Theory Relation

(*  Title:      HOL/Relation.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Author:     Stefan Berghofer, TU Muenchen
*)

section ‹Relations -- as sets of pairs, and binary predicates›

theory Relation
  imports Finite_Set
begin

text ‹A preliminary: classical rules for reasoning on predicates›

declare predicate1I [Pure.intro!, intro!]
declare predicate1D [Pure.dest, dest]
declare predicate2I [Pure.intro!, intro!]
declare predicate2D [Pure.dest, dest]
declare bot1E [elim!]
declare bot2E [elim!]
declare top1I [intro!]
declare top2I [intro!]
declare inf1I [intro!]
declare inf2I [intro!]
declare inf1E [elim!]
declare inf2E [elim!]
declare sup1I1 [intro?]
declare sup2I1 [intro?]
declare sup1I2 [intro?]
declare sup2I2 [intro?]
declare sup1E [elim!]
declare sup2E [elim!]
declare sup1CI [intro!]
declare sup2CI [intro!]
declare Inf1_I [intro!]
declare INF1_I [intro!]
declare Inf2_I [intro!]
declare INF2_I [intro!]
declare Inf1_D [elim]
declare INF1_D [elim]
declare Inf2_D [elim]
declare INF2_D [elim]
declare Inf1_E [elim]
declare INF1_E [elim]
declare Inf2_E [elim]
declare INF2_E [elim]
declare Sup1_I [intro]
declare SUP1_I [intro]
declare Sup2_I [intro]
declare SUP2_I [intro]
declare Sup1_E [elim!]
declare SUP1_E [elim!]
declare Sup2_E [elim!]
declare SUP2_E [elim!]


subsection ‹Fundamental›

subsubsection ‹Relations as sets of pairs›

type_synonym 'a rel = "('a × 'a) set"

lemma subrelI: "(⋀x y. (x, y) ∈ r ⟹ (x, y) ∈ s) ⟹ r ⊆ s"
  ― ‹Version of @{thm [source] subsetI} for binary relations›
  by auto

lemma lfp_induct2:
  "(a, b) ∈ lfp f ⟹ mono f ⟹
    (⋀a b. (a, b) ∈ f (lfp f ∩ {(x, y). P x y}) ⟹ P a b) ⟹ P a b"
  ― ‹Version of @{thm [source] lfp_induct} for binary relations›
  using lfp_induct_set [of "(a, b)" f "case_prod P"] by auto


subsubsection ‹Conversions between set and predicate relations›

lemma pred_equals_eq [pred_set_conv]: "(λx. x ∈ R) = (λx. x ∈ S) ⟷ R = S"
  by (simp add: set_eq_iff fun_eq_iff)

lemma pred_equals_eq2 [pred_set_conv]: "(λx y. (x, y) ∈ R) = (λx y. (x, y) ∈ S) ⟷ R = S"
  by (simp add: set_eq_iff fun_eq_iff)

lemma pred_subset_eq [pred_set_conv]: "(λx. x ∈ R) ≤ (λx. x ∈ S) ⟷ R ⊆ S"
  by (simp add: subset_iff le_fun_def)

lemma pred_subset_eq2 [pred_set_conv]: "(λx y. (x, y) ∈ R) ≤ (λx y. (x, y) ∈ S) ⟷ R ⊆ S"
  by (simp add: subset_iff le_fun_def)

lemma bot_empty_eq [pred_set_conv]: "⊥ = (λx. x ∈ {})"
  by (auto simp add: fun_eq_iff)

lemma bot_empty_eq2 [pred_set_conv]: "⊥ = (λx y. (x, y) ∈ {})"
  by (auto simp add: fun_eq_iff)

lemma top_empty_eq [pred_set_conv]: "⊤ = (λx. x ∈ UNIV)"
  by (auto simp add: fun_eq_iff)

lemma top_empty_eq2 [pred_set_conv]: "⊤ = (λx y. (x, y) ∈ UNIV)"
  by (auto simp add: fun_eq_iff)

lemma inf_Int_eq [pred_set_conv]: "(λx. x ∈ R) ⊓ (λx. x ∈ S) = (λx. x ∈ R ∩ S)"
  by (simp add: inf_fun_def)

lemma inf_Int_eq2 [pred_set_conv]: "(λx y. (x, y) ∈ R) ⊓ (λx y. (x, y) ∈ S) = (λx y. (x, y) ∈ R ∩ S)"
  by (simp add: inf_fun_def)

