Theory Affine_Form
section ‹Affine Form›
theory Affine_Form
imports
"HOL-Analysis.Multivariate_Analysis"
"HOL-Combinatorics.List_Permutation"
Affine_Arithmetic_Auxiliarities
Executable_Euclidean_Space
begin
subsection ‹Auxiliary developments›
lemma sum_list_mono:
fixes xs ys::"'a::ordered_ab_group_add list"
shows
"length xs = length ys ⟹ (⋀x y. (x, y) ∈ set (zip xs ys) ⟹ x ≤ y) ⟹
sum_list xs ≤ sum_list ys"
by (induct xs ys rule: list_induct2) (auto simp: algebra_simps intro: add_mono)
lemma
fixes xs::"'a::ordered_comm_monoid_add list"
shows sum_list_nonneg: "(⋀x. x ∈ set xs ⟹ x ≥ 0) ⟹ sum_list xs ≥ 0"
by (induct xs) (auto intro!: add_nonneg_nonneg)
lemma map_filter:
"map f (filter (λx. P (f x)) xs) = filter P (map f xs)"
by (induct xs) simp_all
lemma
map_of_zip_upto2_length_eq_nth:
assumes "distinct B"
assumes "i < length B"
shows "(map_of (zip B [0..<length B]) (B ! i)) = Some i"
proof -
have "length [0..<length B] = length B"
by simp
from map_of_zip_is_Some[OF this, of i] assms
have "map_of (zip B [0..<length B]) (B ! i) = Some i"
using assms by (auto simp: in_set_zip)
thus ?thesis by simp
qed
lemma distinct_map_fst_snd_eqD:
"distinct (map fst xs) ⟹ (i, a) ∈ set xs ⟹ (i, b) ∈ set xs ⟹ a = b"
by (metis (lifting) map_of_is_SomeI option.inject)
lemma length_filter_snd_zip:
"length ys = length xs ⟹ length (filter (p ∘ snd) (zip ys xs)) = length (filter p xs)"
by (induct ys xs rule: list_induct2) (auto )
lemma filter_snd_nth:
"length ys = length xs ⟹ n < length (filter p xs) ⟹
snd (filter (p ∘ snd) (zip ys xs) ! n) = filter p xs ! n"
by (induct ys xs arbitrary: n rule: list_induct2) (auto simp: o_def nth_Cons split: nat.split)
lemma distinct_map_snd_fst_eqD:
"distinct (map snd xs) ⟹ (i, a) ∈ set xs ⟹ (j, a) ∈ set xs ⟹ i = j"
by (metis Pair_inject inj_on_contraD snd_conv distinct_map)
lemma map_of_mapk_inj_on_SomeI:
"inj_on f (fst ` (set t)) ⟹ map_of t k = Some x ⟹
map_of (map (case_prod (λk. Pair (f k))) t) (f k) = Some x"
by (induct t) (auto simp add: inj_on_def dest!: map_of_SomeD split: if_split_asm)
lemma map_abs_nonneg[simp]:
fixes xs::"'a::ordered_ab_group_add_abs list"
shows "list_all (λx. x ≥ 0) xs ⟹ map abs xs = xs"
by (induct xs) auto
lemma the_inv_into_image_eq: "inj_on f A ⟹ Y ⊆ f ` A ⟹ the_inv_into A f ` Y = f -` Y ∩ A"
using f_the_inv_into_f the_inv_into_f_f[where f = f and A = A]
by force
lemma image_fst_zip: "length ys = length xs ⟹ fst ` set (zip ys xs) = set ys"
by (metis dom_map_of_conv_image_fst dom_map_of_zip)
lemma inj_on_fst_set_zip_distinct[simp]:
"distinct xs ⟹ length xs = length ys ⟹ inj_on fst (set (zip xs ys))"
by (force simp add: in_set_zip distinct_conv_nth intro!: inj_onI)
lemma mem_greaterThanLessThan_absI:
fixes x::real
assumes "abs x < 1"
shows "x ∈ {-1 <..< 1}"
using assms by (auto simp: abs_real_def split: if_split_asm)
lemma minus_one_less_divideI: "b > 0 ⟹ -b < a ⟹ -1 < a / (b::real)"
by (auto simp: field_simps)
lemma divide_less_oneI: "b > 0 ⟹ b > a ⟹ a / (b::real) < 1"
by (auto simp: field_simps)
lemma closed_segment_real:
fixes a b::real
shows "closed_segment a b = (if a ≤ b then {a .. b} else {b .. a})" (is "_ = ?if")
proof safe
fix x assume "x ∈ closed_segment a b"
from segment_bound[OF this]
show "x ∈ ?if" by (auto simp: abs_real_def split: if_split_asm)
next
fix x
assume "x ∈ ?if"
thus "x ∈ closed_segment a b"
by (auto simp: closed_segment_def intro!: exI[where x="(x - a)/(b - a)"]
simp: divide_simps algebra_simps)
qed
subsection ‹Partial Deviations›
typedef (overloaded) 'a pdevs = "{x::nat ⇒ 'a::zero. finite {i. x i ≠ 0}}"
morphisms pdevs_apply Abs_pdev
by (auto intro!: exI[where x="λx. 0"])
setup_lifting type_definition_pdevs
lemma pdevs_eqI: "(⋀i. pdevs_apply x i = pdevs_apply y i) ⟹ x = y"
by transfer auto
definition pdevs_val :: "(nat ⇒ real) ⇒ 'a::real_normed_vector pdevs ⇒ 'a"
where "pdevs_val e x = (∑i. e i *⇩R pdevs_apply x i)"
definition valuate:: "((nat ⇒ real) ⇒ 'a) ⇒ 'a set"
where "valuate x = x ` (UNIV → {-1 .. 1})"
lemma valuate_ex: "x ∈ valuate f ⟷ (∃e. (∀i. e i ∈ {-1 .. 1}) ∧ x = f e)"
unfolding valuate_def
by (auto simp add: valuate_def Pi_iff) blast
instantiation pdevs :: (equal) equal
begin
definition equal_pdevs::"'a pdevs ⇒ 'a pdevs ⇒ bool"
where "equal_pdevs a b ⟷ a = b"
instance proof qed (simp add: equal_pdevs_def)
end
subsection ‹Affine Forms›
text ‹The data structure of affine forms represents particular sets, zonotopes›
type_synonym 'a aform = "'a × 'a pdevs"
subsection ‹Evaluation, Range, Joint Range›
definition aform_val :: "(nat ⇒ real) ⇒ 'a::real_normed_vector aform ⇒ 'a"
where "aform_val e X = fst X + pdevs_val e (snd X)"
definition Affine ::
"'a::real_normed_vector aform ⇒ 'a set"
where "Affine X = valuate (λe. aform_val e X)"
definition Joints ::
"'a::real_normed_vector aform list ⇒ 'a list set"
where "Joints XS = valuate (λe. map (aform_val e) XS)"
lemma Joints_nthE:
assumes "zs ∈ Joints ZS"
obtains e where
"⋀i. i < length zs ⟹ zs ! i = aform_val e (ZS ! i)"
"⋀i. e i ∈ {-1..1}"
using assms
by atomize_elim (auto simp: Joints_def Pi_iff valuate_ex)
lemma Joints_mapE:
assumes "ys ∈ Joints YS"
obtains e where
"ys = map (λx. aform_val e x) YS"
"⋀i. e i ∈ {-1 .. 1}"
using assms
by (force simp: Joints_def valuate_def)
lemma
zipped_subset_mapped_Elem:
assumes "xs = map (aform_val e) XS"
assumes e: "⋀i. e i ∈ {-1 .. 1}"
assumes [simp]: "length xs = length XS"
assumes [simp]: "length ys = length YS"
assumes "set (zip ys YS) ⊆ set (zip xs XS)"
shows "ys = map (aform_val e) YS"
proof -
from assms have ys: "⋀i. i < length xs ⟹ xs ! i = aform_val e (XS ! i)"
by auto
from assms have set_eq: "{(ys ! i, YS ! i) |i. i < length ys ∧ i < length YS} ⊆
{(xs ! i, XS ! i) |i. i < length xs ∧ i < length XS}"
using assms(2)
by (auto simp: set_zip)
hence "∀i<length YS. ∃j<length XS. ys ! i = xs ! j ∧ YS ! i = XS ! j"
by auto
then obtain j where j: "⋀i. i < length YS ⟹ ys ! i = xs ! (j i)"
"⋀i. i < length YS ⟹ YS ! i = XS ! (j i)"
"⋀i. i < length YS ⟹ j i < length XS"
by metis
show ?thesis
using assms
by (auto simp: Joints_def j ys intro!: exI[where x=e] nth_equalityI)
qed
lemma Joints_set_zip_subset:
assumes "xs ∈ Joints XS"
assumes "length xs = length XS"
assumes "length ys = length YS"
assumes "set (zip ys YS) ⊆ set (zip xs XS)"
shows "ys ∈ Joints YS"
proof -
from Joints_mapE assms obtain e where
ys: "xs = map (λx. aform_val e x) XS"
and e: "⋀i. e i ∈ {-1 .. 1}"
by blast
show "ys ∈ Joints YS"
using e zipped_subset_mapped_Elem[OF ys e assms(2-4)]
by (auto simp: Joints_def valuate_def intro!: exI[where x=e])
qed
lemma Joints_set_zip:
assumes "ys ∈ Joints YS"
assumes "length xs = length XS"
assumes "length YS = length XS"
assumes sets_eq: "set (zip xs XS) = set (zip ys YS)"
shows "xs ∈ Joints XS"
proof -
from assms have "length ys = length YS"
by (auto simp: Joints_def valuate_def)
from assms(1) this assms(2) show ?thesis
by (rule Joints_set_zip_subset) (simp add: assms)
qed
definition Joints2 ::
"'a::real_normed_vector aform list ⇒'b::real_normed_vector aform ⇒ ('a list × 'b) set"
where "Joints2 XS Y = valuate (λe. (map (aform_val e) XS, aform_val e Y))"
lemma Joints2E:
assumes "zs_y ∈ Joints2 ZS Y"
obtains e where
"⋀i. i < length (fst zs_y) ⟹ (fst zs_y) ! i = aform_val e (ZS ! i)"
"snd (zs_y) = aform_val e Y"
"⋀i. e i ∈ {-1..1}"
using assms
by atomize_elim (auto simp: Joints2_def Pi_iff valuate_ex)
lemma nth_in_AffineI:
assumes "xs ∈ Joints XS"
assumes "i < length XS"
shows "xs ! i ∈ Affine (XS ! i)"
using assms by (force simp: Affine_def Joints_def valuate_def)
lemma Cons_nth_in_Joints1:
assumes "xs ∈ Joints XS"
assumes "i < length XS"
shows "((xs ! i) # xs) ∈ Joints ((XS ! i) # XS)"
using assms by (force simp: Joints_def valuate_def)
lemma Cons_nth_in_Joints2:
assumes "xs ∈ Joints XS"
assumes "i < length XS"
assumes "j < length XS"
shows "((xs ! i) #(xs ! j) # xs) ∈ Joints ((XS ! i)#(XS ! j) # XS)"
using assms by (force simp: Joints_def valuate_def)
lemma Joints_swap:
"x#y#xs∈Joints (X#Y#XS) ⟷ y#x#xs ∈ Joints (Y#X#XS)"
