› Suc.prems inV in ‹auto simp: scaleR_right.sum›)
moreover have "(∑a∈S. u a *⇩R a) = u x *⇩R x + (∑a∈S - {x}. u a *⇩R a)"
by (meson Suc.prems(3) sum.remove ‹x ∈ S›)
ultimately show "(∑x∈S. u x *⇩R x) ∈ V"
by (simp add: x)
qed
qed (use ‹S≠{}› ‹finite S› in auto)
qed
ultimately show ?thesis
unfolding affine_def by meson
qed

lemma affine_hull_explicit:
"affine hull p = {y. ∃S u. finite S ∧ S ≠ {} ∧ S ⊆ p ∧ sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
(is "_ = ?rhs")
proof (rule hull_unique)
have "⋀x. sum (λz. 1) {x} = 1"
by auto
show "p ⊆ ?rhs"
proof (intro subsetI CollectI exI conjI)
show "⋀x. sum (λz. 1) {x} = 1"
by auto
qed auto
show "?rhs ⊆ T" if "p ⊆ T" "affine T" for T
using that unfolding affine by blast
show "affine ?rhs"
unfolding affine_def
proof clarify
fix u v :: real and sx ux sy uy
assume uv: "u + v = 1"
and x: "finite sx" "sx ≠ {}" "sx ⊆ p" "sum ux sx = (1::real)"
and y: "finite sy" "sy ≠ {}" "sy ⊆ p" "sum uy sy = (1::real)"
have **: "(sx ∪ sy) ∩ sx = sx" "(sx ∪ sy) ∩ sy = sy"
by auto
show "∃S w. finite S ∧ S ≠ {} ∧ S ⊆ p ∧
sum w S = 1 ∧ (∑v∈S. w v *⇩R v) = u *⇩R (∑v∈sx. ux v *⇩R v) + v *⇩R (∑v∈sy. uy v *⇩R v)"
proof (intro exI conjI)
show "finite (sx ∪ sy)"
using x y by auto
show "sum (λi. (if i∈sx then u * ux i else 0) + (if i∈sy then v * uy i else 0)) (sx ∪ sy) = 1"
using x y uv
by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **)
have "(∑i∈sx ∪ sy. ((if i ∈ sx then u * ux i else 0) + (if i ∈ sy then v * uy i else 0)) *⇩R i)
= (∑i∈sx. (u * ux i) *⇩R i) + (∑i∈sy. (v * uy i) *⇩R i)"
using x y
unfolding scaleR_left_distrib scaleR_zero_left if_smult
by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric]  **)
also have "… = u *⇩R (∑v∈sx. ux v *⇩R v) + v *⇩R (∑v∈sy. uy v *⇩R v)"
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast
finally show "(∑i∈sx ∪ sy. ((if i ∈ sx then u * ux i else 0) + (if i ∈ sy then v * uy i else 0)) *⇩R i)
= u *⇩R (∑v∈sx. ux v *⇩R v) + v *⇩R (∑v∈sy. uy v *⇩R v)" .
qed (use x y in auto)
qed
qed

lemma affine_hull_finite:
assumes "finite S"
shows "affine hull S = {y. ∃u. sum u S = 1 ∧ sum (λv. u v *⇩R v) S = y}"
proof -
have *: "∃h. sum h S = 1 ∧ (∑v∈S. h v *⇩R v) = x"
if "F ⊆ S" "finite F" "F ≠ {}" and sum: "sum u F = 1" and x: "(∑v∈F. u v *⇩R v) = x" for x F u
proof -
have "S ∩ F = F"
using that by auto
show ?thesis
proof (intro exI conjI)
show "(∑x∈S. if x ∈ F then u x else 0) = 1"
by (metis (mono_tags, lifting) ‹S ∩ F = F› assms sum.inter_restrict sum)
show "(∑v∈S. (if v ∈ F then u v else 0) *⇩R v) = x"
by (simp add: if_smult cong: if_cong) (metis (no_types) ‹S ∩ F = F› assms sum.inter_restrict x)
qed
qed
show ?thesis
unfolding affine_hull_explicit using assms
by (fastforce dest: *)
qed

subsubsection✐‹tag unimportant› ‹Stepping theorems and hence small special cases›

lemma affine_hull_empty[simp]: "affine hull {} = {}"
by simp

lemma affine_hull_finite_step:
fixes y :: "'a::real_vector"
shows "finite S ⟹
(∃u. sum u (insert a S) = w ∧ sum (λx. u x *⇩R x) (insert a S) = y) ⟷
(∃v u. sum u S = w - v ∧ sum (λx. u x *⇩R x) S = y - v *⇩R a)" (is "_ ⟹ ?lhs = ?rhs")
proof -
assume fin: "finite S"
show "?lhs = ?rhs"
proof
assume ?lhs
then obtain u where u: "sum u (insert a S) = w ∧ (∑x∈insert a S. u x *⇩R x) = y"
by auto
show ?rhs
proof (cases "a ∈ S")
case True
then show ?thesis
using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left)
next
case False
show ?thesis
by (rule exI [where x="u a"]) (use u fin False in auto)
qed
next
assume ?rhs
then obtain v u where vu: "sum u S = w - v"  "(∑x∈S. u x *⇩R x) = y - v *⇩R a"
by auto
have *: "⋀x M. (if x = a then v else M) *⇩R x = (if x = a then v *⇩R x else M *⇩R x)"
by auto
show ?lhs
proof (cases "a ∈ S")
case True
show ?thesis
by (rule exI [where x="λx. (if x=a then v else 0) + u x"])
(simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong)
next
case False
then show ?thesis
apply (rule_tac x="λx. if x=a then v else u x" in exI)
apply (simp add: vu sum_clauses(2)[OF fin] *)
by (simp add: sum_delta_notmem(3) vu)
qed
qed
qed

lemma affine_hull_2:
fixes a b :: "'a::real_vector"
shows "affine hull {a,b} = {u *⇩R a + v *⇩R b| u v. (u + v = 1)}"
(is "?lhs = ?rhs")
proof -
have *:
"⋀x y z. z = x - y ⟷ y + z = (x::real)"
"⋀x y z. z = x - y ⟷ y + z = (x::'a)" by auto
have "?lhs = {y. ∃u. sum u {a, b} = 1 ∧ (∑v∈{a, b}. u v *⇩R v) = y}"
using affine_hull_finite[of "{a,b}"] by auto
also have "… = {y. ∃v u. u b = 1 - v ∧ u b *⇩R b = y - v *⇩R a}"
by (simp add: affine_hull_finite_step[of "{b}" a])
also have "… = ?rhs" unfolding * by auto
finally show ?thesis by auto
qed

