(*
 * Rational Preference Relations (RPRs).
 * (C)opyright Peter Gammie, peteg42 at gmail.com, commenced July 2006.
 * License: BSD
 *)

(*<*)
theory RPRs

imports FSext

begin
(*>*)

(* **************************************** *)

section‹Preliminaries›

text‹

  auxiliary concepts defined here are standard cite‹"Routley:79" and "Sen:70a" and "Taylor:2005"›. Throughout we make use of a fixed set @{term "A"} of
 , drawn from some arbitrary type @{typ "'a"} of suitable
 . Taylor cite‹"Taylor:2005"› terms this set an \emph{agenda}. Similarly
  have a type @{typ "'i"} of individuals and a population @{term "Is"}.

 ›

(* **************************************** *)

subsection‹Rational Preference Relations (RPRs)›

text‹

  for rational preference relations (RPRs), which represent
  or strict preference amongst some set of alternatives. These
  also called \emph{weak orders} or (ambiguously) \emph{ballots}.

  Isabelle's standard ordering operators and lemmas are
 -based, and as introducing new types is painful and we need several
  per type, we need to repeat some things.

 ›

type_synonym 'a RPR = "('a * 'a) set"

abbreviation rpr_eq_syntax :: "'a → 'a RPR → 'a → bool" (‹_ ⪯ _› [50, 1000, 51] 50) where
  "x ⪯ y == (x, y) ∈ r"

definition indifferent_pref :: "'a → 'a RPR → 'a → bool" (‹_ ≈ _› [50, 1000, 51] 50) where
  "x ≈ y ≡ (x ⪯ y ∧ y ⪯ x)"

lemma indifferent_prefI[intro]: "[ x ⪯ y; y ⪯ x ] ==> x ≈ y"
  unfolding indifferent_pref_def by simp

lemma indifferent_prefD[dest]: "x ≈ y ==> x ⪯ y ∧ y ⪯ x"
  unfolding indifferent_pref_def by simp

definition strict_pref :: "'a → 'a RPR → 'a → bool" (‹_ ≺ _› [50, 1000, 51] 50) where
  "x ≺ y ≡ (x ⪯ y ∧ ¬(y ⪯ x))"

lemma strict_pref_def_irrefl[simp]: "¬ (x ≺ x)" unfolding strict_pref_def by blast

lemma strict_prefI[intro]: "[ x ⪯ y; ¬(y ⪯ x) ] ==> x ≺ y"
  unfolding strict_pref_def by simp

text‹

 , @{term "x ⪯ y"} would be written $x\ R\ y$,
 {term "x ≈ y"} as $x\ I\ y$ and @{term "x
 ≺ y"} as $x\ P\ y$, where the relation $r$ is implicit, and
  are indexed by subscripting.

 ›

text‹

 emph{Complete} means that every pair of distinct alternatives is
 . The "distinct" part is a matter of taste, as it makes sense to
  an alternative as as good as itself. Here I take reflexivity
 .

 ›

definition complete :: "'a set → 'a RPR → bool" where
  "complete A r ≡ (∀x ∈ A. ∀y ∈ A - {x}. x _{<esub>\eqy<or> y <bsbr<^eu><rcqx"}

lemma completeI[intro]:    java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 79
  "(∧x y. [ x ∈ A; y ∈ A; x ≠ y ] ==> x ⪯ y ∨ y ⪯ x) ==> complete A r"
  unfolding complete_def by auto

lemma completeD[dest]:
  "[ complete A r; x ∈ A; y ∈ A; x ≠ y ] ==> x ⪯ y ∨ y ⪯ x"
  unfolding complete_def by auto

lemma complete_less_not: "[ complete A r; hasw [x,y] A; ¬ x ≺ y ] ==> y ⪯ x"
  unfolding complete_def strict_pref_def by auto

lemma complete_indiff_not: "[ complete A r; hasw [x,y] A; ¬ x ≈ y ] ==> x ≺ y ∨ y ≺ x"
  unfolding complete_def indifferent_pref_def strict_pref_def by auto

lemma complete_exh:
  assumes "complete A r"
      and "hasw [x,y] A"
  obtains (xPy) "x ≺ y"
    | (yPx) "y ≺ x"
    | (xIy) "x ≈ y"
  using assms unfolding complete_def strict_pref_def indifferent_pref_def by auto

text‹

  the standard @{term "refl"}. Also define \emph{irreflexivity}
  to how @{term "refl"} is defined in the standard library.