lemma sup_Un_eq [pred_set_conv]: "(λx. x ∈ R) ⊔ (λx. x ∈ S) = (λx. x ∈ R ∪ S)"
  by (simp add: sup_fun_def)

lemma sup_Un_eq2 [pred_set_conv]: "(λx y. (x, y) ∈ R) ⊔ (λx y. (x, y) ∈ S) = (λx y. (x, y) ∈ R ∪ S)"
  by (simp add: sup_fun_def)

lemma INF_INT_eq [pred_set_conv]: "(⨅i∈S. (λx. x ∈ r i)) = (λx. x ∈ (⋂i∈S. r i))"
  by (simp add: fun_eq_iff)

lemma INF_INT_eq2 [pred_set_conv]: "(⨅i∈S. (λx y. (x, y) ∈ r i)) = (λx y. (x, y) ∈ (⋂i∈S. r i))"
  by (simp add: fun_eq_iff)

lemma SUP_UN_eq [pred_set_conv]: "(⨆i∈S. (λx. x ∈ r i)) = (λx. x ∈ (⋃i∈S. r i))"
  by (simp add: fun_eq_iff)

lemma SUP_UN_eq2 [pred_set_conv]: "(⨆i∈S. (λx y. (x, y) ∈ r i)) = (λx y. (x, y) ∈ (⋃i∈S. r i))"
  by (simp add: fun_eq_iff)

lemma Inf_INT_eq [pred_set_conv]: "⨅S = (λx. x ∈ (⋂(Collect ` S)))"
  by (simp add: fun_eq_iff)

lemma INF_Int_eq [pred_set_conv]: "(⨅i∈S. (λx. x ∈ i)) = (λx. x ∈ ⋂S)"
  by (simp add: fun_eq_iff)

lemma Inf_INT_eq2 [pred_set_conv]: "⨅S = (λx y. (x, y) ∈ (⋂(Collect ` case_prod ` S)))"
  by (simp add: fun_eq_iff)

lemma INF_Int_eq2 [pred_set_conv]: "(⨅i∈S. (λx y. (x, y) ∈ i)) = (λx y. (x, y) ∈ ⋂S)"
  by (simp add: fun_eq_iff)

lemma Sup_SUP_eq [pred_set_conv]: "⨆S = (λx. x ∈ ⋃(Collect ` S))"
  by (simp add: fun_eq_iff)

lemma SUP_Sup_eq [pred_set_conv]: "(⨆i∈S. (λx. x ∈ i)) = (λx. x ∈ ⋃S)"
  by (simp add: fun_eq_iff)

lemma Sup_SUP_eq2 [pred_set_conv]: "⨆S = (λx y. (x, y) ∈ (⋃(Collect ` case_prod ` S)))"
  by (simp add: fun_eq_iff)

lemma SUP_Sup_eq2 [pred_set_conv]: "(⨆i∈S. (λx y. (x, y) ∈ i)) = (λx y. (x, y) ∈ ⋃S)"
  by (simp add: fun_eq_iff)


subsection ‹Properties of relations›

subsubsection ‹Reflexivity›

definition refl_on :: "'a set ⇒ 'a rel ⇒ bool"
  where "refl_on A r ⟷ r ⊆ A × A ∧ (∀x∈A. (x, x) ∈ r)"

abbreviation refl :: "'a rel ⇒ bool" ― ‹reflexivity over a type›
  where "refl ≡ refl_on UNIV"

definition reflp_on :: "'a set ⇒ ('a ⇒ 'a ⇒ bool) ⇒ bool"
  where "reflp_on A R ⟷ (∀x∈A. R x x)"

abbreviation reflp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
  where "reflp ≡ reflp_on UNIV"

lemma reflp_def[no_atp]: "reflp R ⟷ (∀x. R x x)"
  by (simp add: reflp_on_def)

text ‹@{thm [source] reflp_def} is for backward compatibility.›

lemma reflp_refl_eq [pred_set_conv]: "reflp (λx y. (x, y) ∈ r) ⟷ refl r"
  by (simp add: refl_on_def reflp_def)

lemma refl_onI [intro?]: "r ⊆ A × A ⟹ (⋀x. x ∈ A ⟹ (x, x) ∈ r) ⟹ refl_on A r"
  unfolding refl_on_def by (iprover intro!: ballI)