lemma affine_hull_3:
fixes a b c :: "'a::real_vector"
shows "affine hull {a,b,c} = { u *⇩R a + v *⇩R b + w *⇩R c| u v w. u + v + w = 1}"
proof -
have *:
"⋀x y z. z = x - y ⟷ y + z = (x::real)"
"⋀x y z. z = x - y ⟷ y + z = (x::'a)" by auto
show ?thesis
apply (simp add: affine_hull_finite affine_hull_finite_step)
unfolding *
apply safe
apply (rule_tac x=u in exI, force)
done
qed

lemma mem_affine:
assumes "affine S" "x ∈ S" "y ∈ S" "u + v = 1"
shows "u *⇩R x + v *⇩R y ∈ S"
using assms affine_def[of S] by auto

lemma mem_affine_3:
assumes "affine S" "x ∈ S" "y ∈ S" "z ∈ S" "u + v + w = 1"
shows "u *⇩R x + v *⇩R y + w *⇩R z ∈ S"
proof -
have "u *⇩R x + v *⇩R y + w *⇩R z ∈ affine hull {x, y, z}"
using affine_hull_3[of x y z] assms by auto
moreover
have "affine hull {x, y, z} ⊆ affine hull S"
using hull_mono[of "{x, y, z}" "S"] assms by auto
moreover
have "affine hull S = S"
using assms affine_hull_eq[of S] by auto
ultimately show ?thesis by auto
qed

lemma mem_affine_3_minus:
assumes "affine S" "x ∈ S" "y ∈ S" "z ∈ S"
shows "x + v *⇩R (y-z) ∈ S"
using mem_affine_3[of S x y z 1 v "-v"] assms
by (simp add: algebra_simps)

corollary%unimportant mem_affine_3_minus2:
"⟦affine S; x ∈ S; y ∈ S; z ∈ S⟧ ⟹ x - v *⇩R (y-z) ∈ S"
by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left)

subsubsection✐‹tag unimportant› ‹Some relations between affine hull and subspaces›

lemma affine_hull_insert_subset_span:
"affine hull (insert a S) ⊆ {a + v| v . v ∈ span {x - a | x . x ∈ S}}"
proof -
have "∃v T u. x = a + v ∧ (finite T ∧ T ⊆ {x - a |x. x ∈ S} ∧ (∑v∈T. u v *⇩R v) = v)"
if "finite F" "F ≠ {}" "F ⊆ insert a S" "sum u F = 1" "(∑v∈F. u v *⇩R v) = x"
for x F u
proof -
have *: "(λx. x - a) ` (F - {a}) ⊆ {x - a |x. x ∈ S}"
using that by auto
show ?thesis
proof (intro exI conjI)
show "finite ((λx. x - a) ` (F - {a}))"
by (simp add: that(1))
show "(∑v∈(λx. x - a) ` (F - {a}). u(v+a) *⇩R v) = x-a"
by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps
sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that)
qed (use ‹F ⊆ insert a S› in auto)
qed
then show ?thesis
unfolding affine_hull_explicit span_explicit by fast
qed

lemma affine_hull_insert_span:
assumes "a ∉ S"
shows "affine hull (insert a S) = {a + v | v . v ∈ span {x - a | x.  x ∈ S}}"
proof -
have *: "∃G u. finite G ∧ G ≠ {} ∧ G ⊆ insert a S ∧ sum u G = 1 ∧ (∑v∈G. u v *⇩R v) = y"
if "v ∈ span {x - a |x. x ∈ S}" "y = a + v" for y v
proof -
from that
obtain T u where u: "finite T" "T ⊆ {x - a |x. x ∈ S}" "a + (∑v∈T. u v *⇩R v) = y"
unfolding span_explicit by auto
define F where "F = (λx. x + a) ` T"
have F: "finite F" "F ⊆ S" "(∑v∈F. u (v - a) *⇩R (v - a)) = y - a"
unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def])
have *: "F ∩ {a} = {}" "F ∩ - {a} = F"
using F assms by auto
show "∃G u. finite G ∧ G ≠ {} ∧ G ⊆ insert a S ∧ sum u G = 1 ∧ (∑v∈G. u v *⇩R v) = y"
apply (rule_tac x = "insert a F" in exI)
apply (rule_tac x = "λx. if x=a then 1 - sum (λx. u (x - a)) F else u (x - a)" in exI)
using assms F
apply (auto simp:  sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *)
done
qed
show ?thesis
by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *)
qed

lemma affine_hull_span:
assumes "a ∈ S"
shows "affine hull S = {a + v | v. v ∈ span {x - a | x. x ∈ S - {a}}}"
using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto

subsubsection✐‹tag unimportant› ‹Parallel affine sets›

definition affine_parallel :: "'a::real_vector set ⇒ 'a::real_vector set ⇒ bool"
where "affine_parallel S T ⟷ (∃a. T = (λx. a + x) ` S)"

lemma affine_parallel_expl_aux:
fixes S T :: "'a::real_vector set"
assumes "⋀x. x ∈ S ⟷ a + x ∈ T"
shows "T = (λx. a + x) ` S"
proof -
have "x ∈ ((λx. a + x) ` S)" if "x ∈ T" for x
using that
moreover have "T ≥ (λx. a + x) ` S"
using assms by auto
ultimately show ?thesis by auto
qed

lemma affine_parallel_expl: "affine_parallel S T ⟷ (∃a. ∀x. x ∈ S ⟷ a + x ∈ T)"
by (auto simp add: affine_parallel_def)
(use affine_parallel_expl_aux [of S _ T] in blast)

lemma affine_parallel_reflex: "affine_parallel S S"
unfolding affine_parallel_def
using image_add_0 by blast

lemma affine_parallel_commute:
assumes "affine_parallel A B"
shows "affine_parallel B A"
by (metis affine_parallel_def assms translation_galois)

lemma affine_parallel_assoc:
assumes "affine_parallel A B"
and "affine_parallel B C"
shows "affine_parallel A C"
by (metis affine_parallel_def assms translation_assoc)

