Theory Lipschitz_Interval_Extension

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chapter‹Lipschitz Continuity of Intervals›

text‹An extension of of Lipschitz Continuity, developed for the Lipschitz Continuity of intervals.›

theory
  Lipschitz_Interval_Extension
  imports
  Inclusion_Isotonicity
  "HOL-Analysis.Lipschitz"
  Interval_Utilities
begin

section ‹Definition of Lipschitz Continuity on Intervals›

text ‹An interval extension F is said to be lipschitz in X if there is a constant L such that 
@{term ‹w(F(X)) ≤ Lw(X)›} for every @{term ‹X ⊆ X›}. Hence the width of 
@{term ‹F(X)›} approaches @{term ‹0›} at least linearly with the width of X.›

definition lipschitzI_on✐‹tag important› :: "'a::{zero,minus,preorder,times} ⇒ 'a interval set ⇒ ('a interval ⇒ 'a interval) ⇒ bool"
  where "lipschitzI_on C U F ⟷ (0 ≤ C ∧ (∀x ∈ U. width (F x) ≤ C * width x))"


text ‹The definition @{term "lipschitzI_on"} refers to definition 6.1 in\<^cite>‹"moore.ea:introduction:2009"››

bundle lipschitzI_syntax begin
notation✐‹tag important› lipschitzI_on ("_-lipschitzI'_on" [1000])
end
bundle no_lipschitzI_syntax begin
no_notation lipschitzI_on ("_-lipschitzI'_on" [1000])
end

unbundle lipschitzI_syntax

lemma lipschitzI_onI: "C-lipschitzI_on U F"
  if "⋀x . x ∈ U ⟹ width (F x) ≤ C * width x" "0 ≤ C"
  using that by (auto simp: lipschitzI_on_def)

lemma lipschitzI_onD:
  "width (F x) ≤ C * width x "
  if "C-lipschitzI_on U F" "x ∈ U"
  using that by (auto simp: lipschitzI_on_def)

lemma lipschitzI_on_nonneg:
  "0 ≤ C" if "C-lipschitzI_on U F"
  using that by (auto simp: lipschitzI_on_def)

lemma lipschitz_on_normD:
  "norm (width (F x)) ≤ C * norm (width x)"
  if "C-lipschitzI_on U F" "x ∈ U"
  using lipschitzI_onD[OF that]
  by (simp add: width_def dist_norm) 

lemma lipschitzI_on_mono: "L-lipschitzI_on D (F:: 'a::{linordered_ring} interval ⇒ 'a interval)" 
  if "M-lipschitzI_on E F" "M ≤ L" "D ⊆ E" 
  using that  diff_ge_0_iff_ge lower_le_upper 
   order_trans[OF _ mult_right_mono]
  unfolding lipschitzI_on_def width_def 
  by (metis (no_types, lifting) order_trans subsetD width_def)

lemmas lipschitzI_on_subset
 = lipschitzI_on_mono[OF _ _ order_refl]
  and lipschitzI_on_le = lipschitzI_on_mono[OF _ order_refl]

lemma lipschitzI_on_leI:
  "C-lipschitzI_on U F"
  if "⋀x. x ∈ U ⟹ width (F x) ≤ C * width x"
     "0 ≤ C"
  for F::"'a::{minus,preorder,times,zero} interval ⇒ 'a interval"
proof (rule lipschitzI_onI)
  fix x assume x: "x ∈ U"
  consider "upper x ≤ lower x" | "lower x ≤ upper x" 
    by simp
  then show "width (F x) ≤ C * width x"
  proof cases
    case 1
    then have "width (F x) ≤ C * width x"
      by (auto intro!: that x)
    then show ?thesis
      by (simp add: dist_commute)
  qed (auto intro!: that x)
qed fact

subsection‹Lipschitz Continuity of Operations›

text ‹Let F and G be inclusion isotonic interval extensions with @{term ‹F›} Lipschitz 
in @{term ‹Y›} and @{term ‹G›} Lipschitz in @{term ‹X›} 
and @{term ‹G(X) ⊆ Y›}. Then the composition @{term ‹H(X) = F(G(X))›} is 
Lipschitz in @{term ‹X›} and is inclusion isotonic›

lemma lipschitzI_on_compose:
  fixes U :: ‹'a::{linordered_ring} interval set›
  assumes f: "C-lipschitzI_on U F" and g: "D-lipschitzI_on (F`U) G"
  shows "(D * C)-lipschitzI_on U (G o F)"
proof (rule lipschitzI_onI)
  show "D * C ≥ 0" using lipschitzI_on_nonneg[OF f] lipschitzI_on_nonneg[OF g] by simp
  fix x assume H: "x ∈ U" 
  have "width (G(F x)) ≤ D * C * width x"
    using lipschitzI_onD[OF g] H assms 
    unfolding width_def lipschitzI_on_def apply simp  
    by (smt (verit, ccfv_SIG) mult.assoc mult_left_mono order_trans)
  then show "width ((G ∘ F) x) ≤ D * C * width x"
    unfolding comp_def by simp
qed


text ‹The definition @{thm "lipschitzI_on_compose"} refers to lemma 6.3 in
\<^cite>‹"moore.ea:introduction:2009"››

lemma lipschitzI_on_compose2:
  fixes U :: ‹'a::{linordered_ring} interval set›
  assumes F: "C-lipschitzI_on U F" and G: "D-lipschitzI_on (F`U) G"
  shows "(D * C)-lipschitzI_on U (λx. G (F x))"
  using lipschitzI_on_compose F G unfolding o_def by blast

lemma lipschitzI_on_cong:
  "C-lipschitzI_on U G ⟷ D-lipschitzI_on V F"
  if "C = D" "U = V" "⋀x. x ∈ V ⟹ G x = F x"
  using that by (auto simp: lipschitzI_on_def)