lemma reflp_onI:
  "(⋀x y. x ∈ A ⟹ R x x) ⟹ reflp_on A R"
  by (simp add: reflp_on_def)

lemma reflpI[intro?]: "(⋀x. R x x) ⟹ reflp R"
  by (rule reflp_onI)

lemma refl_onD: "refl_on A r ⟹ a ∈ A ⟹ (a, a) ∈ r"
  unfolding refl_on_def by blast

lemma refl_onD1: "refl_on A r ⟹ (x, y) ∈ r ⟹ x ∈ A"
  unfolding refl_on_def by blast

lemma refl_onD2: "refl_on A r ⟹ (x, y) ∈ r ⟹ y ∈ A"
  unfolding refl_on_def by blast

lemma reflp_onD:
  "reflp_on A R ⟹ x ∈ A ⟹ R x x"
  by (simp add: reflp_on_def)

lemma reflpD[dest?]: "reflp R ⟹ R x x"
  by (simp add: reflp_onD)

lemma reflpE:
  assumes "reflp r"
  obtains "r x x"
  using assms by (auto dest: refl_onD simp add: reflp_def)

lemma reflp_on_subset: "reflp_on A R ⟹ B ⊆ A ⟹ reflp_on B R"
  by (auto intro: reflp_onI dest: reflp_onD)

lemma refl_on_Int: "refl_on A r ⟹ refl_on B s ⟹ refl_on (A ∩ B) (r ∩ s)"
  unfolding refl_on_def by blast

lemma reflp_on_inf: "reflp_on A R ⟹ reflp_on B S ⟹ reflp_on (A ∩ B) (R ⊓ S)"
  by (auto intro: reflp_onI dest: reflp_onD)

lemma reflp_inf: "reflp r ⟹ reflp s ⟹ reflp (r ⊓ s)"
  by (rule reflp_on_inf[of UNIV _ UNIV, unfolded Int_absorb])

lemma refl_on_Un: "refl_on A r ⟹ refl_on B s ⟹ refl_on (A ∪ B) (r ∪ s)"
  unfolding refl_on_def by blast

lemma reflp_on_sup: "reflp_on A R ⟹ reflp_on B S ⟹ reflp_on (A ∪ B) (R ⊔ S)"
  by (auto intro: reflp_onI dest: reflp_onD)

lemma reflp_sup: "reflp r ⟹ reflp s ⟹ reflp (r ⊔ s)"
  by (rule reflp_on_sup[of UNIV _ UNIV, unfolded Un_absorb])

lemma refl_on_INTER: "∀x∈S. refl_on (A x) (r x) ⟹ refl_on (⋂(A ` S)) (⋂(r ` S))"
  unfolding refl_on_def by fast

lemma reflp_on_Inf: "∀x∈S. reflp_on (A x) (R x) ⟹ reflp_on (⋂(A ` S)) (⨅(R ` S))"
  by (auto intro: reflp_onI dest: reflp_onD)

lemma refl_on_UNION: "∀x∈S. refl_on (A x) (r x) ⟹ refl_on (⋃(A ` S)) (⋃(r ` S))"
  unfolding refl_on_def by blast

lemma reflp_on_Sup: "∀x∈S. reflp_on (A x) (R x) ⟹ reflp_on (⋃(A ` S)) (⨆(R ` S))"
  by (auto intro: reflp_onI dest: reflp_onD)

lemma refl_on_empty [simp]: "refl_on {} {}"
  by (simp add: refl_on_def)

lemma reflp_on_empty [simp]: "reflp_on {} R"
  by (auto intro: reflp_onI)

lemma refl_on_singleton [simp]: "refl_on {x} {(x, x)}"
by (blast intro: refl_onI)

lemma refl_on_def' [nitpick_unfold, code]:
  "refl_on A r ⟷ (∀(x, y) ∈ r. x ∈ A ∧ y ∈ A) ∧ (∀x ∈ A. (x, x) ∈ r)"
  by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)

lemma reflp_equality [simp]: "reflp (=)"
  by (simp add: reflp_def)

lemma reflp_on_mono:
  "reflp_on A R ⟹ (⋀x y. x ∈ A ⟹ y ∈ A ⟹ R x y ⟹ Q x y) ⟹ reflp_on A Q"
  by (auto intro: reflp_onI dest: reflp_onD)