lemma affine_translation_aux:
fixes a :: "'a::real_vector"
assumes "affine ((λx. a + x) ` S)"
shows "affine S"
proof -
{
fix x y u v
assume xy: "x ∈ S" "y ∈ S" "(u :: real) + v = 1"
then have "(a + x) ∈ ((λx. a + x) ` S)" "(a + y) ∈ ((λx. a + x) ` S)"
by auto
then have h1: "u *⇩R  (a + x) + v *⇩R (a + y) ∈ (λx. a + x) ` S"
using xy assms unfolding affine_def by auto
have "u *⇩R (a + x) + v *⇩R (a + y) = (u + v) *⇩R a + (u *⇩R x + v *⇩R y)"
by (simp add: algebra_simps)
also have "… = a + (u *⇩R x + v *⇩R y)"
using ‹u + v = 1› by auto
ultimately have "a + (u *⇩R x + v *⇩R y) ∈ (λx. a + x) ` S"
using h1 by auto
then have "u *⇩R x + v *⇩R y ∈ S" by auto
}
then show ?thesis unfolding affine_def by auto
qed

lemma affine_translation:
"affine S ⟷ affine ((+) a ` S)" for a :: "'a::real_vector"
by (metis affine_translation_aux translation_galois)

lemma parallel_is_affine:
fixes S T :: "'a::real_vector set"
assumes "affine S" "affine_parallel S T"
shows "affine T"
by (metis affine_parallel_def affine_translation assms)

lemma subspace_imp_affine: "subspace s ⟹ affine s"
unfolding subspace_def affine_def by auto

lemma affine_hull_subset_span: "(affine hull s) ⊆ (span s)"
by (metis hull_minimal span_superset subspace_imp_affine subspace_span)

subsubsection✐‹tag unimportant› ‹Subspace parallel to an affine set›

lemma subspace_affine: "subspace S ⟷ affine S ∧ 0 ∈ S"
by (metis add_cancel_right_left affine_alt diff_add_cancel mem_affine_3 scaleR_eq_0_iff subspace_def vector_space_assms(4))

lemma affine_diffs_subspace:
assumes "affine S" "a ∈ S"
shows "subspace ((λx. (-a)+x) ` S)"
by (metis ab_left_minus affine_translation assms image_eqI subspace_affine)

lemma affine_diffs_subspace_subtract:
"subspace ((λx. x - a) ` S)" if "affine S" "a ∈ S"
using that affine_diffs_subspace [of _ a] by simp

lemma parallel_subspace_explicit:
assumes "affine S"
and "a ∈ S"
assumes "L ≡ {y. ∃x ∈ S. (-a) + x = y}"
shows "subspace L ∧ affine_parallel S L"
by (smt (verit) Collect_cong ab_left_minus affine_parallel_def assms image_def mem_Collect_eq parallel_is_affine subspace_affine)

lemma parallel_subspace_aux:
assumes "subspace A"
and "subspace B"
and "affine_parallel A B"
shows "A ⊇ B"
by (metis add.right_neutral affine_parallel_expl assms subsetI subspace_def)

lemma parallel_subspace:
assumes "subspace A"
and "subspace B"
and "affine_parallel A B"
shows "A = B"
by (simp add: affine_parallel_commute assms parallel_subspace_aux subset_antisym)

lemma affine_parallel_subspace:
assumes "affine S" "S ≠ {}"
shows "∃!L. subspace L ∧ affine_parallel S L"
by (meson affine_parallel_assoc affine_parallel_commute assms equals0I parallel_subspace parallel_subspace_explicit)

subsection ‹Affine Dependence›

text "Formalized by Lars Schewe."

definition✐‹tag important› affine_dependent :: "'a::real_vector set ⇒ bool"
where "affine_dependent S ⟷ (∃x∈S. x ∈ affine hull (S - {x}))"

lemma affine_dependent_imp_dependent: "affine_dependent S ⟹ dependent S"
unfolding affine_dependent_def dependent_def
using affine_hull_subset_span by auto

lemma affine_dependent_subset:
"⟦affine_dependent S; S ⊆ T⟧ ⟹ affine_dependent T"
using hull_mono [OF Diff_mono [OF _ subset_refl]]
by (smt (verit) affine_dependent_def subsetD)

lemma affine_independent_subset:
shows "⟦¬ affine_dependent T; S ⊆ T⟧ ⟹ ¬ affine_dependent S"
by (metis affine_dependent_subset)

lemma affine_independent_Diff:
"¬ affine_dependent S ⟹ ¬ affine_dependent(S - T)"
by (meson Diff_subset affine_dependent_subset)

proposition affine_dependent_explicit:
"affine_dependent p ⟷
(∃S U. finite S ∧ S ⊆ p ∧ sum U S = 0 ∧ (∃v∈S. U v ≠ 0) ∧ sum (λv. U v *⇩R v) S = 0)"
proof -
have "∃S U. finite S ∧ S ⊆ p ∧ sum U S = 0 ∧ (∃v∈S. U v ≠ 0) ∧ (∑w∈S. U w *⇩R w) = 0"
if "(∑w∈S. U w *⇩R w) = x" "x ∈ p" "finite S" "S ≠ {}" "S ⊆ p - {x}" "sum U S = 1" for x S U
proof (intro exI conjI)
have "x ∉ S"
using that by auto
then show "(∑v ∈ insert x S. if v = x then - 1 else U v) = 0"
using that by (simp add: sum_delta_notmem)
show "(∑w ∈ insert x S. (if w = x then - 1 else U w) *⇩R w) = 0"
using that ‹x ∉ S› by (simp add: if_smult sum_delta_notmem cong: if_cong)
qed (use that in auto)
moreover have "∃x∈p. ∃S U. finite S ∧ S ≠ {} ∧ S ⊆ p - {x} ∧ sum U S = 1 ∧ (∑v∈S. U v *⇩R v) = x"
if "(∑v∈S. U v *⇩R v) = 0" "finite S" "S ⊆ p" "sum U S = 0" "v ∈ S" "U v ≠ 0" for S U v
proof (intro bexI exI conjI)
have "S ≠ {v}"
using that by auto
then show "S - {v} ≠ {}"
using that by auto
show "(∑x ∈ S - {v}. - (1 / U v) * U x) = 1"
unfolding sum_distrib_left[symmetric] sum_diff1[OF ‹finite S›] by (simp add: that)
show "(∑x∈S - {v}. (- (1 / U v) * U x) *⇩R x) = v"
unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric]
scaleR_right.sum [symmetric] sum_diff1[OF ‹finite S›]
using that by auto
show "S - {v} ⊆ p - {v}"
using that by auto
qed (use that in auto)
ultimately show ?thesis
unfolding affine_dependent_def affine_hull_explicit by auto
qed

