Theory Metric_Arith

(*  Title:      HOL/Analysis/Metric_Arith.thy
    Author:     Maximilian Schäffeler (port from HOL Light)
*)

chapter ‹Functional Analysis›

section✐‹tag unimportant› ‹A decision procedure for metric spaces›

theory Metric_Arith
  imports HOL.Real_Vector_Spaces
begin

text✐‹tag unimportant› ‹
A port of the decision procedure ``Formalization of metric spaces in HOL Light''
\<^cite>‹"DBLP:journals/jar/Maggesi18"› for the type class @{class metric_space},
with the ‹Argo› solver as backend.
›

named_theorems metric_prenex
named_theorems metric_nnf
named_theorems metric_unfold
named_theorems metric_pre_arith

lemmas pre_arith_simps =
  max.bounded_iff max_less_iff_conj
  le_max_iff_disj less_max_iff_disj
  simp_thms HOL.eq_commute
declare pre_arith_simps [metric_pre_arith]

lemmas unfold_simps =
  Un_iff subset_iff disjoint_iff_not_equal
  Ball_def Bex_def
declare unfold_simps [metric_unfold]

declare HOL.nnf_simps(4) [metric_prenex]

lemma imp_prenex [metric_prenex]:
  "⋀P Q. (∃x. P x) ⟶ Q ≡ ∀x. (P x ⟶ Q)"
  "⋀P Q. P ⟶ (∃x. Q x) ≡ ∃x. (P ⟶ Q x)"
  "⋀P Q. (∀x. P x) ⟶ Q ≡ ∃x. (P x ⟶ Q)"
  "⋀P Q. P ⟶ (∀x. Q x) ≡ ∀x. (P ⟶ Q x)"
  by simp_all

lemma ex_prenex [metric_prenex]:
  "⋀P Q. (∃x. P x) ∧ Q ≡ ∃x. (P x ∧ Q)"
  "⋀P Q. P ∧ (∃x. Q x) ≡ ∃x. (P ∧ Q x)"
  "⋀P Q. (∃x. P x) ∨ Q ≡ ∃x. (P x ∨ Q)"
  "⋀P Q. P ∨ (∃x. Q x) ≡ ∃x. (P ∨ Q x)"
  "⋀P. ¬(∃x. P x) ≡ ∀x. ¬P x"
  by simp_all

lemma all_prenex [metric_prenex]:
  "⋀P Q. (∀x. P x) ∧ Q ≡ ∀x. (P x ∧ Q)"
  "⋀P Q. P ∧ (∀x. Q x) ≡ ∀x. (P ∧ Q x)"
  "⋀P Q. (∀x. P x) ∨ Q ≡ ∀x. (P x ∨ Q)"
  "⋀P Q. P ∨ (∀x. Q x) ≡ ∀x. (P ∨ Q x)"
  "⋀P. ¬(∀x. P x) ≡ ∃x. ¬P x"
  by simp_all

lemma nnf_thms [metric_nnf]:
  "(¬ (P ∧ Q)) = (¬ P ∨ ¬ Q)"
  "(¬ (P ∨ Q)) = (¬ P ∧ ¬ Q)"
  "(P ⟶ Q) = (¬ P ∨ Q)"
  "(P = Q) ⟷ (P ∨ ¬ Q) ∧ (¬ P ∨ Q)"
  "(¬ (P = Q)) ⟷ (¬ P ∨ ¬ Q) ∧ (P ∨ Q)"
  "(¬ ¬ P) = P"
  by blast+

lemmas nnf_simps = nnf_thms linorder_not_less linorder_not_le
declare nnf_simps[metric_nnf]

lemma ball_insert: "(∀x∈insert a B. P x) = (P a ∧ (∀x∈B. P x))"
  by blast

lemma Sup_insert_insert:
  fixes a::real
  shows "Sup (insert a (insert b s)) = Sup (insert (max a b) s)"
  by (simp add: Sup_real_def)

lemma real_abs_dist: "¦dist x y¦ = dist x y"
  by simp

lemma maxdist_thm [THEN HOL.eq_reflection]:
  assumes "finite s" "x ∈ s" "y ∈ s"
  defines "⋀a. f a ≡ ¦dist x a - dist a y¦"
  shows "dist x y = Sup (f ` s)"
proof -
  have "dist x y ≤ Sup (f ` s)"
  proof -
    have "finite (f ` s)"
      by (simp add: ‹finite s›)
    moreover have "¦dist x y - dist y y¦ ∈ f ` s"
      by (metis ‹y ∈ s› f_def imageI)
    ultimately show ?thesis
      using le_cSup_finite by simp
  qed
  also have "Sup (f ` s) ≤ dist x y"
    using ‹x ∈ s› cSUP_least[of s f] abs_dist_diff_le
    unfolding f_def
    by blast
  finally show ?thesis .
qed

theorem metric_eq_thm [THEN HOL.eq_reflection]:
  "x ∈ s ⟹ y ∈ s ⟹ x = y ⟷ (∀a∈s. dist x a = dist y a)"
  by auto

ML_file ‹metric_arith.ML›

method_setup metric = ‹
  Scan.succeed (SIMPLE_METHOD' o Metric_Arith.metric_arith_tac)
› "prove simple linear statements in metric spaces (∀∃⇩p fragment)"

end