by (force simp: Joints_def valuate_def)
lemma Joints_swap_Cons_append:
"length xs = length XS ⟹ x#ys@xs∈Joints (X#YS@XS) ⟷ ys@x#xs ∈ Joints (YS@X#XS)"
by (auto simp: Joints_def valuate_def)
lemma Joints_ConsD:
"x#xs∈Joints (X#XS) ⟹ xs ∈ Joints XS"
by (force simp: Joints_def valuate_def)
lemma Joints_appendD1:
"ys@xs∈Joints (YS@XS) ⟹ length xs = length XS ⟹ xs ∈ Joints XS"
by (force simp: Joints_def valuate_def)
lemma Joints_appendD2:
"ys@xs∈Joints (YS@XS) ⟹ length ys = length YS ⟹ ys ∈ Joints YS"
by (force simp: Joints_def valuate_def)
lemma Joints_imp_length_eq: "xs ∈ Joints XS ⟹ length xs = length XS"
by (auto simp: Joints_def valuate_def)
lemma Joints_rotate[simp]: "xs@[x] ∈ Joints (XS @[X]) ⟷ x#xs ∈ Joints (X#XS)"
by (auto simp: Joints_def valuate_def)
subsection ‹Domain›
definition "pdevs_domain x = {i. pdevs_apply x i ≠ 0}"
lemma finite_pdevs_domain[intro, simp]: "finite (pdevs_domain x)"
unfolding pdevs_domain_def by transfer
lemma in_pdevs_domain[simp]: "i ∈ pdevs_domain x ⟷ pdevs_apply x i ≠ 0"
by (auto simp: pdevs_domain_def)
subsection ‹Least Fresh Index›
definition degree::"'a::real_vector pdevs ⇒ nat"
where "degree x = (LEAST i. ∀j≥i. pdevs_apply x j = 0)"
lemma degree[rule_format, intro, simp]:
shows "∀j≥degree x. pdevs_apply x j = 0"
unfolding degree_def
proof (rule LeastI_ex)
have "⋀j. j > Max (pdevs_domain x) ⟹ j ∉ (pdevs_domain x)"
by (metis Max_less_iff all_not_in_conv less_irrefl_nat finite_pdevs_domain)
then show "∃xa. ∀j≥xa. pdevs_apply x j = 0"
by (auto intro!: exI[where x="Max (pdevs_domain x) + 1"])
qed
lemma degree_le:
assumes d: "∀j ≥ d. pdevs_apply x j = 0"
shows "degree x ≤ d"
unfolding degree_def
by (rule Least_le) (rule d)
lemma degree_gt: "pdevs_apply x j ≠ 0 ⟹ degree x > j"
by auto
lemma pdevs_val_pdevs_domain: "pdevs_val e X = (∑i∈pdevs_domain X. e i *⇩R pdevs_apply X i)"
by (auto simp: pdevs_val_def intro!: suminf_finite)
lemma pdevs_val_sum_le: "degree X ≤ d ⟹ pdevs_val e X = (∑i < d. e i *⇩R pdevs_apply X i)"
by (force intro!: degree_gt sum.mono_neutral_cong_left simp: pdevs_val_pdevs_domain)
lemmas pdevs_val_sum = pdevs_val_sum_le[OF order_refl]
lemma pdevs_val_zero[simp]: "pdevs_val (λ_. 0) x = 0"
by (auto simp: pdevs_val_sum)
lemma degree_eqI:
assumes "pdevs_apply x d ≠ 0"
assumes "⋀j. j > d ⟹ pdevs_apply x j = 0"
shows "degree x = Suc d"
unfolding eq_iff
by (auto intro!: degree_gt degree_le assms simp: Suc_le_eq)
lemma finite_degree_nonzero[intro, simp]: "finite {i. pdevs_apply x i ≠ 0}"
by transfer (auto simp: vimage_def Collect_neg_eq)
lemma degree_eq_Suc_max:
"degree x = (if (∀i. pdevs_apply x i = 0) then 0 else Suc (Max {i. pdevs_apply x i ≠ 0}))"
proof -
{
assume "⋀i. pdevs_apply x i = 0"
hence ?thesis
by auto (metis degree_le le_0_eq)
} moreover {
fix i assume "pdevs_apply x i ≠ 0"
hence ?thesis
using Max_in[OF finite_degree_nonzero, of x]
by (auto intro!: degree_eqI) (metis Max.coboundedI[OF finite_degree_nonzero] in_pdevs_domain
le_eq_less_or_eq less_asym pdevs_domain_def)
} ultimately show ?thesis
by blast
qed
lemma pdevs_val_degree_cong:
assumes "b = d"
assumes "⋀i. i < degree b ⟹ a i = c i"
shows "pdevs_val a b = pdevs_val c d"
using assms
by (auto simp: pdevs_val_sum)
abbreviation degree_aform::"'a::real_vector aform ⇒ nat"
where "degree_aform X ≡ degree (snd X)"
lemma degree_cong: "(⋀i. (pdevs_apply x i = 0) = (pdevs_apply y i = 0)) ⟹ degree x = degree y"
unfolding degree_def
by auto
lemma Least_True_nat[intro, simp]: "(LEAST i::nat. True) = 0"
by (metis (lifting) One_nat_def less_one not_less_Least not_less_eq)
lemma sorted_list_of_pdevs_domain_eq:
"sorted_list_of_set (pdevs_domain X) = filter ((≠) 0 o pdevs_apply X) [0..<degree X]"
by (auto simp: degree_gt intro!: sorted_distinct_set_unique sorted_filter[of "λx. x", simplified])
subsection ‹Total Deviation›
definition tdev::"'a::ordered_euclidean_space pdevs ⇒ 'a" where
"tdev x = (∑i<degree x. ¦pdevs_apply x i¦)"
lemma abs_pdevs_val_le_tdev: "e ∈ UNIV → {-1 .. 1} ⟹ ¦pdevs_val e x¦ ≤ tdev x"
by (force simp: pdevs_val_sum tdev_def abs_scaleR Pi_iff
intro!: order_trans[OF sum_abs] sum_mono scaleR_left_le_one_le
intro: abs_leI)
subsection ‹Binary Pointwise Operations›
definition binop_pdevs_raw::"('a::zero ⇒ 'b::zero ⇒ 'c::zero) ⇒
(nat ⇒ 'a) ⇒ (nat ⇒ 'b) ⇒ nat ⇒ 'c"
where "binop_pdevs_raw f x y i = (if x i = 0 ∧ y i = 0 then 0 else f (x i) (y i))"
lemma nonzeros_binop_pdevs_subset:
"{i. binop_pdevs_raw f x y i ≠ 0} ⊆ {i. x i ≠ 0} ∪ {i. y i ≠ 0}"
by (auto simp: binop_pdevs_raw_def)
lift_definition binop_pdevs::
"('a ⇒ 'b ⇒ 'c) ⇒ 'a::zero pdevs ⇒ 'b::zero pdevs ⇒ 'c::zero pdevs"
is binop_pdevs_raw
using nonzeros_binop_pdevs_subset
by (rule finite_subset) auto
lemma pdevs_apply_binop_pdevs[simp]: "pdevs_apply (binop_pdevs f x y) i =
(if pdevs_apply x i = 0 ∧ pdevs_apply y i = 0 then 0 else f (pdevs_apply x i) (pdevs_apply y i))"
by transfer (auto simp: binop_pdevs_raw_def)
subsection ‹Addition›
definition add_pdevs::"'a::real_vector pdevs ⇒ 'a pdevs ⇒ 'a pdevs"
where "add_pdevs = binop_pdevs (+)"
lemma pdevs_apply_add_pdevs[simp]:
"pdevs_apply (add_pdevs X Y) n = pdevs_apply X n + pdevs_apply Y n"
by (auto simp: add_pdevs_def)
lemma pdevs_val_add_pdevs[simp]:
fixes x y::"'a::euclidean_space"
shows "pdevs_val e (add_pdevs X Y) = pdevs_val e X + pdevs_val e Y"
proof -
let ?sum = "λm X. ∑i < m. e i *⇩R pdevs_apply X i"
let ?m = "max (degree X) (degree Y)"
have "pdevs_val e X + pdevs_val e Y = ?sum (degree X) X + ?sum (degree Y) Y"
by (simp add: pdevs_val_sum)
also have "?sum (degree X) X = ?sum ?m X"
by (rule sum.mono_neutral_cong_left) auto
also have "?sum (degree Y) Y = ?sum ?m Y"
by (rule sum.mono_neutral_cong_left) auto
also have "?sum ?m X + ?sum ?m Y = (∑i < ?m. e i *⇩R (pdevs_apply X i + pdevs_apply Y i))"
by (simp add: scaleR_right_distrib sum.distrib)
also have "… = (∑i. e i *⇩R (pdevs_apply X i + pdevs_apply Y i))"
by (rule suminf_finite[symmetric]) auto
also have "… = pdevs_val e (add_pdevs X Y)"
by (simp add: pdevs_val_def)
finally show "pdevs_val e (add_pdevs X Y) = pdevs_val e X + pdevs_val e Y" by simp
qed
subsection ‹Total Deviation›
lemma tdev_eq_zero_iff: fixes X::"real pdevs" shows "tdev X = 0 ⟷ (∀e. pdevs_val e X = 0)"
by (force simp add: pdevs_val_sum tdev_def sum_nonneg_eq_0_iff
dest!: spec[where x="λi. if pdevs_apply X i ≥ 0 then 1 else -1"] split: if_split_asm)
lemma tdev_nonneg[intro, simp]: "tdev X ≥ 0"
by (auto simp: tdev_def)
lemma tdev_nonpos_iff[simp]: "tdev X ≤ 0 ⟷ tdev X = 0"
by (auto simp: order.antisym)
subsection ‹Unary Operations›
definition unop_pdevs_raw::
"('a::zero ⇒ 'b::zero) ⇒ (nat ⇒ 'a) ⇒ nat ⇒ 'b"
where "unop_pdevs_raw f x i = (if x i = 0 then 0 else f (x i))"
lemma nonzeros_unop_pdevs_subset: "{i. unop_pdevs_raw f x i ≠ 0} ⊆ {i. x i ≠ 0}"
by (auto simp: unop_pdevs_raw_def)
lift_definition unop_pdevs::
"('a ⇒ 'b) ⇒ 'a::zero pdevs ⇒ 'b::zero pdevs"
is unop_pdevs_raw
using nonzeros_unop_pdevs_subset
by (rule finite_subset) auto
lemma pdevs_apply_unop_pdevs[simp]: "pdevs_apply (unop_pdevs f x) i =
(if pdevs_apply x i = 0 then 0 else f (pdevs_apply x i))"
by transfer (auto simp: unop_pdevs_raw_def)
lemma pdevs_domain_unop_linear:
assumes "linear f"
shows "pdevs_domain (unop_pdevs f x) ⊆ pdevs_domain x"
proof -
interpret f: linear f by fact
show ?thesis
by (auto simp: f.zero)
qed
lemma
pdevs_val_unop_linear:
assumes "linear f"
shows "pdevs_val e (unop_pdevs f x) = f (pdevs_val e x)"
proof -
interpret f: linear f by fact
have *: "⋀i. (if pdevs_apply x i = 0 then 0 else f (pdevs_apply x i)) = f (pdevs_apply x i)"
by (auto simp: f.zero)
have "pdevs_val e (unop_pdevs f x) =
(∑i∈pdevs_domain (unop_pdevs f x). e i *⇩R f (pdevs_apply x i))"
by (auto simp add: pdevs_val_pdevs_domain *)
also have "… = (∑xa∈pdevs_domain x. e xa *⇩R f (pdevs_apply x xa))"
by (auto intro!: sum.mono_neutral_cong_left)
also have "… = f (pdevs_val e x)"
by (auto simp add: pdevs_val_pdevs_domain f.sum f.scaleR)
finally show ?thesis .