lemma affine_dependent_explicit_finite:
fixes S :: "'a::real_vector set"
assumes "finite S"
shows "affine_dependent S ⟷
(∃U. sum U S = 0 ∧ (∃v∈S. U v ≠ 0) ∧ sum (λv. U v *⇩R v) S = 0)"
(is "?lhs = ?rhs")
proof
have *: "⋀vt U v. (if vt then U v else 0) *⇩R v = (if vt then (U v) *⇩R v else 0::'a)"
by auto
assume ?lhs
then obtain T U v where
"finite T" "T ⊆ S" "sum U T = 0" "v∈T" "U v ≠ 0"  "(∑v∈T. U v *⇩R v) = 0"
unfolding affine_dependent_explicit by auto
then show ?rhs
apply (rule_tac x="λx. if x∈T then U x else 0" in exI)
apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF ‹T⊆S›])
done
next
assume ?rhs
then obtain U v where "sum U S = 0"  "v∈S" "U v ≠ 0" "(∑v∈S. U v *⇩R v) = 0"
by auto
then show ?lhs unfolding affine_dependent_explicit
using assms by auto
qed

lemma dependent_imp_affine_dependent:
assumes "dependent {x - a| x . x ∈ S}"
and "a ∉ S"
shows "affine_dependent (insert a S)"
proof -
from assms(1)[unfolded dependent_explicit] obtain S' U v
where S: "finite S'" "S' ⊆ {x - a |x. x ∈ S}" "v∈S'" "U v  ≠ 0" "(∑v∈S'. U v *⇩R v) = 0"
by auto
define T where "T = (λx. x + a) ` S'"
have inj: "inj_on (λx. x + a) S'"
unfolding inj_on_def by auto
have "0 ∉ S'"
using S(2) assms(2) unfolding subset_eq by auto
have fin: "finite T" and "T ⊆ S"
unfolding T_def using S(1,2) by auto
then have "finite (insert a T)" and "insert a T ⊆ insert a S"
by auto
moreover have *: "⋀P Q. (∑x∈T. (if x = a then P x else Q x)) = (∑x∈T. Q x)"
by (smt (verit, best) ‹T ⊆ S› assms(2) subsetD sum.cong)
have "(∑x∈insert a T. if x = a then - (∑x∈T. U (x - a)) else U (x - a)) = 0"
by (smt (verit) ‹T ⊆ S› assms(2) fin insert_absorb insert_subset sum.insert sum_mono)
moreover have "∃v∈insert a T. (if v = a then - (∑x∈T. U (x - a)) else U (v - a)) ≠ 0"
using S(3,4) ‹0∉S'›
by (rule_tac x="v + a" in bexI) (auto simp: T_def)
moreover have *: "⋀P Q. (∑x∈T. (if x = a then P x else Q x) *⇩R x) = (∑x∈T. Q x *⇩R x)"
using ‹a∉S› ‹T⊆S› by (auto intro!: sum.cong)
have "(∑x∈T. U (x - a)) *⇩R a = (∑v∈T. U (v - a) *⇩R v)"
unfolding scaleR_left.sum
unfolding T_def and sum.reindex[OF inj] and o_def
using S(5)
by (auto simp: sum.distrib scaleR_right_distrib)
then have "(∑v∈insert a T. (if v = a then - (∑x∈T. U (x - a)) else U (v - a)) *⇩R v) = 0"
unfolding sum_clauses(2)[OF fin] using ‹a∉S› ‹T⊆S› by (auto simp: *)
ultimately show ?thesis
unfolding affine_dependent_explicit
by (force intro!: exI[where x="insert a T"])
qed

lemma affine_dependent_biggerset:
fixes S :: "'a::euclidean_space set"
assumes "finite S" "card S ≥ DIM('a) + 2"
shows "affine_dependent S"
proof -
have "S ≠ {}" using assms by auto
then obtain a where "a∈S" by auto
have *: "{x - a |x. x ∈ S - {a}} = (λx. x - a) ` (S - {a})"
by auto
have "card {x - a |x. x ∈ S - {a}} = card (S - {a})"
unfolding * by (simp add: card_image inj_on_def)
also have "… > DIM('a)" using assms(2)
unfolding card_Diff_singleton[OF ‹a∈S›] by auto
finally  have "affine_dependent (insert a (S - {a}))"
using dependent_biggerset dependent_imp_affine_dependent by blast
then show ?thesis
by (simp add: ‹a ∈ S› insert_absorb)
qed

lemma affine_dependent_biggerset_general:
assumes "finite (S :: 'a::euclidean_space set)"
and "card S ≥ dim S + 2"
shows "affine_dependent S"
proof -
from assms(2) have "S ≠ {}" by auto
then obtain a where "a∈S" by auto
have *: "{x - a |x. x ∈ S - {a}} = (λx. x - a) ` (S - {a})"
by auto
have **: "card {x - a |x. x ∈ S - {a}} = card (S - {a})"
by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def)
have "dim {x - a |x. x ∈ S - {a}} ≤ dim S"
using ‹a∈S› by (auto simp: span_base span_diff intro: subset_le_dim)
also have "… < dim S + 1" by auto
also have "… ≤ card (S - {a})"
using assms card_Diff_singleton[OF ‹a∈S›] by auto
finally have "affine_dependent (insert a (S - {a}))"
by (smt (verit) Collect_cong dependent_imp_affine_dependent dependent_biggerset_general ** Diff_iff insertCI)
then show ?thesis
by (simp add: ‹a ∈ S› insert_absorb)
qed

subsection✐‹tag unimportant› ‹Some Properties of Affine Dependent Sets›

lemma affine_independent_0 [simp]: "¬ affine_dependent {}"
by (simp add: affine_dependent_def)

lemma affine_independent_1 [simp]: "¬ affine_dependent {a}"
by (simp add: affine_dependent_def)

lemma affine_independent_2 [simp]: "¬ affine_dependent {a,b}"
by (simp add: affine_dependent_def insert_Diff_if hull_same)