 ›

declare refl_onI[intro] refl_onD[dest]

lemma complete_refl_on:
  "[ complete A r; refl_on A r; x ∈ A; y ∈ A ] ==> x ⪯ y ∨ y ⪯ x"
  unfolding complete_def by auto

definition irrefl :: "'a set → 'a RPR → bool" where
  "irrefl A r ≡ r ⊆ A × A ∧ (∀x ∈ A. ¬ x ⪯ x)"

lemma irreflI[intro]: "[ r ⊆ A × A; ∧x. x ∈ A ==> ¬ x ⪯ x ] ==> irrefl A r"
  unfolding irrefl_def by simp

lemma irreflD[dest]: "[ irrefl A r; (x, y) ∈ r ] ==> hasw [x,y] A"
  unfolding irrefl_def by auto

lemma irreflD'[dest]:
  "[ irrefl A r; r ≠ {} ] ==> ∃x y. hasw [x,y] A ∧ (x, y) ∈ r"
  unfolding irrefl_def by auto

text‹Rational preference relations, also known as weak orders and (I
 ) complete pre-orders.›

definition rpr :: "'a set → 'a RPR → bool" where
  "rpr A r ≡ complete A r ∧ r ⊆ A × A ∧ refl_on A r ∧ trans r"

lemma rprI[intro]: "[ complete A r; r ⊆ A × A; refl_on A r; trans r ] ==> rpr A r"
  unfolding rpr_def by simp

lemma rprD: "rpr A r ==> complete A r ∧ refl_on A r ∧ trans r"
  unfolding rpr_def by simp

lemma rpr_in_set[dest]: "[ rpr A r; x ⪯ y ] ==> {x,y} ⊆ A"
  unfolding rpr_def refl_on_def by auto

lemma rpr_refl[dest]: "[ rpr A r; x ∈ A ] ==> x ⪯ x"
  unfolding rpr_def by blast

lemma rpr_less_not: "[ rpr A r; hasw [x,y] A; ¬ x ≺ y ] ==> y ⪯ x"
  unfolding rpr_def by (auto simp add: complete_less_not)

lemma rpr_less_imp_le[simp]: "[ x ≺ y ] ==> x ⪯ y"
  unfolding strict_pref_def by simp

lemma rpr_less_imp_neq[simp]: "[ x ≺ y ] ==> x ≠ y"
  unfolding strict_pref_def by blast

lemma rpr_less_trans[trans]: "[ x ≺ y; y ≺ z; rpr A r ] ==> x ≺ z"
  unfolding rpr_def strict_pref_def trans_def by blast

lemma rpr_le_trans[trans]: "[ x ⪯ y; y ⪯ z; rpr A r ] ==> x ⪯ z"
                  (_cket 🚫now_serving ≤ 🚫now_serving ≠ 🚫now_serving = 🚫now_serving := 🚫myticket i < \🚫now_serving = 1 ∧ 🚫myticket = (λj. 0) }

lemma rpr_le_less_trans[trans]: "[ x ⪯ y; y ≺ z; rpr A r ] ==> x ≺ z"
  unfolding rpr_def strict_pref_def trans_def by blast

lemma rpr_less_le_trans[trans]: "[ x ≺ y; y ⪯ z; rpr A r ] ==> x ≺ z"
  unfolding rpr_def strict_pref_def trans_def by blast

lemma rpr_complete: "[ rpr A r; x ∈ A; y ∈ A ] ==> x ⪯ y ∨ y ⪯ x"
  unfolding rpr_def by (blast dest: complete_refl_on)

(* **************************************** *)

subsection‹Profiles›

text ‹A \emph{profile} (also termed a collection of \emph{ballots}) maps
  individual to an RPR for that individual.›

type_synonym ('a, 'i) Profile = "'i → 'a RPR"

definition profile :: "'a set → 'i set → ('a, 'i) Profile → bool" where
  "profile A Is P ≡ Is ≠ {} ∧ (∀i ∈ Is. rpr A (P i))"

lemma profileI[intro]: "[ ∧i. i ∈ Is ==> rpr A (P i); Is ≠ {} ] ==> profile A Is P"
  unfolding profile_def by simp

lemma profile_rprD[dest]: "[ profile A Is P; i ∈ Is ] ==> rpr A (P i)"
  unfolding profile_def by simp

lemma profile_non_empty: "profile A Is P ==> Is ≠ {}"
  unfolding profile_def by simp

(*<*)
end
(*>*)