lemma reflp_mono: "reflp R ⟹ (⋀x y. R x y ⟹ Q x y) ⟹ reflp Q"
  by (rule reflp_on_mono[of UNIV R Q]) simp_all


subsubsection ‹Irreflexivity›

definition irrefl :: "'a rel ⇒ bool"
  where "irrefl r ⟷ (∀a. (a, a) ∉ r)"

definition irreflp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
  where "irreflp R ⟷ (∀a. ¬ R a a)"

lemma irreflp_irrefl_eq [pred_set_conv]: "irreflp (λa b. (a, b) ∈ R) ⟷ irrefl R"
  by (simp add: irrefl_def irreflp_def)

lemma irreflI [intro?]: "(⋀a. (a, a) ∉ R) ⟹ irrefl R"
  by (simp add: irrefl_def)

lemma irreflpI [intro?]: "(⋀a. ¬ R a a) ⟹ irreflp R"
  by (fact irreflI [to_pred])

lemma irrefl_distinct [code]: "irrefl r ⟷ (∀(a, b) ∈ r. a ≠ b)"
  by (auto simp add: irrefl_def)

lemma (in preorder) irreflp_less[simp]: "irreflp (<)"
  by (simp add: irreflpI)

lemma (in preorder) irreflp_greater[simp]: "irreflp (>)"
  by (simp add: irreflpI)

subsubsection ‹Asymmetry›

inductive asym :: "'a rel ⇒ bool"
  where asymI: "(⋀a b. (a, b) ∈ R ⟹ (b, a) ∉ R) ⟹ asym R"

inductive asymp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
  where asympI: "(⋀a b. R a b ⟹ ¬ R b a) ⟹ asymp R"

lemma asymp_asym_eq [pred_set_conv]: "asymp (λa b. (a, b) ∈ R) ⟷ asym R"
  by (auto intro!: asymI asympI elim: asym.cases asymp.cases simp add: irreflp_irrefl_eq)

lemma asymD: "⟦asym R; (x,y) ∈ R⟧ ⟹ (y,x) ∉ R"
  by (simp add: asym.simps)

lemma asympD: "asymp R ⟹ R x y ⟹ ¬ R y x"
  by (rule asymD[to_pred])

lemma asym_iff: "asym R ⟷ (∀x y. (x,y) ∈ R ⟶ (y,x) ∉ R)"
  by (blast intro: asymI dest: asymD)

lemma (in preorder) asymp_less[simp]: "asymp (<)"
  by (auto intro: asympI dual_order.asym)

lemma (in preorder) asymp_greater[simp]: "asymp (>)"
  by (auto intro: asympI dual_order.asym)


subsubsection ‹Symmetry›

definition sym :: "'a rel ⇒ bool"
  where "sym r ⟷ (∀x y. (x, y) ∈ r ⟶ (y, x) ∈ r)"

definition symp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
  where "symp r ⟷ (∀x y. r x y ⟶ r y x)"

lemma symp_sym_eq [pred_set_conv]: "symp (λx y. (x, y) ∈ r) ⟷ sym r"
  by (simp add: sym_def symp_def)

lemma symI [intro?]: "(⋀a b. (a, b) ∈ r ⟹ (b, a) ∈ r) ⟹ sym r"
  by (unfold sym_def) iprover

lemma sympI [intro?]: "(⋀a b. r a b ⟹ r b a) ⟹ symp r"
  by (fact symI [to_pred])

lemma symE:
  assumes "sym r" and "(b, a) ∈ r"
  obtains "(a, b) ∈ r"
  using assms by (simp add: sym_def)

lemma sympE:
  assumes "symp r" and "r b a"
  obtains "r a b"
  using assms by (rule symE [to_pred])

lemma symD [dest?]:
  assumes "sym r" and "(b, a) ∈ r"
  shows "(a, b) ∈ r"
  using assms by (rule symE)

lemma sympD [dest?]:
  assumes "symp r" and "r b a"
  shows "r a b"
  using assms by (rule symD [to_pred])

lemma sym_Int: "sym r ⟹ sym s ⟹ sym (r ∩ s)"
  by (fast intro: symI elim: symE)

lemma symp_inf: "symp r ⟹ symp s ⟹ symp (r ⊓ s)"
  by (fact sym_Int [to_pred])

lemma sym_Un: "sym r ⟹ sym s ⟹ sym (r ∪ s)"
  by (fast intro: symI elim: symE)