qed
subsection ‹Pointwise Scaling of Partial Deviations›
definition scaleR_pdevs::"real ⇒ 'a::real_vector pdevs ⇒ 'a pdevs"
where "scaleR_pdevs r x = unop_pdevs ((*⇩R) r) x"
lemma pdevs_apply_scaleR_pdevs[simp]:
"pdevs_apply (scaleR_pdevs x Y) n = x *⇩R pdevs_apply Y n"
by (auto simp: scaleR_pdevs_def)
lemma degree_scaleR_pdevs[simp]: "degree (scaleR_pdevs r x) = (if r = 0 then 0 else degree x)"
unfolding degree_def
by auto
lemma pdevs_val_scaleR_pdevs[simp]:
fixes x::real and Y::"'a::real_normed_vector pdevs"
shows "pdevs_val e (scaleR_pdevs x Y) = x *⇩R pdevs_val e Y"
by (auto simp: pdevs_val_sum scaleR_sum_right ac_simps)
subsection ‹Partial Deviations Scale Pointwise›
definition pdevs_scaleR::"real pdevs ⇒ 'a::real_vector ⇒ 'a pdevs"
where "pdevs_scaleR r x = unop_pdevs (λr. r *⇩R x) r"
lemma pdevs_apply_pdevs_scaleR[simp]:
"pdevs_apply (pdevs_scaleR X y) n = pdevs_apply X n *⇩R y"
by (auto simp: pdevs_scaleR_def)
lemma degree_pdevs_scaleR[simp]: "degree (pdevs_scaleR r x) = (if x = 0 then 0 else degree r)"
unfolding degree_def
by auto
lemma pdevs_val_pdevs_scaleR[simp]:
fixes X::"real pdevs" and y::"'a::real_normed_vector"
shows "pdevs_val e (pdevs_scaleR X y) = pdevs_val e X *⇩R y"
by (auto simp: pdevs_val_sum scaleR_sum_left)
subsection ‹Pointwise Unary Minus›
definition uminus_pdevs::"'a::real_vector pdevs ⇒ 'a pdevs"
where "uminus_pdevs = unop_pdevs uminus"
lemma pdevs_apply_uminus_pdevs[simp]: "pdevs_apply (uminus_pdevs x) = - pdevs_apply x"
by (auto simp: uminus_pdevs_def)
lemma degree_uminus_pdevs[simp]: "degree (uminus_pdevs x) = degree x"
by (rule degree_cong) simp
lemma pdevs_val_uminus_pdevs[simp]: "pdevs_val e (uminus_pdevs x) = - pdevs_val e x"
unfolding pdevs_val_sum
by (auto simp: sum_negf)
definition "uminus_aform X = (- fst X, uminus_pdevs (snd X))"
lemma fst_uminus_aform[simp]: "fst (uminus_aform Y) = - fst Y"
by (simp add: uminus_aform_def)
lemma aform_val_uminus_aform[simp]: "aform_val e (uminus_aform X) = - aform_val e X"
by (auto simp: uminus_aform_def aform_val_def)
subsection ‹Constant›
lift_definition zero_pdevs::"'a::zero pdevs" is "λ_. 0" by simp
lemma pdevs_apply_zero_pdevs[simp]: "pdevs_apply zero_pdevs i = 0"
by transfer simp
lemma pdevs_val_zero_pdevs[simp]: "pdevs_val e zero_pdevs = 0"
by (auto simp: pdevs_val_def)
definition "num_aform f = (f, zero_pdevs)"
subsection ‹Inner Product›
definition pdevs_inner::"'a::euclidean_space pdevs ⇒ 'a ⇒ real pdevs"
where "pdevs_inner x b = unop_pdevs (λx. x ∙ b) x"
lemma pdevs_apply_pdevs_inner[simp]: "pdevs_apply (pdevs_inner p a) i = pdevs_apply p i ∙ a"
by (simp add: pdevs_inner_def)
lemma pdevs_val_pdevs_inner[simp]: "pdevs_val e (pdevs_inner p a) = pdevs_val e p ∙ a"
by (auto simp add: inner_sum_left pdevs_val_pdevs_domain intro!: sum.mono_neutral_cong_left)
definition inner_aform::"'a::euclidean_space aform ⇒ 'a ⇒ real aform"
where "inner_aform X b = (fst X ∙ b, pdevs_inner (snd X) b)"
subsection ‹Inner Product Pair›
definition inner2::"'a::euclidean_space ⇒ 'a ⇒ 'a ⇒ real*real"
where "inner2 x n l = (x ∙ n, x ∙ l)"
definition pdevs_inner2::"'a::euclidean_space pdevs ⇒ 'a ⇒ 'a ⇒ (real*real) pdevs"
where "pdevs_inner2 X n l = unop_pdevs (λx. inner2 x n l) X"
lemma pdevs_apply_pdevs_inner2[simp]: "pdevs_apply (pdevs_inner2 p a b) i = (pdevs_apply p i ∙ a, pdevs_apply p i ∙ b)"
by (simp add: pdevs_inner2_def inner2_def zero_prod_def)
definition inner2_aform::"'a::euclidean_space aform ⇒ 'a ⇒ 'a ⇒ (real*real) aform"
where "inner2_aform X a b = (inner2 (fst X) a b, pdevs_inner2 (snd X) a b)"
lemma linear_inner2[intro, simp]: "linear (λx. inner2 x n i)"
by unfold_locales (auto simp: inner2_def algebra_simps)
lemma aform_val_inner2_aform[simp]: "aform_val e (inner2_aform Z n i) = inner2 (aform_val e Z) n i"
proof -
have "aform_val e (inner2_aform Z n i) = inner2 (fst Z) n i + inner2 (pdevs_val e (snd Z)) n i"
by (auto simp: aform_val_def inner2_aform_def pdevs_inner2_def pdevs_val_unop_linear)
also have "… = inner2 (aform_val e Z) n i"
by (simp add: inner2_def algebra_simps aform_val_def)
finally show ?thesis .
qed
subsection ‹Update›
lemma pdevs_val_upd[simp]:
"pdevs_val (e(n := e')) X = pdevs_val e X - e n * pdevs_apply X n + e' * pdevs_apply X n"
unfolding pdevs_val_def
by (subst suminf_finite[OF finite.insertI[OF finite_degree_nonzero], of n X],
auto simp: pdevs_val_def sum.insert_remove)+
lemma nonzeros_fun_upd:
"{i. (f(n := a)) i ≠ 0} ⊆ {i. f i ≠ 0} ∪ {n}"
by (auto split: if_split_asm)
lift_definition pdev_upd::"'a::real_vector pdevs ⇒ nat ⇒ 'a ⇒ 'a pdevs"
is "λx n a. x(n:=a)"
by (rule finite_subset[OF nonzeros_fun_upd]) simp
lemma pdevs_apply_pdev_upd[simp]:
"pdevs_apply (pdev_upd X n x) = (pdevs_apply X)(n:=x)"
by transfer simp
lemma pdevs_val_pdev_upd[simp]:
"pdevs_val e (pdev_upd X n x) = pdevs_val e X + e n *⇩R x - e n *⇩R pdevs_apply X n"
unfolding pdevs_val_def
by (subst suminf_finite[OF finite.insertI[OF finite_degree_nonzero], of n X],
auto simp: pdevs_val_def sum.insert_remove)+
lemma degree_pdev_upd:
assumes "x = 0 ⟷ pdevs_apply X n = 0"
shows "degree (pdev_upd X n x) = degree X"
using assms
by (auto intro!: degree_cong split: if_split_asm)
lemma degree_pdev_upd_le:
assumes "degree X ≤ n"
shows "degree (pdev_upd X n x) ≤ Suc n"
using assms
by (auto intro!: degree_le)
subsection ‹Inf/Sup›
definition "Inf_aform X = fst X - tdev (snd X)"
definition "Sup_aform X = fst X + tdev (snd X)"
lemma Inf_aform:
assumes "e ∈ UNIV → {-1 .. 1}"
shows "Inf_aform X ≤ aform_val e X"
using order_trans[OF abs_ge_minus_self abs_pdevs_val_le_tdev[OF assms]]
by (auto simp: Inf_aform_def aform_val_def minus_le_iff)
lemma Sup_aform:
assumes "e ∈ UNIV → {-1 .. 1}"
shows "aform_val e X ≤ Sup_aform X"
using order_trans[OF abs_ge_self abs_pdevs_val_le_tdev[OF assms]]
by (auto simp: Sup_aform_def aform_val_def)
subsection ‹Minkowski Sum›
definition msum_pdevs_raw::"nat⇒(nat ⇒ 'a::real_vector)⇒(nat ⇒ 'a)⇒(nat⇒'a)" where
"msum_pdevs_raw n x y i = (if i < n then x i else y (i - n))"
lemma nonzeros_msum_pdevs_raw:
"{i. msum_pdevs_raw n f g i ≠ 0} = ({0..<n} ∩ {i. f i ≠ 0}) ∪ (+) n ` ({i. g i ≠ 0})"
by (force simp: msum_pdevs_raw_def not_less split: if_split_asm)
lift_definition msum_pdevs::"nat⇒'a::real_vector pdevs⇒'a pdevs⇒'a pdevs" is msum_pdevs_raw
unfolding nonzeros_msum_pdevs_raw by simp
lemma pdevs_apply_msum_pdevs: "pdevs_apply (msum_pdevs n f g) i =
(if i < n then pdevs_apply f i else pdevs_apply g (i - n))"
by transfer (auto simp: msum_pdevs_raw_def)
lemma degree_least_nonzero:
assumes "degree f ≠ 0"
shows "pdevs_apply f (degree f - 1) ≠ 0"
proof
assume H: "pdevs_apply f (degree f - 1) = 0"
{
fix j
assume "j≥degree f - 1"
with H have "pdevs_apply f j = 0"
by (cases "degree f - 1 = j") auto
}
from degree_le[rule_format, OF this]
have "degree f ≤ degree f - 1"
by blast
with assms show False by simp
qed
lemma degree_leI:
assumes "(⋀i. pdevs_apply y i = 0 ⟹ pdevs_apply x i = 0)"
shows "degree x ≤ degree y"
proof cases
assume "degree x ≠ 0"
from degree_least_nonzero[OF this]
have "pdevs_apply y (degree x - 1) ≠ 0"
by (auto simp: assms split: if_split_asm)
from degree_gt[OF this] show ?thesis
by simp
qed simp
lemma degree_msum_pdevs_ge1:
shows "degree f ≤ n ⟹ degree f ≤ degree (msum_pdevs n f g)"
by (rule degree_leI) (auto simp: pdevs_apply_msum_pdevs split: if_split_asm)
lemma degree_msum_pdevs_ge2:
assumes "degree f ≤ n"
shows "degree g ≤ degree (msum_pdevs n f g) - n"
proof cases
assume "degree g ≠ 0"
hence "pdevs_apply g (degree g - 1) ≠ 0" by (rule degree_least_nonzero)
hence "pdevs_apply (msum_pdevs n f g) (n + degree g - 1) ≠ 0"
using assms
by (auto simp: pdevs_apply_msum_pdevs)
from degree_gt[OF this]
show ?thesis
by simp
qed simp
lemma degree_msum_pdevs_le:
shows "degree (msum_pdevs n f g) ≤ n + degree g"
by (auto intro!: degree_le simp: pdevs_apply_msum_pdevs)
lemma
sum_msum_pdevs_cases:
assumes "degree f ≤ n"
assumes [simp]: "⋀i. e i 0 = 0"
shows
"(∑i <degree (msum_pdevs n f g).
e i (if i < n then pdevs_apply f i else pdevs_apply g (i - n))) =
(∑i <degree f. e i (pdevs_apply f i)) + (∑i <degree g. e (i + n) (pdevs_apply g i))"
(is "?lhs = ?rhs")
proof -
have "?lhs = (∑i∈{..<degree (msum_pdevs n f g)} ∩ {i. i < n}. e i (pdevs_apply f i)) +
(∑i∈{..<degree (msum_pdevs n f g)} ∩ - {i. i < n}. e i (pdevs_apply g (i - n)))"
(is "_ = ?sum_f + ?sum_g")
by (simp add: sum.If_cases if_distrib)
also have "?sum_f = (∑i = 0..<degree f. e i (pdevs_apply f i))"
using assms degree_msum_pdevs_ge1[of f n g]
by (intro sum.mono_neutral_cong_right) auto
also
have "?sum_g = (∑i∈{0 + n..<degree (msum_pdevs n f g) - n + n}. e i (pdevs_apply g (i - n)))"
by (rule sum.cong) auto
also have "… = (∑i = 0..<degree (msum_pdevs n f g) - n. e (i + n) (pdevs_apply g (i + n - n)))"
by (rule sum.shift_bounds_nat_ivl)
also have "… = (∑i = 0..<degree g. e (i + n) (pdevs_apply g i))"
using assms degree_msum_pdevs_ge2[of f n]
by (intro sum.mono_neutral_cong_right) (auto intro!: sum.mono_neutral_cong_right)
finally show ?thesis
by (simp add: atLeast0LessThan)
qed
lemma tdev_msum_pdevs: "degree f ≤ n ⟹ tdev (msum_pdevs n f g) = tdev f + tdev g"
by (auto simp: tdev_def pdevs_apply_msum_pdevs intro!: sum_msum_pdevs_cases)
lemma pdevs_val_msum_pdevs:
"degree f ≤ n ⟹ pdevs_val e (msum_pdevs n f g) = pdevs_val e f + pdevs_val (λi. e (i + n)) g"
by (auto simp: pdevs_val_sum pdevs_apply_msum_pdevs intro!: sum_msum_pdevs_cases)
definition msum_aform::"nat ⇒ 'a::real_vector aform ⇒ 'a aform ⇒ 'a aform"
where "msum_aform n f g = (fst f + fst g, msum_pdevs n (snd f) (snd g))"
lemma fst_msum_aform[simp]: "fst (msum_aform n f g) = fst f + fst g"
by (simp add: msum_aform_def)
lemma snd_msum_aform[simp]: "snd (msum_aform n f g) = msum_pdevs n (snd f) (snd g)"
by (simp add: msum_aform_def)
lemma finite_nonzero_summable: "finite {i. f i ≠ 0} ⟹ summable f"
by (auto intro!: sums_summable sums_finite)
lemma aform_val_msum_aform:
assumes "degree_aform f ≤ n"
shows "aform_val e (msum_aform n f g) = aform_val e f + aform_val (λi. e (i + n)) g"
using assms
by (auto simp: pdevs_val_msum_pdevs aform_val_def)
lemma Inf_aform_msum_aform:
"degree_aform X ≤ n ⟹ Inf_aform (msum_aform n X Y) = Inf_aform X + Inf_aform Y"
by (simp add: Inf_aform_def tdev_msum_pdevs)
lemma Sup_aform_msum_aform:
"degree_aform X ≤ n ⟹ Sup_aform (msum_aform n X Y) = Sup_aform X + Sup_aform Y"
by (simp add: Sup_aform_def tdev_msum_pdevs)