lemma affine_hull_translation: "affine hull ((λx. a + x) `  S) = (λx. a + x) ` (affine hull S)"
proof -
have "affine ((λx. a + x) ` (affine hull S))"
using affine_translation affine_affine_hull by blast
moreover have "(λx. a + x) `  S ⊆ (λx. a + x) ` (affine hull S)"
using hull_subset[of S] by auto
ultimately have h1: "affine hull ((λx. a + x) `  S) ⊆ (λx. a + x) ` (affine hull S)"
by (metis hull_minimal)
have "affine((λx. -a + x) ` (affine hull ((λx. a + x) `  S)))"
using affine_translation affine_affine_hull by blast
moreover have "(λx. -a + x) ` (λx. a + x) `  S ⊆ (λx. -a + x) ` (affine hull ((λx. a + x) `  S))"
using hull_subset[of "(λx. a + x) `  S"] by auto
moreover have "S = (λx. -a + x) ` (λx. a + x) `  S"
using translation_assoc[of "-a" a] by auto
ultimately have "(λx. -a + x) ` (affine hull ((λx. a + x) `  S)) >= (affine hull S)"
by (metis hull_minimal)
then have "affine hull ((λx. a + x) ` S) >= (λx. a + x) ` (affine hull S)"
by auto
then show ?thesis using h1 by auto
qed

lemma affine_dependent_translation:
assumes "affine_dependent S"
shows "affine_dependent ((λx. a + x) ` S)"
proof -
obtain x where x: "x ∈ S ∧ x ∈ affine hull (S - {x})"
using assms affine_dependent_def by auto
have "(+) a ` (S - {x}) = (+) a ` S - {a + x}"
by auto
then have "a + x ∈ affine hull ((λx. a + x) ` S - {a + x})"
using affine_hull_translation[of a "S - {x}"] x by auto
moreover have "a + x ∈ (λx. a + x) ` S"
using x by auto
ultimately show ?thesis
unfolding affine_dependent_def by auto
qed

lemma affine_dependent_translation_eq:
"affine_dependent S ⟷ affine_dependent ((λx. a + x) ` S)"
by (metis affine_dependent_translation translation_galois)

lemma affine_hull_0_dependent:
assumes "0 ∈ affine hull S"
shows "dependent S"
proof -
obtain s u where s_u: "finite s ∧ s ≠ {} ∧ s ⊆ S ∧ sum u s = 1 ∧ (∑v∈s. u v *⇩R v) = 0"
using assms affine_hull_explicit[of S] by auto
then have "∃v∈s. u v ≠ 0" by auto
then have "finite s ∧ s ⊆ S ∧ (∃v∈s. u v ≠ 0 ∧ (∑v∈s. u v *⇩R v) = 0)"
using s_u by auto
then show ?thesis
unfolding dependent_explicit[of S] by auto
qed

lemma affine_dependent_imp_dependent2:
assumes "affine_dependent (insert 0 S)"
shows "dependent S"
proof -
obtain x where x: "x ∈ insert 0 S ∧ x ∈ affine hull (insert 0 S - {x})"
using affine_dependent_def[of "(insert 0 S)"] assms by blast
then have "x ∈ span (insert 0 S - {x})"
using affine_hull_subset_span by auto
moreover have "span (insert 0 S - {x}) = span (S - {x})"
using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
ultimately have "x ∈ span (S - {x})" by auto
then have "x ≠ 0 ⟹ dependent S"
using x dependent_def by auto
moreover have "dependent S" if "x = 0"
by (metis that affine_hull_0_dependent Diff_insert_absorb dependent_zero x)
ultimately show ?thesis by auto
qed

lemma affine_dependent_iff_dependent:
assumes "a ∉ S"
shows "affine_dependent (insert a S) ⟷ dependent ((λx. -a + x) ` S)"
proof -
have "((+) (- a) ` S) = {x - a| x . x ∈ S}" by auto
then show ?thesis
using affine_dependent_translation_eq[of "(insert a S)" "-a"]
affine_dependent_imp_dependent2 assms
dependent_imp_affine_dependent[of a S]
by (auto simp del: uminus_add_conv_diff)
qed

lemma affine_dependent_iff_dependent2:
assumes "a ∈ S"
shows "affine_dependent S ⟷ dependent ((λx. -a + x) ` (S-{a}))"
by (metis Diff_iff affine_dependent_iff_dependent assms insert_Diff singletonI)

lemma affine_hull_insert_span_gen:
"affine hull (insert a S) = (λx. a + x) ` span ((λx. - a + x) ` S)"
proof -
have h1: "{x - a |x. x ∈ S} = ((λx. -a+x) ` S)"
by auto
{
assume "a ∉ S"
then have ?thesis
using affine_hull_insert_span[of a S] h1 by auto
}
moreover
{
assume a1: "a ∈ S"
then have "insert 0 ((λx. -a+x) ` (S - {a})) = (λx. -a+x) ` S"
by auto
then have "span ((λx. -a+x) ` (S - {a})) = span ((λx. -a+x) ` S)"
using span_insert_0[of "(+) (- a) ` (S - {a})"]
by presburger
moreover have "{x - a |x. x ∈ (S - {a})} = ((λx. -a+x) ` (S - {a}))"
by auto
moreover have "insert a (S - {a}) = insert a S"
by auto
ultimately have ?thesis
using affine_hull_insert_span[of "a" "S-{a}"] by auto
}
ultimately show ?thesis by auto
qed

lemma affine_hull_span2:
assumes "a ∈ S"
shows "affine hull S = (λx. a+x) ` span ((λx. -a+x) ` (S-{a}))"
by (metis affine_hull_insert_span_gen assms insert_Diff)

lemma affine_hull_span_gen:
assumes "a ∈ affine hull S"
shows "affine hull S = (λx. a+x) ` span ((λx. -a+x) ` S)"
by (metis affine_hull_insert_span_gen assms hull_redundant)

lemma affine_hull_span_0:
assumes "0 ∈ affine hull S"
shows "affine hull S = span S"
using affine_hull_span_gen[of "0" S] assms by auto