lemma symp_sup: "symp r ⟹ symp s ⟹ symp (r ⊔ s)"
  by (fact sym_Un [to_pred])

lemma sym_INTER: "∀x∈S. sym (r x) ⟹ sym (⋂(r ` S))"
  by (fast intro: symI elim: symE)

lemma symp_INF: "∀x∈S. symp (r x) ⟹ symp (⨅(r ` S))"
  by (fact sym_INTER [to_pred])

lemma sym_UNION: "∀x∈S. sym (r x) ⟹ sym (⋃(r ` S))"
  by (fast intro: symI elim: symE)

lemma symp_SUP: "∀x∈S. symp (r x) ⟹ symp (⨆(r ` S))"
  by (fact sym_UNION [to_pred])


subsubsection ‹Antisymmetry›

definition antisym :: "'a rel ⇒ bool"
  where "antisym r ⟷ (∀x y. (x, y) ∈ r ⟶ (y, x) ∈ r ⟶ x = y)"

definition antisymp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
  where "antisymp r ⟷ (∀x y. r x y ⟶ r y x ⟶ x = y)"

lemma antisymp_antisym_eq [pred_set_conv]: "antisymp (λx y. (x, y) ∈ r) ⟷ antisym r"
  by (simp add: antisym_def antisymp_def)

lemma antisymI [intro?]:
  "(⋀x y. (x, y) ∈ r ⟹ (y, x) ∈ r ⟹ x = y) ⟹ antisym r"
  unfolding antisym_def by iprover

lemma antisympI [intro?]:
  "(⋀x y. r x y ⟹ r y x ⟹ x = y) ⟹ antisymp r"
  by (fact antisymI [to_pred])
    
lemma antisymD [dest?]:
  "antisym r ⟹ (a, b) ∈ r ⟹ (b, a) ∈ r ⟹ a = b"
  unfolding antisym_def by iprover

lemma antisympD [dest?]:
  "antisymp r ⟹ r a b ⟹ r b a ⟹ a = b"
  by (fact antisymD [to_pred])

lemma antisym_subset:
  "r ⊆ s ⟹ antisym s ⟹ antisym r"
  unfolding antisym_def by blast

lemma antisymp_less_eq:
  "r ≤ s ⟹ antisymp s ⟹ antisymp r"
  by (fact antisym_subset [to_pred])
    
lemma antisym_empty [simp]:
  "antisym {}"
  unfolding antisym_def by blast

lemma antisym_bot [simp]:
  "antisymp ⊥"
  by (fact antisym_empty [to_pred])
    
lemma antisymp_equality [simp]:
  "antisymp HOL.eq"
  by (auto intro: antisympI)

lemma antisym_singleton [simp]:
  "antisym {x}"
  by (blast intro: antisymI)


subsubsection ‹Transitivity›

definition trans :: "'a rel ⇒ bool"
  where "trans r ⟷ (∀x y z. (x, y) ∈ r ⟶ (y, z) ∈ r ⟶ (x, z) ∈ r)"

definition transp :: "('a ⇒ 'a ⇒ bool) ⇒ bool"
  where "transp r ⟷ (∀x y z. r x y ⟶ r y z ⟶ r x z)"

lemma transp_trans_eq [pred_set_conv]: "transp (λx y. (x, y) ∈ r) ⟷ trans r"
  by (simp add: trans_def transp_def)

lemma transI [intro?]: "(⋀x y z. (x, y) ∈ r ⟹ (y, z) ∈ r ⟹ (x, z) ∈ r) ⟹ trans r"
  by (unfold trans_def) iprover

lemma transpI [intro?]: "(⋀x y z. r x y ⟹ r y z ⟹ r x z) ⟹ transp r"
  by (fact transI [to_pred])

lemma transE:
  assumes "trans r" and "(x, y) ∈ r" and "(y, z) ∈ r"
  obtains "(x, z) ∈ r"
  using assms by (unfold trans_def) iprover

lemma transpE:
  assumes "transp r" and "r x y" and "r y z"
  obtains "r x z"
  using assms by (rule transE [to_pred])

lemma transD [dest?]:
  assumes "trans r" and "(x, y) ∈ r" and "(y, z) ∈ r"
  shows "(x, z) ∈ r"
  using assms by (rule transE)