definition "independent_from d Y = msum_aform d (0, zero_pdevs) Y"
definition "independent_aform X Y = independent_from (degree_aform X) Y"
lemma degree_zero_pdevs[simp]: "degree zero_pdevs = 0"
by (metis degree_least_nonzero pdevs_apply_zero_pdevs)
lemma independent_aform_Joints:
assumes "x ∈ Affine X"
assumes "y ∈ Affine Y"
shows "[x, y] ∈ Joints [X, independent_aform X Y]"
using assms
unfolding Affine_def valuate_def Joints_def
apply safe
subgoal premises prems for e ea
using prems
by (intro image_eqI[where x="λi. if i < degree_aform X then e i else ea (i - degree_aform X)"])
(auto simp: aform_val_def pdevs_val_msum_pdevs Pi_iff
independent_aform_def independent_from_def intro!: pdevs_val_degree_cong)
done
lemma msum_aform_Joints:
assumes "d ≥ degree_aform X"
assumes "⋀X. X ∈ set XS ⟹ d ≥ degree_aform X"
assumes "(x#xs) ∈ Joints (X#XS)"
assumes "y ∈ Affine Y"
shows "((x + y)#x#xs) ∈ Joints (msum_aform d X Y#X#XS)"
using assms
unfolding Joints_def valuate_def Affine_def
proof (safe, goal_cases)
case (1 e ea a b zs)
then show ?case
by (intro image_eqI[where x = "λi. if i < d then e i else ea (i - d)"])
(force simp: aform_val_def pdevs_val_msum_pdevs intro!: intro!: pdevs_val_degree_cong)+
qed
lemma Joints_msum_aform:
assumes "d ≥ degree_aform X"
assumes "⋀Y. Y ∈ set YS ⟹ d ≥ degree_aform Y"
shows "Joints (msum_aform d X Y#YS) = {((x + y)#ys) |x y ys. y ∈ Affine Y ∧ x#ys ∈ Joints (X#YS)}"
unfolding Affine_def valuate_def Joints_def
proof (safe, goal_cases)
case (1 x e)
thus ?case
using assms
by (intro exI[where x = "aform_val e X"] exI[where x = "aform_val ((λi. e (i + d))) Y"])
(auto simp add: aform_val_def pdevs_val_msum_pdevs)
next
case (2 x xa y ys e ea)
thus ?case using assms
by (intro image_eqI[where x="λi. if i < d then ea i else e (i - d)"])
(force simp: aform_val_def pdevs_val_msum_pdevs Pi_iff intro!: pdevs_val_degree_cong)+
qed
lemma Joints_singleton_image: "Joints [x] = (λx. [x]) ` Affine x"
by (auto simp: Joints_def Affine_def valuate_def)
lemma : "{g (f x y) |x y. P x y} = g ` {f x y |x y. P x y}"
by auto
lemma inj_Cons: "inj (λx. x#xs)"
by (auto intro!: injI)
lemma Joints_Nil[simp]: "Joints [] = {[]}"
by (force simp: Joints_def valuate_def)
lemma msum_pdevs_zero_ident[simp]: "msum_pdevs 0 zero_pdevs x = x"
by transfer (auto simp: msum_pdevs_raw_def)
lemma msum_aform_zero_ident[simp]: "msum_aform 0 (0, zero_pdevs) x = x"
by (simp add: msum_aform_def)
lemma mem_Joints_singleton: "(x ∈ Joints [X]) = (∃y. x = [y] ∧ y ∈ Affine X)"
by (auto simp: Affine_def valuate_def Joints_def)
lemma singleton_mem_Joints[simp]: "[x] ∈ Joints [X] ⟷ x ∈ Affine X"
by (auto simp: mem_Joints_singleton)
lemma msum_aform_Joints_without_first:
assumes "d ≥ degree_aform X"
assumes "⋀X. X ∈ set XS ⟹ d ≥ degree_aform X"
assumes "(x#xs) ∈ Joints (X#XS)"
assumes "y ∈ Affine Y"
assumes "z = x + y"
shows "z#xs ∈ Joints (msum_aform d X Y#XS)"
unfolding ‹z = x + y›
using msum_aform_Joints[OF assms(1-4)]
by (force simp: Joints_def valuate_def)
lemma Affine_msum_aform:
assumes "d ≥ degree_aform X"
shows "Affine (msum_aform d X Y) = {x + y |x y. x ∈ Affine X ∧ y ∈ Affine Y}"
using Joints_msum_aform[OF assms, of Nil Y, simplified, unfolded mem_Joints_singleton]
by (auto simp add: Joints_singleton_image Collect_extract_image[where g="λx. [x]"]
inj_image_eq_iff[OF inj_Cons] )
lemma Affine_zero_pdevs[simp]: "Affine (0, zero_pdevs) = {0}"
by (force simp: Affine_def valuate_def aform_val_def)
lemma Affine_independent_aform:
"Affine (independent_aform X Y) = Affine Y"
by (auto simp: independent_aform_def independent_from_def Affine_msum_aform)
lemma
abs_diff_eq1:
fixes l u::"'a::ordered_euclidean_space"
shows "l ≤ u ⟹ ¦u - l¦ = u - l"
by (metis abs_of_nonneg diff_add_cancel le_add_same_cancel2)
lemma compact_sum:
fixes f :: "'a ⇒ 'b::topological_space ⇒ 'c::real_normed_vector"
assumes "finite I"
assumes "⋀i. i ∈ I ⟹ compact (S i)"
assumes "⋀i. i ∈ I ⟹ continuous_on (S i) (f i)"
assumes "I ⊆ J"
shows "compact {∑i∈I. f i (x i) | x. x ∈ Pi J S}"
using assms
proof (induct I)
case empty
thus ?case
proof (cases "∃x. x ∈ Pi J S")
case False
hence *: "{∑i∈{}. f i (x i) |x. x ∈ Pi J S} = {}"
by (auto simp: Pi_iff)
show ?thesis unfolding * by simp
qed auto
next
case (insert a I)
hence "{∑i∈insert a I. f i (xa i) |xa. xa ∈ Pi J S}
= {x + y |x y. x ∈ f a ` S a ∧ y ∈ {∑i∈I. f i (x i) |x. x ∈ Pi J S}}"
proof safe
fix s x
assume "s ∈ S a" "x ∈ Pi J S"
thus "∃xa. f a s + (∑i∈I. f i (x i)) = (∑i∈insert a I. f i (xa i)) ∧ xa ∈ Pi J S"
using insert
by (auto intro!: exI[where x="x(a:=s)"] sum.cong)
qed force
also have "compact …"
using insert
by (intro compact_sums) (auto intro!: compact_continuous_image)
finally show ?case .
qed
lemma compact_Affine:
fixes X::"'a::ordered_euclidean_space aform"
shows "compact (Affine X)"
proof -
have "Affine X = {x + y|x y. x ∈ {fst X} ∧
y ∈ {(∑i ∈ {0..<degree_aform X}. e i *⇩R pdevs_apply (snd X) i) | e. e ∈ UNIV → {-1 .. 1}}}"
by (auto simp: Affine_def valuate_def aform_val_def pdevs_val_sum atLeast0LessThan)
also have "compact …"
by (rule compact_sums) (auto intro!: compact_sum continuous_intros)
finally show ?thesis .
qed
lemma Joints2_JointsI:
"(xs, x) ∈ Joints2 XS X ⟹ x#xs ∈ Joints (X#XS)"
by (auto simp: Joints_def Joints2_def valuate_def)
subsection ‹Splitting›
definition "split_aform X i =
(let xi = pdevs_apply (snd X) i /⇩R 2
in ((fst X - xi, pdev_upd (snd X) i xi), (fst X + xi, pdev_upd (snd X) i xi)))"
lemma split_aformE:
assumes "e ∈ UNIV → {-1 .. 1}"
assumes "x = aform_val e X"
obtains err where "x = aform_val (e(i:=err)) (fst (split_aform X i))" "err ∈ {-1 .. 1}"
| err where "x = aform_val (e(i:=err)) (snd (split_aform X i))" "err ∈ {-1 .. 1}"
proof (atomize_elim)
let ?thesis = "(∃err. x = aform_val (e(i := err)) (fst (split_aform X i)) ∧ err ∈ {- 1..1}) ∨
(∃err. x = aform_val (e(i := err)) (snd (split_aform X i)) ∧ err ∈ {- 1..1})"
{
assume "pdevs_apply (snd X) i = 0"
hence "X = fst (split_aform X i)"
by (auto simp: split_aform_def intro!: prod_eqI pdevs_eqI)
with assms have ?thesis by (auto intro!: exI[where x="e i"])
} moreover {
assume "pdevs_apply (snd X) i ≠ 0"
hence [simp]: "degree_aform X > i"
by (rule degree_gt)
note assms(2)
also
have "aform_val e X = fst X + (∑i<degree_aform X. e i *⇩R pdevs_apply (snd X) i)"
by (simp add: aform_val_def pdevs_val_sum)
also
have rewr: "{..<degree_aform X} = {0..<degree_aform X} - {i} ∪ {i}"
by auto
have "(∑i<degree_aform X. e i *⇩R pdevs_apply (snd X) i) =
(∑i ∈ {0..<degree_aform X} - {i}. e i *⇩R pdevs_apply (snd X) i) +
e i *⇩R pdevs_apply (snd X) i"
by (subst rewr, subst sum.union_disjoint) auto
finally have "x = fst X + …" .
hence "x = aform_val (e(i:=2 * e i - 1)) (snd (split_aform X i))"
"x = aform_val (e(i:=2 * e i + 1)) (fst (split_aform X i))"
by (auto simp: aform_val_def split_aform_def Let_def pdevs_val_sum atLeast0LessThan
Diff_eq degree_pdev_upd if_distrib sum.If_cases field_simps
scaleR_left_distrib[symmetric])
moreover
have "2 * e i - 1 ∈ {-1 .. 1} ∨ 2 * e i + 1 ∈ {-1 .. 1}"
using assms by (auto simp: not_le Pi_iff dest!: spec[where x=i])
ultimately have ?thesis by blast
} ultimately show ?thesis by blast
qed
lemma pdevs_val_add: "pdevs_val (λi. e i + f i) xs = pdevs_val e xs + pdevs_val f xs"
by (auto simp: pdevs_val_pdevs_domain algebra_simps sum.distrib)
lemma pdevs_val_minus: "pdevs_val (λi. e i - f i) xs = pdevs_val e xs - pdevs_val f xs"
by (auto simp: pdevs_val_pdevs_domain algebra_simps sum_subtractf)
lemma pdevs_val_cmul: "pdevs_val (λi. u * e i) xs = u *⇩R pdevs_val e xs"
by (auto simp: pdevs_val_pdevs_domain scaleR_sum_right)
lemma atLeastAtMost_absI: "- a ≤ a ⟹ ¦x::real¦ ≤ ¦a¦ ⟹ x ∈ atLeastAtMost (- a) a"
by auto
lemma divide_atLeastAtMost_1_absI: "¦x::real¦ ≤ ¦a¦ ⟹ x/a ∈ {-1 .. 1}"
by (intro atLeastAtMost_absI) (auto simp: divide_le_eq_1)
lemma convex_scaleR_aux: "u + v = 1 ⟹ u *⇩R x + v *⇩R x = (x::'a::real_vector)"
by (metis scaleR_add_left scaleR_one)
lemma convex_mult_aux: "u + v = 1 ⟹ u * x + v * x = (x::real)"
using convex_scaleR_aux[of u v x] by simp
lemma convex_Affine: "convex (Affine X)"
proof (rule convexI)
fix x y::'a and u v::real
assume "x ∈ Affine X" "y ∈ Affine X" and convex: "0 ≤ u" "0 ≤ v" "u + v = 1"
then obtain e f where x: "x = aform_val e X" "e ∈ UNIV → {-1 .. 1}"
and y: "y = aform_val f X" "f ∈ UNIV → {-1 .. 1}"
by (auto simp: Affine_def valuate_def)
let ?conv = "λi. u * e i + v * f i"
{
fix i
have "¦?conv i¦ ≤ u * ¦e i¦ + v * ¦f i¦"
using convex by (intro order_trans[OF abs_triangle_ineq]) (simp add: abs_mult)
also have "… ≤ 1"
using convex x y
by (intro convex_bound_le) (auto simp: Pi_iff abs_real_def)
finally have "?conv i ≤ 1" "-1 ≤ ?conv i"
by (auto simp: abs_real_def split: if_split_asm)
}
thus "u *⇩R x + v *⇩R y ∈ Affine X"
using convex x y
by (auto simp: Affine_def valuate_def aform_val_def pdevs_val_add pdevs_val_cmul algebra_simps
convex_scaleR_aux intro!: image_eqI[where x="?conv"])
qed
lemma segment_in_aform_val:
assumes "e ∈ UNIV → {-1 .. 1}"
assumes "f ∈ UNIV → {-1 .. 1}"
shows "closed_segment (aform_val e X) (aform_val f X) ⊆ Affine X"
proof -
have "aform_val e X ∈ Affine X" "aform_val f X ∈ Affine X"
using assms by (auto simp: Affine_def valuate_def)
with convex_Affine[of X, simplified convex_contains_segment]
show ?thesis
by simp
qed
subsection ‹From List of Generators›
lift_definition pdevs_of_list::"'a::zero list ⇒ 'a pdevs"
is "λxs i. if i < length xs then xs ! i else 0"
by auto
lemma pdevs_apply_pdevs_of_list:
"pdevs_apply (pdevs_of_list xs) i = (if i < length xs then xs ! i else 0)"
by transfer simp
lemma pdevs_apply_pdevs_of_list_Nil[simp]:
"pdevs_apply (pdevs_of_list []) i = 0"
by transfer auto
lemma pdevs_apply_pdevs_of_list_Cons:
"pdevs_apply (pdevs_of_list (x # xs)) i =
(if i = 0 then x else pdevs_apply (pdevs_of_list xs) (i - 1))"
by transfer auto
lemma pdevs_domain_pdevs_of_list_Cons[simp]: "pdevs_domain (pdevs_of_list (x # xs)) =
(if x = 0 then {} else {0}) ∪ (+) 1 ` pdevs_domain (pdevs_of_list xs)"
by (force simp: pdevs_apply_pdevs_of_list_Cons split: if_split_asm)
lemma pdevs_val_pdevs_of_list_eq[simp]:
"pdevs_val e (pdevs_of_list (x # xs)) = e 0 *⇩R x + pdevs_val (e o (+) 1) (pdevs_of_list xs)"
proof -
have "pdevs_val e (pdevs_of_list (x # xs)) =
(∑i∈pdevs_domain (pdevs_of_list (x # xs)) ∩ {0}. e i *⇩R x) +
(∑i∈pdevs_domain (pdevs_of_list (x # xs)) ∩ - {0}.