lemma extend_to_affine_basis_nonempty:
fixes S V :: "'n::real_vector set"
assumes "¬ affine_dependent S" "S ⊆ V" "S ≠ {}"
shows "∃T. ¬ affine_dependent T ∧ S ⊆ T ∧ T ⊆ V ∧ affine hull T = affine hull V"
proof -
obtain a where a: "a ∈ S"
using assms by auto
then have h0: "independent  ((λx. -a + x) ` (S-{a}))"
using affine_dependent_iff_dependent2 assms by auto
obtain B where B:
"(λx. -a+x) ` (S - {a}) ⊆ B ∧ B ⊆ (λx. -a+x) ` V ∧ independent B ∧ (λx. -a+x) ` V ⊆ span B"
using assms
by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(λx. -a + x) ` V"])
define T where "T = (λx. a+x) ` insert 0 B"
then have "T = insert a ((λx. a+x) ` B)"
by auto
then have "affine hull T = (λx. a+x) ` span B"
using affine_hull_insert_span_gen[of a "((λx. a+x) ` B)"] translation_assoc[of "-a" a B]
by auto
then have "V ⊆ affine hull T"
using B assms translation_inverse_subset[of a V "span B"]
by auto
moreover have "T ⊆ V"
using T_def B a assms by auto
ultimately have "affine hull T = affine hull V"
by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono)
moreover have "S ⊆ T"
using T_def B translation_inverse_subset[of a "S-{a}" B]
by auto
moreover have "¬ affine_dependent T"
using T_def affine_dependent_translation_eq[of "insert 0 B"]
affine_dependent_imp_dependent2 B
by auto
ultimately show ?thesis using ‹T ⊆ V› by auto
qed

lemma affine_basis_exists:
fixes V :: "'n::real_vector set"
shows "∃B. B ⊆ V ∧ ¬ affine_dependent B ∧ affine hull V = affine hull B"
by (metis affine_dependent_def affine_independent_1 empty_subsetI extend_to_affine_basis_nonempty insert_subset order_refl)

proposition extend_to_affine_basis:
fixes S V :: "'n::real_vector set"
assumes "¬ affine_dependent S" "S ⊆ V"
obtains T where "¬ affine_dependent T" "S ⊆ T" "T ⊆ V" "affine hull T = affine hull V"
by (metis affine_basis_exists assms(1) assms(2) bot.extremum extend_to_affine_basis_nonempty)

subsection ‹Affine Dimension of a Set›

definition✐‹tag important› aff_dim :: "('a::euclidean_space) set ⇒ int"
where "aff_dim V =
(SOME d :: int.
∃B. affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ of_nat (card B) = d + 1)"

lemma aff_dim_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "∃B. affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ of_nat (card B) = aff_dim V + 1"
proof -
obtain B where "¬ affine_dependent B ∧ affine hull B = affine hull V"
using affine_basis_exists[of V] by auto
then show ?thesis
unfolding aff_dim_def
some_eq_ex[of "λd. ∃B. affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ of_nat (card B) = d + 1"]
apply auto
apply (rule exI[of _ "int (card B) - (1 :: int)"])
apply (rule exI[of _ "B"], auto)
done
qed

lemma affine_hull_eq_empty [simp]: "affine hull S = {} ⟷ S = {}"
by (metis affine_empty subset_empty subset_hull)

lemma empty_eq_affine_hull[simp]: "{} = affine hull S ⟷ S = {}"
by (metis affine_hull_eq_empty)

lemma aff_dim_parallel_subspace_aux:
fixes B :: "'n::euclidean_space set"
assumes "¬ affine_dependent B" "a ∈ B"
shows "finite B ∧ ((card B) - 1 = dim (span ((λx. -a+x) ` (B-{a}))))"
proof -
have "independent ((λx. -a + x) ` (B-{a}))"
using affine_dependent_iff_dependent2 assms by auto
then have fin: "dim (span ((λx. -a+x) ` (B-{a}))) = card ((λx. -a + x) ` (B-{a}))"
"finite ((λx. -a + x) ` (B - {a}))"
using indep_card_eq_dim_span[of "(λx. -a+x) ` (B-{a})"] by auto
show ?thesis
proof (cases "(λx. -a + x) ` (B - {a}) = {}")
case True
have "B = insert a ((λx. a + x) ` (λx. -a + x) ` (B - {a}))"
using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
then have "B = {a}" using True by auto
then show ?thesis using assms fin by auto
next
case False
then have "card ((λx. -a + x) ` (B - {a})) > 0"
using fin by auto
moreover have h1: "card ((λx. -a + x) ` (B-{a})) = card (B-{a})"
by (rule card_image) (use translate_inj_on in blast)
ultimately have "card (B-{a}) > 0" by auto
then have *: "finite (B - {a})"
using card_gt_0_iff[of "(B - {a})"] by auto
then have "card (B - {a}) = card B - 1"
using card_Diff_singleton assms by auto
with * show ?thesis using fin h1 by auto
qed
qed

lemma aff_dim_parallel_subspace:
fixes V L :: "'n::euclidean_space set"
assumes "V ≠ {}"
and "subspace L"
and "affine_parallel (affine hull V) L"
shows "aff_dim V = int (dim L)"
proof -
obtain B where
B: "affine hull B = affine hull V ∧ ¬ affine_dependent B ∧ int (card B) = aff_dim V + 1"
using aff_dim_basis_exists by auto
then have "B ≠ {}"
using assms B
by auto
then obtain a where a: "a ∈ B" by auto
define Lb where "Lb = span ((λx. -a+x) ` (B-{a}))"
moreover have "affine_parallel (affine hull B) Lb"
using Lb_def B assms affine_hull_span2[of a B] a
affine_parallel_commute[of "Lb" "(affine hull B)"]
unfolding affine_parallel_def
by auto
moreover have "subspace Lb"
using Lb_def subspace_span by auto
moreover have "affine hull B ≠ {}"
using assms B by auto
ultimately have "L = Lb"
using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B
by auto
then have "dim L = dim Lb"
by auto
moreover have "card B - 1 = dim Lb" and "finite B"
using Lb_def aff_dim_parallel_subspace_aux a B by auto
ultimately show ?thesis
using B ‹B ≠ {}› card_gt_0_iff[of B] by auto
qed

lemma aff_independent_finite:
fixes B :: "'n::euclidean_space set"
assumes "¬ affine_dependent B"
shows "finite B"
using aff_dim_parallel_subspace_aux assms finite.simps by fastforce

lemma aff_dim_empty:
fixes S :: "'n::euclidean_space set"
shows "S = {} ⟷ aff_dim S = -1"
proof -
obtain B where *: "affine hull B = affine hull S"
and "¬ affine_dependent B"
and "int (card B) = aff_dim S + 1"
using aff_dim_basis_exists by auto
moreover
from * have "S = {} ⟷ B = {}"
by auto
ultimately show ?thesis
using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
qed

lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1"
using aff_dim_empty by blast

lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S"
unfolding aff_dim_def using hull_hull[of _ S] by auto

lemma aff_dim_affine_hull2:
assumes "affine hull S = affine hull T"
shows "aff_dim S = aff_dim T"
unfolding aff_dim_def using assms by auto