lemma transpD [dest?]:
  assumes "transp r" and "r x y" and "r y z"
  shows "r x z"
  using assms by (rule transD [to_pred])

lemma trans_Int: "trans r ⟹ trans s ⟹ trans (r ∩ s)"
  by (fast intro: transI elim: transE)

lemma transp_inf: "transp r ⟹ transp s ⟹ transp (r ⊓ s)"
  by (fact trans_Int [to_pred])

lemma trans_INTER: "∀x∈S. trans (r x) ⟹ trans (⋂(r ` S))"
  by (fast intro: transI elim: transD)

lemma transp_INF: "∀x∈S. transp (r x) ⟹ transp (⨅(r ` S))"
  by (fact trans_INTER [to_pred])
    
lemma trans_join [code]: "trans r ⟷ (∀(x, y1) ∈ r. ∀(y2, z) ∈ r. y1 = y2 ⟶ (x, z) ∈ r)"
  by (auto simp add: trans_def)

lemma transp_trans: "transp r ⟷ trans {(x, y). r x y}"
  by (simp add: trans_def transp_def)

lemma transp_equality [simp]: "transp (=)"
  by (auto intro: transpI)

lemma trans_empty [simp]: "trans {}"
  by (blast intro: transI)

lemma transp_empty [simp]: "transp (λx y. False)"
  using trans_empty[to_pred] by (simp add: bot_fun_def)

lemma trans_singleton [simp]: "trans {(a, a)}"
  by (blast intro: transI)

lemma transp_singleton [simp]: "transp (λx y. x = a ∧ y = a)"
  by (simp add: transp_def)

context preorder
begin

lemma transp_le[simp]: "transp (≤)"
by(auto simp add: transp_def intro: order_trans)

lemma transp_less[simp]: "transp (<)"
by(auto simp add: transp_def intro: less_trans)

lemma transp_ge[simp]: "transp (≥)"
by(auto simp add: transp_def intro: order_trans)

lemma transp_gr[simp]: "transp (>)"
by(auto simp add: transp_def intro: less_trans)

end

subsubsection ‹Totality›

definition total_on :: "'a set ⇒ 'a rel ⇒ bool"
  where "total_on A r ⟷ (∀x∈A. ∀y∈A. x ≠ y ⟶ (x, y) ∈ r ∨ (y, x) ∈ r)"

lemma total_onI [intro?]:
  "(⋀x y. ⟦x ∈ A; y ∈ A; x ≠ y⟧ ⟹ (x, y) ∈ r ∨ (y, x) ∈ r) ⟹ total_on A r"
  unfolding total_on_def by blast

abbreviation "total ≡ total_on UNIV"

definition totalp_on where
  "totalp_on A R ⟷ (∀x ∈ A. ∀y ∈ A. x ≠ y ⟶ R x y ∨ R y x)"

abbreviation totalp where
  "totalp ≡ totalp_on UNIV"

lemma totalp_on_refl_on_eq[pred_set_conv]: "totalp_on A (λx y. (x, y) ∈ r) ⟷ total_on A r"
  by (simp add: totalp_on_def total_on_def)

lemma totalp_onI:
  "(⋀x y. x ∈ A ⟹ y ∈ A ⟹ x ≠ y ⟹ R x y ∨ R y x) ⟹ totalp_on A R"
  by (simp add: totalp_on_def)

lemma totalpI: "(⋀x y. x ≠ y ⟹ R x y ∨ R y x) ⟹ totalp R"
  by (rule totalp_onI)

lemma totalp_onD:
  "totalp_on A R ⟹ x ∈ A ⟹ y ∈ A ⟹ x ≠ y ⟹ R x y ∨ R y x"
  by (simp add: totalp_on_def)

lemma totalpD: "totalp R ⟹ x ≠ y ⟹ R x y ∨ R y x"
  by (simp add: totalp_onD)

lemma total_on_subset: "total_on A r ⟹ B ⊆ A ⟹ total_on B r"
  by (auto simp: total_on_def)

lemma totalp_on_subset: "totalp_on A R ⟹ B ⊆ A ⟹ totalp_on B R"
  by (auto intro: totalp_onI dest: totalp_onD)

lemma total_on_empty [simp]: "total_on {} r"
  by (simp add: total_on_def)

lemma totalp_on_empty [simp]: "totalp_on {} R"
  by (auto intro: totalp_onI)

lemma total_on_singleton [simp]: "total_on {x} {(x, x)}"
  unfolding total_on_def by blast