e i *⇩R pdevs_apply (pdevs_of_list xs) (i - Suc 0))"
(is "_ = ?l + ?r")
by (simp add: pdevs_val_pdevs_domain if_distrib sum.If_cases pdevs_apply_pdevs_of_list_Cons)
also
have "?r = (∑i∈pdevs_domain (pdevs_of_list xs). e (Suc i) *⇩R pdevs_apply (pdevs_of_list xs) i)"
by (rule sum.reindex_cong[of "λi. i + 1"]) auto
also have "… = pdevs_val (e o (+) 1) (pdevs_of_list xs)"
by (simp add: pdevs_val_pdevs_domain )
also have "?l = (∑i∈{0}. e i *⇩R x)"
by (rule sum.mono_neutral_cong_left) auto
also have "… = e 0 *⇩R x" by simp
finally show ?thesis .
qed
lemma
less_degree_pdevs_of_list_imp_less_length:
assumes "i < degree (pdevs_of_list xs)"
shows "i < length xs"
proof -
from assms have "pdevs_apply (pdevs_of_list xs) (degree (pdevs_of_list xs) - 1) ≠ 0"
by (metis degree_least_nonzero less_nat_zero_code)
hence "degree (pdevs_of_list xs) - 1 < length xs"
by (simp add: pdevs_apply_pdevs_of_list split: if_split_asm)
with assms show ?thesis
by simp
qed
lemma tdev_pdevs_of_list[simp]: "tdev (pdevs_of_list xs) = sum_list (map abs xs)"
by (auto simp: tdev_def pdevs_apply_pdevs_of_list sum_list_sum_nth
less_degree_pdevs_of_list_imp_less_length
intro!: sum.mono_neutral_cong_left degree_gt)
lemma pdevs_of_list_Nil[simp]: "pdevs_of_list [] = zero_pdevs"
by (auto intro!: pdevs_eqI)
lemma pdevs_val_inj_sumI:
fixes K::"'a set" and g::"'a ⇒ nat"
assumes "finite K"
assumes "inj_on g K"
assumes "pdevs_domain x ⊆ g ` K"
assumes "⋀i. i ∈ K ⟹ g i ∉ pdevs_domain x ⟹ f i = 0"
assumes "⋀i. i ∈ K ⟹ g i ∈ pdevs_domain x ⟹ f i = e (g i) *⇩R pdevs_apply x (g i)"
shows "pdevs_val e x = (∑i∈K. f i)"
proof -
have [simp]: "inj_on (the_inv_into K g) (pdevs_domain x)"
using assms
by (auto simp: intro!: subset_inj_on[OF inj_on_the_inv_into])
{
fix y assume y: "y ∈ pdevs_domain x"
have g_inv: "g (the_inv_into K g y) = y"
by (meson assms(2) assms(3) y f_the_inv_into_f subset_eq)
have inv_in: "the_inv_into K g y ∈ K"
by (meson assms(2) assms(3) y subset_iff in_pdevs_domain the_inv_into_into)
have inv3: "the_inv_into (pdevs_domain x) (the_inv_into K g) (the_inv_into K g y) =
g (the_inv_into K g y)"
using assms y
by (subst the_inv_into_f_f) (auto simp: f_the_inv_into_f[OF assms(2)])
note g_inv inv_in inv3
} note this[simp]
have "pdevs_val e x = (∑i∈pdevs_domain x. e i *⇩R pdevs_apply x i)"
by (simp add: pdevs_val_pdevs_domain)
also have "… = (∑i ∈ the_inv_into K g ` pdevs_domain x. e (g i) *⇩R pdevs_apply x (g i))"
by (rule sum.reindex_cong[OF inj_on_the_inv_into]) auto
also have "… = (∑i∈K. f i)"
using assms
by (intro sum.mono_neutral_cong_left) (auto simp: the_inv_into_image_eq)
finally show ?thesis .
qed
lemma pdevs_domain_pdevs_of_list_le: "pdevs_domain (pdevs_of_list xs) ⊆ {0..<length xs}"
by (auto simp: pdevs_apply_pdevs_of_list split: if_split_asm)
lemma pdevs_val_zip: "pdevs_val e (pdevs_of_list xs) = (∑(i,x)←zip [0..<length xs] xs. e i *⇩R x)"
by (auto simp: sum_list_distinct_conv_sum_set
in_set_zip image_fst_zip pdevs_apply_pdevs_of_list distinct_zipI1
intro!: pdevs_val_inj_sumI[of _ fst]
split: if_split_asm)
lemma pdevs_val_map:
‹pdevs_val e (pdevs_of_list xs)
= (∑n←[0..<length xs]. e n *⇩R xs ! n)›
proof -
have ‹map2 (λi. (*⇩R) (e i)) [0..<length xs] xs =
map (λn. e n *⇩R xs ! n) [0..<length xs]›
by (rule nth_equalityI) simp_all
then show ?thesis
by (simp add: pdevs_val_zip)
qed
lemma scaleR_sum_list:
fixes xs::"'a::real_vector list"
shows "a *⇩R sum_list xs = sum_list (map (scaleR a) xs)"
by (induct xs) (auto simp: algebra_simps)
lemma pdevs_val_const_pdevs_of_list: "pdevs_val (λ_. c) (pdevs_of_list xs) = c *⇩R sum_list xs"
unfolding pdevs_val_zip split_beta' scaleR_sum_list
by (rule arg_cong) (auto intro!: nth_equalityI)
lemma pdevs_val_partition:
assumes "e ∈ UNIV → I"
obtains f g where "pdevs_val e (pdevs_of_list xs) =
pdevs_val f (pdevs_of_list (filter p xs)) +
pdevs_val g (pdevs_of_list (filter (Not o p) xs))"
"f ∈ UNIV → I"
"g ∈ UNIV → I"
proof -
obtain i where i: "i ∈ I"
by (metis assms funcset_mem iso_tuple_UNIV_I)
let ?zip = "zip [0..<length xs] xs"
define part where "part = partition (p ∘ snd) ?zip"
let ?f =
"(λn. if n < degree (pdevs_of_list (filter p xs)) then e (map fst (fst part) ! n) else i)"
let ?g =
"(λn. if n < degree (pdevs_of_list (filter (Not ∘ p) xs))
then e (map fst (snd part) ! n)
else i)"
show ?thesis
proof
have "pdevs_val e (pdevs_of_list xs) = (∑(i,x)←?zip. e i *⇩R x)"
by (rule pdevs_val_zip)
also have "… = (∑(i, x)∈set ?zip. e i *⇩R x)"
by (simp add: sum_list_distinct_conv_sum_set distinct_zipI1)
also
have [simp]: "set (fst part) ∩ set (snd part) = {}"
by (auto simp: part_def)
from partition_set[of "p o snd" ?zip "fst part" "snd part"]
have "set ?zip = set (fst part) ∪ set (snd part)"
by (auto simp: part_def)
also have "(∑a∈set (fst part) ∪ set (snd part). case a of (i, x) ⇒ e i *⇩R x) =
(∑(i, x)∈set (fst part). e i *⇩R x) + (∑(i, x)∈set (snd part). e i *⇩R x)"
by (auto simp: split_beta sum_Un)
also
have "(∑(i, x)∈set (fst part). e i *⇩R x) = (∑(i, x)←(fst part). e i *⇩R x)"
by (simp add: sum_list_distinct_conv_sum_set distinct_zipI1 part_def)
also have "… = (∑i<length (fst part). case (fst part ! i) of (i, x) ⇒ e i *⇩R x)"
by (subst sum_list_sum_nth) (simp add: split_beta' atLeast0LessThan)
also have "… =
pdevs_val (λn. e (map fst (fst part) ! n)) (pdevs_of_list (map snd (fst part)))"
by (force
simp: pdevs_val_zip sum_list_distinct_conv_sum_set distinct_zipI1 split_beta' in_set_zip
intro!:
sum.reindex_cong[where l=fst] image_eqI[where x = "(x, map snd (fst part) ! x)" for x])
also
have "(∑(i, x)∈set (snd part). e i *⇩R x) = (∑(i, x)←(snd part). e i *⇩R x)"
by (simp add: sum_list_distinct_conv_sum_set distinct_zipI1 part_def)
also have "… = (∑i<length (snd part). case (snd part ! i) of (i, x) ⇒ e i *⇩R x)"
by (subst sum_list_sum_nth) (simp add: split_beta' atLeast0LessThan)
also have "… =
pdevs_val (λn. e (map fst (snd part) ! n)) (pdevs_of_list (map snd (snd part)))"
by (force simp: pdevs_val_zip sum_list_distinct_conv_sum_set distinct_zipI1 split_beta'
in_set_zip
intro!: sum.reindex_cong[where l=fst]
image_eqI[where x = "(x, map snd (snd part) ! x)" for x])
also
have "pdevs_val (λn. e (map fst (fst part) ! n)) (pdevs_of_list (map snd (fst part))) =
pdevs_val (λn.
if n < degree (pdevs_of_list (map snd (fst part))) then e (map fst (fst part) ! n) else i)
(pdevs_of_list (map snd (fst part)))"
by (rule pdevs_val_degree_cong) simp_all
also
have "pdevs_val (λn. e (map fst (snd part) ! n)) (pdevs_of_list (map snd (snd part))) =
pdevs_val (λn.
if n < degree (pdevs_of_list (map snd (snd part))) then e (map fst (snd part) ! n) else i)
(pdevs_of_list (map snd (snd part)))"
by (rule pdevs_val_degree_cong) simp_all
also have "map snd (snd part) = filter (Not o p) xs"
by (simp add: part_def filter_map[symmetric] o_assoc)
also have "map snd (fst part) = filter p xs"
by (simp add: part_def filter_map[symmetric])
finally
show
"pdevs_val e (pdevs_of_list xs) =
pdevs_val ?f (pdevs_of_list (filter p xs)) +
pdevs_val ?g (pdevs_of_list (filter (Not ∘ p) xs))" .
show "?f ∈ UNIV → I" "?g ∈ UNIV → I"
using assms ‹i∈I›
by (auto simp: Pi_iff)
qed
qed
lemma pdevs_apply_pdevs_of_list_append:
"pdevs_apply (pdevs_of_list (xs @ zs)) i =
(if i < length xs
then pdevs_apply (pdevs_of_list xs) i else pdevs_apply (pdevs_of_list zs) (i - length xs))"
by (auto simp: pdevs_apply_pdevs_of_list nth_append)
lemma degree_pdevs_of_list_le_length[intro, simp]: "degree (pdevs_of_list xs) ≤ length xs"
by (metis less_irrefl_nat le_less_linear less_degree_pdevs_of_list_imp_less_length)
lemma degree_pdevs_of_list_append:
"degree (pdevs_of_list (xs @ ys)) ≤ length xs + degree (pdevs_of_list ys)"
by (rule degree_le) (auto simp: pdevs_apply_pdevs_of_list_append)
lemma pdevs_val_pdevs_of_list_append:
assumes "f ∈ UNIV → I"
assumes "g ∈ UNIV → I"
obtains e where
"pdevs_val f (pdevs_of_list xs) + pdevs_val g (pdevs_of_list ys) =
pdevs_val e (pdevs_of_list (xs @ ys))"
"e ∈ UNIV → I"
proof
let ?e = "(λi. if i < length xs then f i else g (i - length xs))"
have f: "pdevs_val f (pdevs_of_list xs) =
(∑i∈{..<length xs}. ?e i *⇩R pdevs_apply (pdevs_of_list (xs @ ys)) i)"
by (auto simp: pdevs_val_sum degree_gt pdevs_apply_pdevs_of_list_append
intro: sum.mono_neutral_cong_left)
have g: "pdevs_val g (pdevs_of_list ys) =
(∑i=length xs ..<length xs + degree (pdevs_of_list ys).