lemma aff_dim_unique:
fixes B V :: "'n::euclidean_space set"
assumes "affine hull B = affine hull V ∧ ¬ affine_dependent B"
shows "of_nat (card B) = aff_dim V + 1"
proof (cases "B = {}")
case True
then have "V = {}"
using assms
by auto
then have "aff_dim V = (-1::int)"
using aff_dim_empty by auto
then show ?thesis
using ‹B = {}› by auto
next
case False
then obtain a where a: "a ∈ B" by auto
define Lb where "Lb = span ((λx. -a+x) ` (B-{a}))"
have "affine_parallel (affine hull B) Lb"
using Lb_def affine_hull_span2[of a B] a
affine_parallel_commute[of "Lb" "(affine hull B)"]
unfolding affine_parallel_def by auto
moreover have "subspace Lb"
using Lb_def subspace_span by auto
ultimately have "aff_dim B = int(dim Lb)"
using aff_dim_parallel_subspace[of B Lb] ‹B ≠ {}› by auto
moreover have "(card B) - 1 = dim Lb" "finite B"
using Lb_def aff_dim_parallel_subspace_aux a assms by auto
ultimately have "of_nat (card B) = aff_dim B + 1"
using ‹B ≠ {}› card_gt_0_iff[of B] by auto
then show ?thesis
using aff_dim_affine_hull2 assms by auto
qed

lemma aff_dim_affine_independent:
fixes B :: "'n::euclidean_space set"
assumes "¬ affine_dependent B"
shows "of_nat (card B) = aff_dim B + 1"
using aff_dim_unique[of B B] assms by auto

lemma affine_independent_iff_card:
fixes S :: "'a::euclidean_space set"
shows "¬ affine_dependent S ⟷ finite S ∧ aff_dim S = int(card S) - 1" (is "?lhs = ?rhs")
proof
show "?lhs ⟹ ?rhs"
by (simp add: aff_dim_affine_independent aff_independent_finite)
show "?rhs ⟹ ?lhs"
by (metis of_nat_eq_iff affine_basis_exists aff_dim_unique card_subset_eq diff_add_cancel)
qed

lemma aff_dim_sing [simp]:
fixes a :: "'n::euclidean_space"
shows "aff_dim {a} = 0"
using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto

lemma aff_dim_2 [simp]:
fixes a :: "'n::euclidean_space"
shows "aff_dim {a,b} = (if a = b then 0 else 1)"
proof (clarsimp)
assume "a ≠ b"
then have "aff_dim{a,b} = card{a,b} - 1"
using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce
also have "… = 1"
using ‹a ≠ b› by simp
finally show "aff_dim {a, b} = 1" .
qed

lemma aff_dim_inner_basis_exists:
fixes V :: "('n::euclidean_space) set"
shows "∃B. B ⊆ V ∧ affine hull B = affine hull V ∧
¬ affine_dependent B ∧ of_nat (card B) = aff_dim V + 1"
by (metis aff_dim_unique affine_basis_exists)

lemma aff_dim_le_card:
fixes V :: "'n::euclidean_space set"
assumes "finite V"
shows "aff_dim V ≤ of_nat (card V) - 1"
by (metis aff_dim_inner_basis_exists assms card_mono le_diff_eq of_nat_le_iff)

lemma aff_dim_parallel_le:
fixes S T :: "'n::euclidean_space set"
assumes "affine_parallel (affine hull S) (affine hull T)"
shows "aff_dim S ≤ aff_dim T"
proof (cases "S={} ∨ T={}")
case True
then show ?thesis
by (smt (verit, best) aff_dim_affine_hull2 affine_hull_empty affine_parallel_def assms empty_is_image)
next
case False
then obtain L where L: "subspace L" "affine_parallel (affine hull T) L"
by (metis affine_affine_hull affine_hull_eq_empty affine_parallel_subspace)
with False show ?thesis
by (metis aff_dim_parallel_subspace affine_parallel_assoc assms dual_order.refl)
qed

lemma aff_dim_parallel_eq:
fixes S T :: "'n::euclidean_space set"
assumes "affine_parallel (affine hull S) (affine hull T)"
shows "aff_dim S = aff_dim T"
by (smt (verit, del_insts) aff_dim_parallel_le affine_parallel_commute assms)

lemma aff_dim_translation_eq:
"aff_dim ((+) a ` S) = aff_dim S" for a :: "'n::euclidean_space"
by (metis aff_dim_parallel_eq affine_hull_translation affine_parallel_def)

lemma aff_dim_translation_eq_subtract:
"aff_dim ((λx. x - a) ` S) = aff_dim S" for a :: "'n::euclidean_space"
using aff_dim_translation_eq [of "- a"] by (simp cong: image_cong_simp)

lemma aff_dim_affine:
fixes S L :: "'n::euclidean_space set"
assumes "affine S" "subspace L" "affine_parallel S L" "S ≠ {}"
shows "aff_dim S = int (dim L)"
by (simp add: aff_dim_parallel_subspace assms hull_same)

lemma dim_affine_hull [simp]:
fixes S :: "'n::euclidean_space set"
shows "dim (affine hull S) = dim S"
by (metis affine_hull_subset_span dim_eq_span dim_mono hull_subset span_eq_dim)

lemma aff_dim_subspace:
fixes S :: "'n::euclidean_space set"
assumes "subspace S"
shows "aff_dim S = int (dim S)"
by (metis aff_dim_affine affine_parallel_subspace assms empty_iff parallel_subspace subspace_affine)

lemma aff_dim_zero:
fixes S :: "'n::euclidean_space set"
assumes "0 ∈ affine hull S"
shows "aff_dim S = int (dim S)"
by (metis aff_dim_affine_hull aff_dim_subspace affine_hull_span_0 assms dim_span subspace_span)

lemma aff_dim_eq_dim:
fixes S :: "'n::euclidean_space set"
assumes "a ∈ affine hull S"
shows "aff_dim S = int (dim ((+) (- a) ` S))"
by (metis ab_left_minus aff_dim_translation_eq aff_dim_zero affine_hull_translation image_eqI assms)

lemma aff_dim_eq_dim_subtract:
fixes S :: "'n::euclidean_space set"
assumes "a ∈ affine hull S"
shows "aff_dim S = int (dim ((λx. x - a) ` S))"
using aff_dim_eq_dim assms by auto

lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))"
by (simp add: aff_dim_subspace)

lemma aff_dim_geq:
fixes V :: "'n::euclidean_space set"
shows "aff_dim V ≥ -1"
by (metis add_le_cancel_right aff_dim_basis_exists diff_self of_nat_0_le_iff uminus_add_conv_diff)