subsubsection ‹Single valued relations›

definition single_valued :: "('a × 'b) set ⇒ bool"
  where "single_valued r ⟷ (∀x y. (x, y) ∈ r ⟶ (∀z. (x, z) ∈ r ⟶ y = z))"

definition single_valuedp :: "('a ⇒ 'b ⇒ bool) ⇒ bool"
  where "single_valuedp r ⟷ (∀x y. r x y ⟶ (∀z. r x z ⟶ y = z))"

lemma single_valuedp_single_valued_eq [pred_set_conv]:
  "single_valuedp (λx y. (x, y) ∈ r) ⟷ single_valued r"
  by (simp add: single_valued_def single_valuedp_def)

lemma single_valuedp_iff_Uniq:
  "single_valuedp r ⟷ (∀x. ∃⇩≤⇩1y. r x y)"
  unfolding Uniq_def single_valuedp_def by auto

lemma single_valuedI:
  "(⋀x y. (x, y) ∈ r ⟹ (⋀z. (x, z) ∈ r ⟹ y = z)) ⟹ single_valued r"
  unfolding single_valued_def by blast

lemma single_valuedpI:
  "(⋀x y. r x y ⟹ (⋀z. r x z ⟹ y = z)) ⟹ single_valuedp r"
  by (fact single_valuedI [to_pred])

lemma single_valuedD:
  "single_valued r ⟹ (x, y) ∈ r ⟹ (x, z) ∈ r ⟹ y = z"
  by (simp add: single_valued_def)

lemma single_valuedpD:
  "single_valuedp r ⟹ r x y ⟹ r x z ⟹ y = z"
  by (fact single_valuedD [to_pred])

lemma single_valued_empty [simp]:
  "single_valued {}"
  by (simp add: single_valued_def)

lemma single_valuedp_bot [simp]:
  "single_valuedp ⊥"
  by (fact single_valued_empty [to_pred])

lemma single_valued_subset:
  "r ⊆ s ⟹ single_valued s ⟹ single_valued r"
  unfolding single_valued_def by blast

lemma single_valuedp_less_eq:
  "r ≤ s ⟹ single_valuedp s ⟹ single_valuedp r"
  by (fact single_valued_subset [to_pred])


subsection ‹Relation operations›

subsubsection ‹The identity relation›

definition Id :: "'a rel"
  where "Id = {p. ∃x. p = (x, x)}"

lemma IdI [intro]: "(a, a) ∈ Id"
  by (simp add: Id_def)

lemma IdE [elim!]: "p ∈ Id ⟹ (⋀x. p = (x, x) ⟹ P) ⟹ P"
  unfolding Id_def by (iprover elim: CollectE)

lemma pair_in_Id_conv [iff]: "(a, b) ∈ Id ⟷ a = b"
  unfolding Id_def by blast

lemma refl_Id: "refl Id"
  by (simp add: refl_on_def)

lemma antisym_Id: "antisym Id"
  ― ‹A strange result, since ‹Id› is also symmetric.›
  by (simp add: antisym_def)

lemma sym_Id: "sym Id"
  by (simp add: sym_def)

lemma trans_Id: "trans Id"
  by (simp add: trans_def)

lemma single_valued_Id [simp]: "single_valued Id"
  by (unfold single_valued_def) blast

lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
  by (simp add: irrefl_def)

lemma trans_diff_Id: "trans r ⟹ antisym r ⟹ trans (r - Id)"
  unfolding antisym_def trans_def by blast

lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
  by (simp add: total_on_def)

lemma Id_fstsnd_eq: "Id = {x. fst x = snd x}"
  by force


subsubsection ‹Diagonal: identity over a set›

definition Id_on :: "'a set ⇒ 'a rel"
  where "Id_on A = (⋃x∈A. {(x, x)})"

lemma Id_on_empty [simp]: "Id_on {} = {}"
  by (simp add: Id_on_def)

lemma Id_on_eqI: "a = b ⟹ a ∈ A ⟹ (a, b) ∈ Id_on A"
  by (simp add: Id_on_def)

lemma Id_onI [intro!]: "a ∈ A ⟹ (a, a) ∈ Id_on A"
  by (rule Id_on_eqI) (rule refl)