?e i *⇩R pdevs_apply (pdevs_of_list (xs @ ys)) i)"
(is "_ = ?sg")
by (auto simp: pdevs_val_sum pdevs_apply_pdevs_of_list_append
intro!: inj_onI image_eqI[where x="length xs + x" for x]
sum.reindex_cong[where l="λi. i - length xs"])
show "pdevs_val f (pdevs_of_list xs) + pdevs_val g (pdevs_of_list ys) =
pdevs_val ?e (pdevs_of_list (xs @ ys))"
unfolding f g
by (subst sum.union_disjoint[symmetric])
(force simp: pdevs_val_sum ivl_disj_un degree_pdevs_of_list_append
intro!: sum.mono_neutral_cong_right
split: if_split_asm)+
show "?e ∈ UNIV → I"
using assms by (auto simp: Pi_iff)
qed
lemma
sum_general_mono:
fixes f::"'a⇒('b::ordered_ab_group_add)"
assumes [simp,intro]: "finite s" "finite t"
assumes f: "⋀x. x ∈ s - t ⟹ f x ≤ 0"
assumes g: "⋀x. x ∈ t - s ⟹ g x ≥ 0"
assumes fg: "⋀x. x ∈ s ∩ t ⟹ f x ≤ g x"
shows "(∑x ∈ s. f x) ≤ (∑x ∈ t. g x)"
proof -
have "s = (s - t) ∪ (s ∩ t)" and [intro, simp]: "(s - t) ∩ (s ∩ t) = {}" by auto
hence "(∑x ∈ s. f x) = (∑x ∈ s - t ∪ s ∩ t. f x)"
using assms by simp
also have "… = (∑x ∈ s - t. f x) + (∑x ∈ s ∩ t. f x)"
by (simp add: sum_Un)
also have "(∑x ∈ s - t. f x) ≤ 0"
by (auto intro!: sum_nonpos f)
also have "0 ≤ (∑x ∈ t - s. g x)"
by (auto intro!: sum_nonneg g)
also have "(∑x ∈ s ∩ t. f x) ≤ (∑x ∈ s ∩ t. g x)"
by (auto intro!: sum_mono fg)
also
have [intro, simp]: "(t - s) ∩ (s ∩ t) = {}" by auto
hence "sum g (t - s) + sum g (s ∩ t) = sum g ((t - s) ∪ (s ∩ t))"
by (simp add: sum_Un)
also have "… = sum g t"
by (auto intro!: sum.cong)
finally show ?thesis by simp
qed
lemma degree_pdevs_of_list_eq':
‹degree (pdevs_of_list xs) = Min {n. n ≤ length xs ∧ (∀m. n ≤ m ⟶ m < length xs ⟶ xs ! m = 0)}›
apply (simp add: degree_def pdevs_apply_pdevs_of_list)
apply (rule Least_equality)
apply auto
apply (subst (asm) Min_le_iff)
apply auto
apply (subst Min_le_iff)
apply auto
apply (metis le_cases not_less)
done
lemma pdevs_val_permuted:
‹pdevs_val (e ∘ p) (pdevs_of_list (permute_list p xs)) = pdevs_val e (pdevs_of_list xs)› (is ‹?r = ?s›)
if perm: ‹p permutes {..<length xs}›
proof -
from that have ‹image_mset p (mset_set {0..<length xs}) = mset_set {0..<length xs}›
by (simp add: permutes_image_mset lessThan_atLeast0)
moreover have ‹map (λn. e (p n) *⇩R permute_list p xs ! n) [0..<length xs] =
map (λn. e n *⇩R xs ! n) (map p [0..<length xs])›
using that by (simp add: permute_list_nth)
ultimately show ?thesis
using that by (simp add: pdevs_apply_pdevs_of_list pdevs_val_map
flip: map_map sum_mset_sum_list)
qed
lemma pdevs_val_perm_ex:
assumes "xs <~~> ys"
assumes mem: "e ∈ UNIV → I"
shows "∃e'. e' ∈ UNIV → I ∧ pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
proof -
from ‹mset xs = mset ys›
have ‹mset ys = mset xs› ..
then obtain p where ‹p permutes {..<length xs}› ‹permute_list p xs = ys›
by (rule mset_eq_permutation)
moreover define e' where ‹e' = e ∘ p›
ultimately have ‹e' ∈ UNIV → I› ‹pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)›
using mem by (auto simp add: pdevs_val_permuted)
then show ?thesis by blast
qed
lemma pdevs_val_perm:
assumes "xs <~~> ys"
assumes mem: "e ∈ UNIV → I"
obtains e' where "e' ∈ UNIV → I"
"pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
using assms
by (metis pdevs_val_perm_ex)
lemma set_distinct_permI: "set xs = set ys ⟹ distinct xs ⟹ distinct ys ⟹ xs <~~> ys"
by (metis eq_set_perm_remdups remdups_id_iff_distinct)
lemmas pdevs_val_permute = pdevs_val_perm[OF set_distinct_permI]
lemma partition_permI:
"filter p xs @ filter (Not o p) xs <~~> xs"
by simp
lemma pdevs_val_eqI:
assumes "⋀i. i ∈ pdevs_domain y ⟹ i ∈ pdevs_domain x ⟹
e i *⇩R pdevs_apply x i = f i *⇩R pdevs_apply y i"
assumes "⋀i. i ∈ pdevs_domain y ⟹ i ∉ pdevs_domain x ⟹ f i *⇩R pdevs_apply y i = 0"
assumes "⋀i. i ∈ pdevs_domain x ⟹ i ∉ pdevs_domain y ⟹ e i *⇩R pdevs_apply x i = 0"
shows "pdevs_val e x = pdevs_val f y"
using assms
by (force simp: pdevs_val_pdevs_domain
intro!:
sum.reindex_bij_witness_not_neutral[where
i=id and j = id and
S'="pdevs_domain x - pdevs_domain y" and
T'="pdevs_domain y - pdevs_domain x"])
definition
filter_pdevs_raw::"(nat ⇒ 'a ⇒ bool) ⇒ (nat ⇒ 'a::real_vector) ⇒ (nat ⇒ 'a)"
where "filter_pdevs_raw I X = (λi. if I i (X i) then X i else 0)"
lemma filter_pdevs_raw_nonzeros: "{i. filter_pdevs_raw s f i ≠ 0} = {i. f i ≠ 0} ∩ {x. s x (f x)}"
by (auto simp: filter_pdevs_raw_def)
lift_definition filter_pdevs::"(nat ⇒ 'a ⇒ bool) ⇒ 'a::real_vector pdevs ⇒ 'a pdevs"
is filter_pdevs_raw
by (simp add: filter_pdevs_raw_nonzeros)
lemma pdevs_apply_filter_pdevs[simp]:
"pdevs_apply (filter_pdevs I x) i = (if I i (pdevs_apply x i) then pdevs_apply x i else 0)"
by transfer (auto simp: filter_pdevs_raw_def)
lemma degree_filter_pdevs_le: "degree (filter_pdevs I x) ≤ degree x"
by (rule degree_leI) (simp split: if_split_asm)
lemma pdevs_val_filter_pdevs:
"pdevs_val e (filter_pdevs I x) =
(∑i ∈ {..<degree x} ∩ {i. I i (pdevs_apply x i)}. e i *⇩R pdevs_apply x i)"
by (auto simp: pdevs_val_sum if_distrib sum.inter_restrict degree_filter_pdevs_le degree_gt
intro!: sum.mono_neutral_cong_left split: if_split_asm)
lemma pdevs_val_filter_pdevs_dom:
"pdevs_val e (filter_pdevs I x) =
(∑i ∈ pdevs_domain x ∩ {i. I i (pdevs_apply x i)}. e i *⇩R pdevs_apply x i)"
by (auto
simp: pdevs_val_pdevs_domain if_distrib sum.inter_restrict degree_filter_pdevs_le degree_gt
intro!: sum.mono_neutral_cong_left split: if_split_asm)
lemma pdevs_val_filter_pdevs_eval:
"pdevs_val e (filter_pdevs p x) = pdevs_val (λi. if p i (pdevs_apply x i) then e i else 0) x"
by (auto split: if_split_asm intro!: pdevs_val_eqI)
definition "pdevs_applys X i = map (λx. pdevs_apply x i) X"
definition "pdevs_vals e X = map (pdevs_val e) X"
definition "aform_vals e X = map (aform_val e) X"
definition "filter_pdevs_list I X = map (filter_pdevs (λi _. I i (pdevs_applys X i))) X"
lemma pdevs_applys_filter_pdevs_list[simp]:
"pdevs_applys (filter_pdevs_list I X) i = (if I i (pdevs_applys X i) then pdevs_applys X i else
map (λ_. 0) X)"
by (auto simp: filter_pdevs_list_def o_def pdevs_applys_def)
definition "degrees X = Max (insert 0 (degree ` set X))"
abbreviation "degree_aforms X ≡ degrees (map snd X)"
lemma degrees_leI:
assumes "⋀x. x ∈ set X ⟹ degree x ≤ K"
shows "degrees X ≤ K"
using assms
by (auto simp: degrees_def intro!: Max.boundedI)
lemma degrees_leD:
assumes "degrees X ≤ K"
shows "⋀x. x ∈ set X ⟹ degree x ≤ K"
using assms
by (auto simp: degrees_def intro!: Max.boundedI)
lemma degree_filter_pdevs_list_le: "degrees (filter_pdevs_list I x) ≤ degrees x"
by (rule degrees_leI) (auto simp: filter_pdevs_list_def intro!: degree_le dest!: degrees_leD)
definition "dense_list_of_pdevs x = map (λi. pdevs_apply x i) [0..<degree x]"
subsubsection ‹(reverse) ordered coefficients as list›
definition "list_of_pdevs x =
map (λi. (i, pdevs_apply x i)) (rev (sorted_list_of_set (pdevs_domain x)))"
lemma list_of_pdevs_zero_pdevs[simp]: "list_of_pdevs zero_pdevs = []"
by (auto simp: list_of_pdevs_def)
lemma sum_list_list_of_pdevs: "sum_list (map snd (list_of_pdevs x)) = sum_list (dense_list_of_pdevs x)"
by (auto intro!: sum.mono_neutral_cong_left
simp add: degree_gt sum_list_distinct_conv_sum_set dense_list_of_pdevs_def list_of_pdevs_def)
lemma sum_list_filter_dense_list_of_pdevs[symmetric]:
"sum_list (map snd (filter (p o snd) (list_of_pdevs x))) =
sum_list (filter p (dense_list_of_pdevs x))"
by (auto intro!: sum.mono_neutral_cong_left
simp add: degree_gt sum_list_distinct_conv_sum_set dense_list_of_pdevs_def list_of_pdevs_def
o_def filter_map)
lemma pdevs_of_list_dense_list_of_pdevs: "pdevs_of_list (dense_list_of_pdevs x) = x"
by (auto simp: pdevs_apply_pdevs_of_list dense_list_of_pdevs_def pdevs_eqI)
lemma pdevs_val_sum_list: "pdevs_val (λ_. c) X = c *⇩R sum_list (map snd (list_of_pdevs X))"
by (auto simp: pdevs_val_sum sum_list_list_of_pdevs pdevs_val_const_pdevs_of_list[symmetric]
pdevs_of_list_dense_list_of_pdevs)
lemma list_of_pdevs_all_nonzero: "list_all (λx. x ≠ 0) (map snd (list_of_pdevs xs))"
by (auto simp: list_of_pdevs_def list_all_iff)
lemma list_of_pdevs_nonzero: "x ∈ set (map snd (list_of_pdevs xs)) ⟹ x ≠ 0"
by (auto simp: list_of_pdevs_def)
lemma pdevs_of_list_scaleR_0[simp]:
fixes xs::"'a::real_vector list"
shows "pdevs_of_list (map ((*⇩R) 0) xs) = zero_pdevs"
by (auto simp: pdevs_apply_pdevs_of_list intro!: pdevs_eqI)
lemma degree_pdevs_of_list_scaleR:
"degree (pdevs_of_list (map ((*⇩R) c) xs)) = (if c ≠ 0 then degree (pdevs_of_list xs) else 0)"
by (auto simp: pdevs_apply_pdevs_of_list intro!: degree_cong)
lemma list_of_pdevs_eq:
"rev (list_of_pdevs X) = (filter ((≠) 0 o snd) (map (λi. (i, pdevs_apply X i)) [0..<degree X]))"
(is "_ = filter ?P (map ?f ?xs)")
using map_filter[of ?f ?P ?xs]
by (auto simp: list_of_pdevs_def o_def sorted_list_of_pdevs_domain_eq rev_map)
lemma sum_list_take_pdevs_val_eq:
"sum_list (take d xs) = pdevs_val (λi. if i < d then 1 else 0) (pdevs_of_list xs)"
proof -
have "sum_list (take d xs) = 1 *⇩R sum_list (take d xs)" by simp
also note pdevs_val_const_pdevs_of_list[symmetric]
also have "pdevs_val (λ_. 1) (pdevs_of_list (take d xs)) =
pdevs_val (λi. if i < d then 1 else 0) (pdevs_of_list xs)"
by (auto simp: pdevs_apply_pdevs_of_list split: if_split_asm intro!: pdevs_val_eqI)
finally show ?thesis .