lemma aff_dim_negative_iff [simp]:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S < 0 ⟷ S = {}"
by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq)

lemma aff_lowdim_subset_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes "aff_dim S < DIM('a)"
obtains a b where "a ≠ 0" "S ⊆ {x. a ∙ x = b}"
proof (cases "S={}")
case True
moreover
have "(SOME b. b ∈ Basis) ≠ 0"
by (metis norm_some_Basis norm_zero zero_neq_one)
ultimately show ?thesis
using that by blast
next
case False
then obtain c S' where "c ∉ S'" "S = insert c S'"
by (meson equals0I mk_disjoint_insert)
have "dim ((+) (-c) ` S) < DIM('a)"
by (metis ‹S = insert c S'› aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less)
then obtain a where "a ≠ 0" "span ((+) (-c) ` S) ⊆ {x. a ∙ x = 0}"
using lowdim_subset_hyperplane by blast
moreover
have "a ∙ w = a ∙ c" if "span ((+) (- c) ` S) ⊆ {x. a ∙ x = 0}" "w ∈ S" for w
proof -
have "w-c ∈ span ((+) (- c) ` S)"
by (simp add: span_base ‹w ∈ S›)
with that have "w-c ∈ {x. a ∙ x = 0}"
by blast
then show ?thesis
by (auto simp: algebra_simps)
qed
ultimately have "S ⊆ {x. a ∙ x = a ∙ c}"
by blast
then show ?thesis
by (rule that[OF ‹a ≠ 0›])
qed

lemma affine_independent_card_dim_diffs:
fixes S :: "'a :: euclidean_space set"
assumes "¬ affine_dependent S" "a ∈ S"
shows "card S = dim ((λx. x - a) ` S) + 1"
using aff_dim_affine_independent aff_dim_eq_dim_subtract assms hull_subset by fastforce

lemma independent_card_le_aff_dim:
fixes B :: "'n::euclidean_space set"
assumes "B ⊆ V"
assumes "¬ affine_dependent B"
shows "int (card B) ≤ aff_dim V + 1"
by (metis aff_dim_unique aff_independent_finite assms card_mono extend_to_affine_basis of_nat_mono)

lemma aff_dim_subset:
fixes S T :: "'n::euclidean_space set"
assumes "S ⊆ T"
shows "aff_dim S ≤ aff_dim T"
by (metis add_le_cancel_right aff_dim_inner_basis_exists assms dual_order.trans independent_card_le_aff_dim)

lemma aff_dim_le_DIM:
fixes S :: "'n::euclidean_space set"
shows "aff_dim S ≤ int (DIM('n))"
by (metis aff_dim_UNIV aff_dim_subset top_greatest)

lemma affine_dim_equal:
fixes S :: "'n::euclidean_space set"
assumes "affine S" "affine T" "S ≠ {}" "S ⊆ T" "aff_dim S = aff_dim T"
shows "S = T"
proof -
obtain a where "a ∈ S" "a ∈ T" "T ≠ {}" using assms by auto
define LS where "LS = {y. ∃x ∈ S. (-a) + x = y}"
then have ls: "subspace LS" "affine_parallel S LS"
using assms parallel_subspace_explicit[of S a LS] ‹a ∈ S› by auto
then have h1: "int(dim LS) = aff_dim S"
using assms aff_dim_affine[of S LS] by auto
define LT where "LT = {y. ∃x ∈ T. (-a) + x = y}"
then have lt: "subspace LT ∧ affine_parallel T LT"
using assms parallel_subspace_explicit[of T a LT] ‹a ∈ T› by auto
then have "dim LS = dim LT"
using assms aff_dim_affine[of T LT] ‹T ≠ {}›  h1 by auto
moreover have "LS ≤ LT"
using LS_def LT_def assms by auto
ultimately have "LS = LT"
using subspace_dim_equal[of LS LT] ls lt by auto
moreover have "S = {x. ∃y ∈ LS. a+y=x}" "T = {x. ∃y ∈ LT. a+y=x}"
using LS_def LT_def by auto
ultimately show ?thesis by auto
qed

lemma aff_dim_eq_0:
fixes S :: "'a::euclidean_space set"
shows "aff_dim S = 0 ⟷ (∃a. S = {a})"
proof (cases "S = {}")
case False
then obtain a where "a ∈ S" by auto
show ?thesis
proof safe
assume 0: "aff_dim S = 0"
have "¬ {a,b} ⊆ S" if "b ≠ a" for b
by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that)
then show "∃a. S = {a}"
using ‹a ∈ S› by blast
qed auto
qed auto

lemma affine_hull_UNIV:
fixes S :: "'n::euclidean_space set"
assumes "aff_dim S = int(DIM('n))"
shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
by (simp add: aff_dim_empty affine_dim_equal assms)

lemma disjoint_affine_hull:
fixes S :: "'n::euclidean_space set"
assumes "¬ affine_dependent S" "T ⊆ S" "U ⊆ S" "T ∩ U = {}"
shows "(affine hull T) ∩ (affine hull U) = {}"
proof -
obtain "finite S" "finite T" "finite U"
using assms by (simp add: aff_independent_finite finite_subset)
have False if "y ∈ affine hull T" and "y ∈ affine hull U" for y
proof -
from that obtain a b
where a1 [simp]: "sum a T = 1"
and [simp]: "sum (λv. a v *⇩R v) T = y"
and [simp]: "sum b U = 1" "sum (λv. b v *⇩R v) U = y"
by (auto simp: affine_hull_finite ‹finite T› ‹finite U›)
define c where "c x = (if x ∈ T then a x else if x ∈ U then -(b x) else 0)" for x
have [simp]: "S ∩ T = T" "S ∩ - T ∩ U = U"
using assms by auto
have "sum c S = 0"
by (simp add: c_def comm_monoid_add_class.sum.If_cases ‹finite S› sum_negf)
moreover have "¬ (∀v∈S. c v = 0)"
by (metis (no_types) IntD1 ‹S ∩ T = T› a1 c_def sum.neutral zero_neq_one)
moreover have "(∑v∈S. c v *⇩R v) = 0"
by (simp add: c_def if_smult sum_negf comm_monoid_add_class.sum.If_cases ‹finite S›)
ultimately show ?thesis
using assms(1) ‹finite S› by (auto simp: affine_dependent_explicit)
qed
then show ?thesis by blast
qed

end```