lemma lipschitzI_on_transform:
  "C-lipschitzI_on U G"
  if "C-lipschitzI_on U F"
    "⋀x. x ∈ U ⟹ G x = F x"
  using that
  unfolding lipschitzI_on_def width_def
  by simp

lemma lipschitz_on_empty_iff: "C-lipschitzI_on {} f ⟷ C ≥ 0"
  by (auto simp: lipschitzI_on_def)

lemma lipschitz_on_id: "(1::real)-lipschitzI_on U (λx. x)"
  by (auto simp: lipschitzI_on_def)

lemma lipschitz_on_constant: 
  assumes ‹lower c = upper c›
  shows "(0::real)-lipschitzI_on U (λx. c)"
  using assms by (auto simp: lipschitzI_on_def width_def)

lemma lipschitzI_on_mult: 
  fixes X :: "'a::{linordered_idom}"
  assumes "lipschitzI_on C U f"
  and "1 ≤ X"
shows "lipschitzI_on (X*C) U f"
  using assms interval_width_positive lower_le_upper mult_le_cancel_right1
  unfolding lipschitzI_on_def width_def  
 by (smt (verit) diff_ge_0_iff_ge linorder_not_le mult.assoc order_trans)  


subsection "Interval bounds on reals"

lemma bounded_image_real:
  fixes X :: "real interval" and f :: "real ⇒ real"
  assumes "∀x∈{lower X..upper X}. 
          f (lower X) - L * (upper X - lower X) ≤ f x ∧ f x ≤ f (lower X) + L * (upper X - lower X)"
  shows "∃x e. ∀y∈f ` {lower X..upper X}. dist x y ≤ e"
proof -
  let ?x = "f (lower X)"
  let ?e = "L * (upper X - lower X)"
  have "∀y∈f ` {lower X..upper X}. dist ?x y ≤ ?e"
  proof
    fix y assume "y ∈ f ` {lower X..upper X}"
    then obtain x where "x ∈ {lower X..upper X}" and "y = f x" by auto
    then have "?x - ?e ≤ y ∧ y ≤ ?x + ?e" using assms by auto
    then show "dist ?x y ≤ ?e" 
      unfolding dist_real_def by force 
  qed
  then show ?thesis by auto
qed

lemma lipschitz_bounded_image_real:
  fixes X :: "real set" and f :: "real ⇒ real"
  assumes "bounded X" and "L-lipschitz_on X f"
  shows "bounded (f ` X)"
  using assms(1) assms(2) bounded_uniformly_continuous_image lipschitz_on_uniformly_continuous by blast 

lemma inf_le_sup_image_real: 
  fixes X :: "real interval" and f :: "real ⇒ real"
  assumes "L-lipschitz_on (set_of X) f"
  shows  "Inf (f ` set_of X) ≤ Sup (f ` set_of X)"
proof -
  have "bounded (f ` set_of X)"
    using assms bounded_set_of lipschitz_bounded_image_real by blast 
  then have "bdd_below (f ` set_of X)" and "bdd_above (f ` set_of X)" 
    using bounded_imp_bdd_below bounded_imp_bdd_above by auto
  then have "Inf (f ` set_of X) ≤ Sup (f ` set_of X)"
    by (metis set_of_nonempty cInf_le_cSup empty_is_image) 
  then show ?thesis by simp
qed

lemma sup_image_le_real:
  fixes f :: "real ⇒ real" and F :: "real interval ⇒ real interval" and X :: "real interval"
  assumes "f ` set_of X ⊆ set_of (F X)"
    and "L-lipschitz_on (set_of X) f"
  shows "Sup (f ` set_of X) ≤ Sup (set_of (F X))"
proof -
  have a0: "bounded (f ` set_of X)" 
    using lipschitz_bounded_image_real[of "set_of X"] assms bounded_set_of[of "X"] by simp
  have a1: "bdd_above (f ` set_of X)" 
    using assms
    using a0 bounded_imp_bdd_above by presburger  
  then have a2: "∀y∈f ` set_of X. y ≤ Sup (f ` set_of X)" using bounded_has_Sup(1) a0
    by blast  
  then have a3: "Sup (f ` set_of X) ≤ Sup (set_of (F X))" 
    using set_of_nonempty assms a0 a1 a2 atLeastAtMost_iff closed_real_atLeastAtMost closed_subset_contains_Sup empty_is_image set_of_eq sup_set_of
    by metis 
  then show ?thesis by simp
qed

lemma inf_image_le_real:
  fixes f :: "real ⇒ real" and F :: "real interval ⇒ real interval" and X :: "real interval"
  assumes "f ` set_of X ⊆ set_of (F X)"
    and "L-lipschitz_on (set_of X) f"
  shows "Inf (set_of (F X)) ≤ Inf (f ` (set_of X))"
proof -
  have a0: "bounded (f ` set_of X)" 
    using lipschitz_bounded_image_real[of "set_of X"] assms bounded_set_of[of "X"] by simp
  have a1: "bdd_above (f ` set_of X)" 
    using assms 
    by (metis atLeastAtMost_iff bdd_above.unfold set_of_eq subset_eq)
  then have a2: "∀y∈f ` set_of X. y ≥ Inf (f ` set_of X)" 
    using bounded_has_Inf(1) a0
    by blast  
  then have a3: "Inf (set_of (F X)) ≤ Inf (f ` (set_of X))" using assms
    by (metis set_of_nonempty a0 bounded_imp_bdd_below closed_real_atLeastAtMost closed_subset_contains_Inf empty_is_image in_bounds inf_set_of set_of_eq)
  then show ?thesis by simp
qed

end