qed
lemma zero_in_range_pdevs_apply[intro, simp]:
fixes X::"'a::real_vector pdevs" shows "0 ∈ range (pdevs_apply X)"
by (metis degree_gt less_irrefl rangeI)
lemma dense_list_in_range: "x ∈ set (dense_list_of_pdevs X) ⟹ x ∈ range (pdevs_apply X)"
by (auto simp: dense_list_of_pdevs_def)
lemma not_in_dense_list_zeroD:
assumes "pdevs_apply X i ∉ set (dense_list_of_pdevs X)"
shows "pdevs_apply X i = 0"
proof (rule ccontr)
assume "pdevs_apply X i ≠ 0"
hence "i < degree X"
by (rule degree_gt)
thus False using assms
by (auto simp: dense_list_of_pdevs_def)
qed
lemma list_all_list_of_pdevsI:
assumes "⋀i. i ∈ pdevs_domain X ⟹ P (pdevs_apply X i)"
shows "list_all (λx. P x) (map snd (list_of_pdevs X))"
using assms by (auto simp: list_all_iff list_of_pdevs_def)
lemma pdevs_of_list_map_scaleR:
"pdevs_of_list (map (scaleR r) xs) = scaleR_pdevs r (pdevs_of_list xs)"
by (auto intro!: pdevs_eqI simp: pdevs_apply_pdevs_of_list)
lemma
map_permI:
assumes "xs <~~> ys"
shows "map f xs <~~> map f ys"
using assms by induct auto
lemma rev_perm: "rev xs <~~> ys ⟷ xs <~~> ys"
by simp
lemma list_of_pdevs_perm_filter_nonzero:
"map snd (list_of_pdevs X) <~~> (filter ((≠) 0) (dense_list_of_pdevs X))"
proof -
have zip_map:
"zip [0..<degree X] (dense_list_of_pdevs X) = map (λi. (i, pdevs_apply X i)) [0..<degree X]"
by (auto simp: dense_list_of_pdevs_def intro!: nth_equalityI)
have "rev (list_of_pdevs X) <~~>
filter ((≠) 0 o snd) (zip [0..<degree X] (dense_list_of_pdevs X))"
by (auto simp: list_of_pdevs_eq o_def zip_map)
from map_permI[OF this, of snd]
have "map snd (list_of_pdevs X) <~~>
map snd (filter ((≠) 0 ∘ snd) (zip [0..<degree X] (dense_list_of_pdevs X)))"
by (simp add: rev_map[symmetric] rev_perm)
also have "map snd (filter ((≠) 0 ∘ snd) (zip [0..<degree X] (dense_list_of_pdevs X))) =
filter ((≠) 0) (dense_list_of_pdevs X)"
using map_filter[of snd "(≠) 0" "(zip [0..<degree X] (dense_list_of_pdevs X))"]
by (simp add: o_def dense_list_of_pdevs_def)
finally
show ?thesis .
qed
lemma pdevs_val_filter:
assumes mem: "e ∈ UNIV → I"
assumes "0 ∈ I"
obtains e' where
"pdevs_val e (pdevs_of_list (filter p xs)) = pdevs_val e' (pdevs_of_list xs)"
"e' ∈ UNIV → I"
unfolding pdevs_val_filter_pdevs_eval
proof -
have "(λ_::nat. 0) ∈ UNIV → I" using assms by simp
have "pdevs_val e (pdevs_of_list (filter p xs)) =
pdevs_val e (pdevs_of_list (filter p xs)) +
pdevs_val (λ_. 0) (pdevs_of_list (filter (Not o p) xs))"
by (simp add: pdevs_val_sum)
also
from pdevs_val_pdevs_of_list_append[OF ‹e ∈ _› ‹(λ_. 0) ∈ _›]
obtain e' where "e' ∈ UNIV → I"
"… = pdevs_val e' (pdevs_of_list (filter p xs @ filter (Not o p) xs))"
by metis
note this(2)
also
from pdevs_val_perm[OF partition_permI ‹e' ∈ _›]
obtain e'' where "… = pdevs_val e'' (pdevs_of_list xs)" "e'' ∈ UNIV → I" by metis
note this(1)
finally show ?thesis using ‹e'' ∈ _› ..
qed
lemma
pdevs_val_of_list_of_pdevs:
assumes "e ∈ UNIV → I"
assumes "0 ∈ I"
obtains e' where
"pdevs_val e (pdevs_of_list (map snd (list_of_pdevs X))) = pdevs_val e' X"
"e' ∈ UNIV → I"
proof -
obtain e' where "e' ∈ UNIV → I"
and "pdevs_val e (pdevs_of_list (map snd (list_of_pdevs X))) =
pdevs_val e' (pdevs_of_list (filter ((≠) 0) (dense_list_of_pdevs X)))"
by (rule pdevs_val_perm[OF list_of_pdevs_perm_filter_nonzero assms(1)])
note this(2)
also from pdevs_val_filter[OF ‹e' ∈ _› ‹0 ∈ I›, of "(≠) 0" "dense_list_of_pdevs X"]
obtain e'' where "e'' ∈ UNIV → I"
and "… = pdevs_val e'' (pdevs_of_list (dense_list_of_pdevs X))"
by metis
note this(2)
also have "… = pdevs_val e'' X" by (simp add: pdevs_of_list_dense_list_of_pdevs)
finally show ?thesis using ‹e'' ∈ UNIV → I› ..
qed
lemma
pdevs_val_of_list_of_pdevs2:
assumes "e ∈ UNIV → I"
obtains e' where
"pdevs_val e X = pdevs_val e' (pdevs_of_list (map snd (list_of_pdevs X)))"
"e' ∈ UNIV → I"
proof -
from list_of_pdevs_perm_filter_nonzero[of X]
have perm: "(filter ((≠) 0) (dense_list_of_pdevs X)) <~~> map snd (list_of_pdevs X)"
by simp
have "pdevs_val e X = pdevs_val e (pdevs_of_list (dense_list_of_pdevs X))"
by (simp add: pdevs_of_list_dense_list_of_pdevs)
also from pdevs_val_partition[OF ‹e ∈ _›, of "dense_list_of_pdevs X" "(≠) 0"]
obtain f g where "f ∈ UNIV → I" "g ∈ UNIV → I"
"… = pdevs_val f (pdevs_of_list (filter ((≠) 0) (dense_list_of_pdevs X))) +
pdevs_val g (pdevs_of_list (filter (Not ∘ (≠) 0) (dense_list_of_pdevs X)))"
(is "_ = ?f + ?g")
by metis
note this(3)
also
have "pdevs_of_list [x←dense_list_of_pdevs X . x = 0] = zero_pdevs"
by (auto intro!: pdevs_eqI simp: pdevs_apply_pdevs_of_list dest!: nth_mem)
hence "?g = 0" by (auto simp: o_def )
also
obtain e' where "e' ∈ UNIV → I"
and "?f = pdevs_val e' (pdevs_of_list (map snd (list_of_pdevs X)))"
by (rule pdevs_val_perm[OF perm ‹f ∈ _›])
note this(2)
finally show ?thesis using ‹e' ∈ UNIV → I› by (auto intro!: that)
qed
lemma dense_list_of_pdevs_scaleR:
"r ≠ 0 ⟹ map ((*⇩R) r) (dense_list_of_pdevs x) = dense_list_of_pdevs (scaleR_pdevs r x)"
by (auto simp: dense_list_of_pdevs_def)
lemma degree_pdevs_of_list_eq:
"(⋀x. x ∈ set xs ⟹ x ≠ 0) ⟹ degree (pdevs_of_list xs) = length xs"
by (cases xs) (auto simp add: pdevs_apply_pdevs_of_list nth_Cons
intro!: degree_eqI
split: nat.split)
lemma dense_list_of_pdevs_pdevs_of_list:
"(⋀x. x ∈ set xs ⟹ x ≠ 0) ⟹ dense_list_of_pdevs (pdevs_of_list xs) = xs"
by (auto simp: dense_list_of_pdevs_def degree_pdevs_of_list_eq pdevs_apply_pdevs_of_list
intro!: nth_equalityI)
lemma pdevs_of_list_sum:
assumes "distinct xs"
assumes "e ∈ UNIV → I"
obtains f where "f ∈ UNIV → I" "pdevs_val e (pdevs_of_list xs) = (∑P∈set xs. f P *⇩R P)"
proof -
define f where "f X = e (the (map_of (zip xs [0..<length xs]) X))" for X
from assms have "f ∈ UNIV → I"
by (auto simp: f_def)
moreover
have "pdevs_val e (pdevs_of_list xs) = (∑P∈set xs. f P *⇩R P)"
by (auto simp add: pdevs_val_zip f_def assms sum_list_distinct_conv_sum_set[symmetric]
in_set_zip map_of_zip_upto2_length_eq_nth
intro!: sum_list_nth_eqI)
ultimately show ?thesis ..
qed
lemma pdevs_domain_eq_pdevs_of_list:
assumes nz: "⋀x. x ∈ set (xs) ⟹ x ≠ 0"
shows "pdevs_domain (pdevs_of_list xs) = {0..<length xs}"
using nz
by (auto simp: pdevs_apply_pdevs_of_list split: if_split_asm)
lemma length_list_of_pdevs_pdevs_of_list:
assumes nz: "⋀x. x ∈ set xs ⟹ x ≠ 0"
shows "length (list_of_pdevs (pdevs_of_list xs)) = length xs"
using nz by (auto simp: list_of_pdevs_def pdevs_domain_eq_pdevs_of_list)
lemma nth_list_of_pdevs_pdevs_of_list:
assumes nz: "⋀x. x ∈ set xs ⟹ x ≠ 0"
assumes l: "n < length xs"
shows "list_of_pdevs (pdevs_of_list xs) ! n = ((length xs - Suc n), xs ! (length xs - Suc n))"
using nz l
by (auto simp: list_of_pdevs_def pdevs_domain_eq_pdevs_of_list rev_nth pdevs_apply_pdevs_of_list)
lemma list_of_pdevs_pdevs_of_list_eq:
"(⋀x. x ∈ set xs ⟹ x ≠ 0) ⟹
list_of_pdevs (pdevs_of_list xs) = zip (rev [0..<length xs]) (rev xs)"
by (auto simp: nth_list_of_pdevs_pdevs_of_list length_list_of_pdevs_pdevs_of_list rev_nth
intro!: nth_equalityI)
lemma sum_list_filter_list_of_pdevs_of_list:
fixes xs::"'a::comm_monoid_add list"
assumes "⋀x. x ∈ set xs ⟹ x ≠ 0"
shows "sum_list (filter p (map snd (list_of_pdevs (pdevs_of_list xs)))) = sum_list (filter p xs)"
using assms
by (auto simp: list_of_pdevs_pdevs_of_list_eq rev_filter[symmetric])
lemma
sum_list_partition:
fixes xs::"'a::comm_monoid_add list"
shows "sum_list (filter p xs) + sum_list (filter (Not o p) xs) = sum_list xs"
by (induct xs) (auto simp: ac_simps)
subsection ‹2d zonotopes›
definition "prod_of_pdevs x y = binop_pdevs Pair x y"
lemma apply_pdevs_prod_of_pdevs[simp]:
"pdevs_apply (prod_of_pdevs x y) i = (pdevs_apply x i, pdevs_apply y i)"
unfolding prod_of_pdevs_def
by (simp add: zero_prod_def)
lemma pdevs_domain_prod_of_pdevs[simp]:
"pdevs_domain (prod_of_pdevs x y) = pdevs_domain x ∪ pdevs_domain y"
by (auto simp: zero_prod_def)
lemma pdevs_val_prod_of_pdevs[simp]:
"pdevs_val e (prod_of_pdevs x y) = (pdevs_val e x, pdevs_val e y)"
proof -
have "pdevs_val e x = (∑i∈pdevs_domain x ∪ pdevs_domain y. e i *⇩R pdevs_apply x i)"
(is "_ = ?x")
unfolding pdevs_val_pdevs_domain
by (rule sum.mono_neutral_cong_left) auto
moreover have "pdevs_val e y = (∑i∈pdevs_domain x ∪ pdevs_domain y. e i *⇩R pdevs_apply y i)"
(is "_ = ?y")
unfolding pdevs_val_pdevs_domain
by (rule sum.mono_neutral_cong_left) auto
ultimately have "(pdevs_val e x, pdevs_val e y) = (?x, ?y)"
by auto
also have "… = pdevs_val e (prod_of_pdevs x y)"
by (simp add: sum_prod pdevs_val_pdevs_domain)
finally show ?thesis by simp
qed
definition prod_of_aforms (infixr "×⇩a" 80)
where "prod_of_aforms x y = ((fst x, fst y), prod_of_pdevs (snd x) (snd y))"
subsection ‹Intervals›
definition One_pdevs_raw::"nat ⇒ 'a::executable_euclidean_space"
where "One_pdevs_raw i = (if i < length (Basis_list::'a list) then Basis_list ! i else 0)"
lemma zeros_One_pdevs_raw:
"One_pdevs_raw -` {0::'a::executable_euclidean_space} = {length (Basis_list::'a list)..}"
by (auto simp: One_pdevs_raw_def nonzero_Basis split: if_split_asm dest!: nth_mem)
lemma nonzeros_One_pdevs_raw:
"{i. One_pdevs_raw i ≠ (0::'a::executable_euclidean_space)} = - {length (Basis_list::'a list)..}"
using zeros_One_pdevs_raw
by blast
lift_definition One_pdevs::"'a::executable_euclidean_space pdevs" is One_pdevs_raw
by (auto simp: nonzeros_One_pdevs_raw)
lemma pdevs_apply_One_pdevs[simp]: "pdevs_apply One_pdevs i =
(if i < length (Basis_list::'a::executable_euclidean_space list) then Basis_list ! i else 0::'a)"
by transfer (simp add: One_pdevs_raw_def)
lemma Max_Collect_less_nat: "Max {i::nat. i < k} = (if k = 0 then Max {} else k - 1)"
by (auto intro!: Max_eqI)
lemma degree_One_pdevs[simp]: "degree (One_pdevs::'a pdevs) =
length (Basis_list::'a::executable_euclidean_space list)"
by (auto simp: degree_eq_Suc_max Basis_list_nth_nonzero Max_Collect_less_nat
intro!: Max_eqI DIM_positive)
definition inner_scaleR_pdevs::"'a::euclidean_space ⇒ 'a pdevs ⇒ 'a pdevs"
where "inner_scaleR_pdevs b x = unop_pdevs (λx. (b ∙ x) *⇩R x) x"
lemma pdevs_apply_inner_scaleR_pdevs[simp]:
"pdevs_apply (inner_scaleR_pdevs a x) i = (a ∙ (pdevs_apply x i)) *⇩R (pdevs_apply x i)"
by (simp add: inner_scaleR_pdevs_def)
lemma degree_inner_scaleR_pdevs_le:
"degree (inner_scaleR_pdevs (l::'a::executable_euclidean_space) One_pdevs) ≤
degree (One_pdevs::'a pdevs)"
by (rule degree_leI) (auto simp: inner_scaleR_pdevs_def One_pdevs_raw_def)
definition "pdevs_of_ivl l u = scaleR_pdevs (1/2) (inner_scaleR_pdevs (u - l) One_pdevs)"
lemma degree_pdevs_of_ivl_le:
"degree (pdevs_of_ivl l u::'a::executable_euclidean_space pdevs) ≤ DIM('a)"
using degree_inner_scaleR_pdevs_le
by (simp add: pdevs_of_ivl_def)
lemma pdevs_apply_pdevs_of_ivl:
defines "B ≡ Basis_list::'a::executable_euclidean_space list"
shows "pdevs_apply (pdevs_of_ivl l u) i = (if i < length B then ((u - l)∙(B!i)/2)*⇩R(B!i) else 0)"
by (auto simp: pdevs_of_ivl_def B_def)
lemma deg_length_less_imp[simp]:
"k < degree (pdevs_of_ivl l u::'a::executable_euclidean_space pdevs) ⟹
k < length (Basis_list::'a list)"
by (metis (no_types, opaque_lifting) degree_One_pdevs degree_inner_scaleR_pdevs_le degree_scaleR_pdevs
dual_order.strict_trans length_Basis_list_pos nat_neq_iff not_le pdevs_of_ivl_def)
lemma tdev_pdevs_of_ivl: "tdev (pdevs_of_ivl l u) = ¦u - l¦ /⇩R 2"
proof -
have "tdev (pdevs_of_ivl l u) =
(∑i <degree (pdevs_of_ivl l u). ¦pdevs_apply (pdevs_of_ivl l u) i¦)"
by (auto simp: tdev_def)
also have "… = (∑i = 0..<length (Basis_list::'a list). ¦pdevs_apply (pdevs_of_ivl l u) i¦)"
using degree_pdevs_of_ivl_le[of l u]
by (intro sum.mono_neutral_cong_left) auto
also have "… = (∑i = 0..<length (Basis_list::'a list).
¦((u - l) ∙ Basis_list ! i / 2) *⇩R Basis_list ! i¦)"
by (auto simp: pdevs_apply_pdevs_of_ivl)
also have "… = (∑b ← Basis_list. ¦((u - l) ∙ b / 2) *⇩R b¦)"
by (auto simp: sum_list_sum_nth)
also have "… = (∑b∈Basis. ¦((u - l) ∙ b / 2) *⇩R b¦)"
by (auto simp: sum_list_distinct_conv_sum_set)
also have "… = ¦u - l¦ /⇩R 2"
by (subst euclidean_representation[symmetric, of "¦u - l¦ /⇩R 2"])
(simp add: abs_inner abs_scaleR)
finally show ?thesis .
qed
definition "aform_of_ivl l u = ((l + u)/⇩R2, pdevs_of_ivl l u)"
definition "aform_of_point x = aform_of_ivl x x"
lemma Elem_affine_of_ivl_le:
assumes "e ∈ UNIV → {-1 .. 1}"
assumes "l ≤ u"
shows "l ≤ aform_val e (aform_of_ivl l u)"
proof -
have "l = (1 / 2) *⇩R l + (1 / 2) *⇩R l"
by (simp add: scaleR_left_distrib[symmetric])
also have "… = (l + u)/⇩R2 - tdev (pdevs_of_ivl l u)"
by (auto simp: assms tdev_pdevs_of_ivl algebra_simps)
also have "… ≤ aform_val e (aform_of_ivl l u)"
using abs_pdevs_val_le_tdev[OF assms(1), of "pdevs_of_ivl l u"]
by (auto simp: aform_val_def aform_of_ivl_def minus_le_iff dest!: abs_le_D2)
finally show ?thesis .
qed
lemma Elem_affine_of_ivl_ge:
assumes "e ∈ UNIV → {-1 .. 1}"
assumes "l ≤ u"
shows "aform_val e (aform_of_ivl l u) ≤ u"
proof -
have "aform_val e (aform_of_ivl l u) ≤ (l + u)/⇩R2 + tdev (pdevs_of_ivl l u)"
using abs_pdevs_val_le_tdev[OF assms(1), of "pdevs_of_ivl l u"]
by (auto simp: aform_val_def aform_of_ivl_def minus_le_iff dest!: abs_le_D1)
also have "… = (1 / 2) *⇩R u + (1 / 2) *⇩R u"
by (auto simp: assms tdev_pdevs_of_ivl algebra_simps)
also have "… = u"
by (simp add: scaleR_left_distrib[symmetric])
finally show ?thesis .
qed
lemma
map_of_zip_upto_length_eq_nth:
assumes "i < length B"
assumes "d = length B"
shows "(map_of (zip [0..<d] B) i) = Some (B ! i)"
proof -
have "length [0..<length B] = length B"
by simp
from map_of_zip_is_Some[OF this, of i] assms
have "map_of (zip [0..<length B] B) i = Some (B ! i)"
by (auto simp: in_set_zip)
thus ?thesis by (simp add: assms)
qed
lemma in_ivl_affine_of_ivlE:
assumes "k ∈ {l .. u}"
obtains e where "e ∈ UNIV → {-1 .. 1}" "k = aform_val e (aform_of_ivl l u)"
proof atomize_elim
define e where [abs_def]: "e i = (let b = if i <length (Basis_list::'a list) then
(the (map_of (zip [0..<length (Basis_list::'a list)] (Basis_list::'a list)) i)) else 0 in
((k - (l + u) /⇩R 2) ∙ b) / (((u - l) /⇩R 2) ∙ b))" for i
let ?B = "Basis_list::'a list"
have "k = (1 / 2) *⇩R (l + u) +
(∑b ∈ Basis. (if (u - l) ∙ b = 0 then 0 else ((k - (1 / 2) *⇩R (l + u)) ∙ b)) *⇩R b)"
(is "_ = _ + ?dots")
using assms
by (force simp add: algebra_simps eucl_le[where 'a='a] intro!: euclidean_eqI[where 'a='a])
also have
"?dots = (∑b ∈ Basis. (if (u - l) ∙ b = 0 then 0 else ((k - (1 / 2) *⇩R (l + u)) ∙ b) *⇩R b))"
by (auto intro!: sum.cong)
also have "… = (∑b ← ?B. (if (u - l) ∙ b = 0 then 0 else ((k - (1 / 2) *⇩R (l + u)) ∙ b) *⇩R b))"
by (auto simp: sum_list_distinct_conv_sum_set)
also have "… =
(∑i = 0..<length ?B.
(if (u - l) ∙ ?B ! i = 0 then 0 else ((k - (1 / 2) *⇩R (l + u)) ∙ ?B ! i) *⇩R ?B ! i))"
by (auto simp: sum_list_sum_nth)
also have "… =
(∑i = 0..<degree (inner_scaleR_pdevs (u - l) One_pdevs).
(if (u - l) ∙ Basis_list ! i = 0 then 0
else ((k - (1 / 2) *⇩R (l + u)) ∙ Basis_list ! i) *⇩R Basis_list ! i))"
using degree_inner_scaleR_pdevs_le[of "u - l"]
by (intro sum.mono_neutral_cong_right) (auto dest!: degree)
also have "(1 / 2) *⇩R (l + u) +
(∑i = 0..<degree (inner_scaleR_pdevs (u - l) One_pdevs).
(if (u - l) ∙ Basis_list ! i = 0 then 0
else ((k - (1 / 2) *⇩R (l + u)) ∙ Basis_list ! i) *⇩R Basis_list ! i)) =
aform_val e (aform_of_ivl l u)"
using degree_inner_scaleR_pdevs_le[of "u - l"]
by (auto simp: aform_val_def aform_of_ivl_def pdevs_of_ivl_def map_of_zip_upto_length_eq_nth
e_def Let_def pdevs_val_sum
intro!: sum.cong)
finally have "k = aform_val e (aform_of_ivl l u)" .
moreover
{
fix k l u::real assume *: "l ≤ k" "k ≤ u"
let ?m = "l / 2 + u / 2"
have "¦k - ?m¦ ≤ ¦if k ≤ ?m then ?m - l else u - ?m¦"
using * by auto
also have "… ≤ ¦u / 2 - l / 2¦"
by (auto simp: abs_real_def)
finally have "¦k - (l / 2 + u / 2)¦ ≤ ¦u / 2 - l/2¦" .
} note midpoint_abs = this
have "e ∈ UNIV → {- 1..1}"
using assms
unfolding e_def Let_def
by (intro Pi_I divide_atLeastAtMost_1_absI)
(auto simp: map_of_zip_upto_length_eq_nth eucl_le[where 'a='a]
divide_le_eq_1 not_less inner_Basis algebra_simps intro!: midpoint_abs
dest!: nth_mem)
ultimately show "∃e. e ∈ UNIV → {- 1..1} ∧ k = aform_val e (aform_of_ivl l u)"
by blast
qed
lemma Inf_aform_aform_of_ivl:
assumes "l ≤ u"
shows "Inf_aform (aform_of_ivl l u) = l"
using assms
by (auto simp: Inf_aform_def aform_of_ivl_def tdev_pdevs_of_ivl abs_diff_eq1 algebra_simps)
(metis field_sum_of_halves scaleR_add_left scaleR_one)
lemma Sup_aform_aform_of_ivl:
assumes "l ≤ u"
shows "Sup_aform (aform_of_ivl l u) = u"
using assms
by (auto simp: Sup_aform_def aform_of_ivl_def tdev_pdevs_of_ivl abs_diff_eq1 algebra_simps)
(metis field_sum_of_halves scaleR_add_left scaleR_one)
lemma Affine_aform_of_ivl:
"a ≤ b ⟹ Affine (aform_of_ivl a b) = {a .. b}"
by (force simp: Affine_def valuate_def intro!: Elem_affine_of_ivl_ge Elem_affine_of_ivl_le
elim!: in_ivl_affine_of_ivlE)
end