lemma Id_onE [elim!]: "c ∈ Id_on A ⟹ (⋀x. x ∈ A ⟹ c = (x, x) ⟹ P) ⟹ P"
  ― ‹The general elimination rule.›
  unfolding Id_on_def by (iprover elim!: UN_E singletonE)

lemma Id_on_iff: "(x, y) ∈ Id_on A ⟷ x = y ∧ x ∈ A"
  by blast

lemma Id_on_def' [nitpick_unfold]: "Id_on {x. A x} = Collect (λ(x, y). x = y ∧ A x)"
  by auto

lemma Id_on_subset_Times: "Id_on A ⊆ A × A"
  by blast

lemma refl_on_Id_on: "refl_on A (Id_on A)"
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])

lemma antisym_Id_on [simp]: "antisym (Id_on A)"
  unfolding antisym_def by blast

lemma sym_Id_on [simp]: "sym (Id_on A)"
  by (rule symI) clarify

lemma trans_Id_on [simp]: "trans (Id_on A)"
  by (fast intro: transI elim: transD)

lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
  unfolding single_valued_def by blast


subsubsection ‹Composition›

inductive_set relcomp  :: "('a × 'b) set ⇒ ('b × 'c) set ⇒ ('a × 'c) set"  (infixr "O" 75)
  for r :: "('a × 'b) set" and s :: "('b × 'c) set"
  where relcompI [intro]: "(a, b) ∈ r ⟹ (b, c) ∈ s ⟹ (a, c) ∈ r O s"

notation relcompp (infixr "OO" 75)

lemmas relcomppI = relcompp.intros

text ‹
  For historic reasons, the elimination rules are not wholly corresponding.
  Feel free to consolidate this.
›

inductive_cases relcompEpair: "(a, c) ∈ r O s"
inductive_cases relcomppE [elim!]: "(r OO s) a c"

lemma relcompE [elim!]: "xz ∈ r O s ⟹
  (⋀x y z. xz = (x, z) ⟹ (x, y) ∈ r ⟹ (y, z) ∈ s  ⟹ P) ⟹ P"
  apply (cases xz)
  apply simp
  apply (erule relcompEpair)
  apply iprover
  done

lemma R_O_Id [simp]: "R O Id = R"
  by fast

lemma Id_O_R [simp]: "Id O R = R"
  by fast

lemma relcomp_empty1 [simp]: "{} O R = {}"
  by blast

lemma relcompp_bot1 [simp]: "⊥ OO R = ⊥"
  by (fact relcomp_empty1 [to_pred])

lemma relcomp_empty2 [simp]: "R O {} = {}"
  by blast

lemma relcompp_bot2 [simp]: "R OO ⊥ = ⊥"
  by (fact relcomp_empty2 [to_pred])

lemma O_assoc: "(R O S) O T = R O (S O T)"
  by blast

lemma relcompp_assoc: "(r OO s) OO t = r OO (s OO t)"
  by (fact O_assoc [to_pred])

lemma trans_O_subset: "trans r ⟹ r O r ⊆ r"
  by (unfold trans_def) blast

lemma transp_relcompp_less_eq: "transp r ⟹ r OO r ≤ r "
  by (fact trans_O_subset [to_pred])

lemma relcomp_mono: "r' ⊆ r ⟹ s' ⊆ s ⟹ r' O s' ⊆ r O s"
  by blast

lemma relcompp_mono: "r' ≤ r ⟹ s' ≤ s ⟹ r' OO s' ≤ r OO s "
  by (fact relcomp_mono [to_pred])

lemma relcomp_subset_Sigma: "r ⊆ A × B ⟹ s ⊆ B × C ⟹ r O s ⊆ A × C"
  by blast

lemma relcomp_distrib [simp]: "R O (S ∪ T) = (R O S) ∪ (R O T)"
  by auto

lemma relcompp_distrib [simp]: "R OO (S ⊔ T) = R OO S ⊔ R OO T"
  by (fact relcomp_distrib [to_pred])

lemma relcomp_distrib2 [simp]: "(S ∪ T) O R = (S O R) ∪ (T O R)"
  by auto

lemma relcompp_distrib2 [simp]: "(S ⊔ T) OO R = S OO R ⊔ T OO R"
  by (fact relcomp_distrib2 [to_pred])

lemma relcomp_UNION_distrib: "s O ⋃(r ` I) = (⋃i∈I. s O r i) "
  by auto

lemma relcompp_SUP_distrib: "s OO ⨆(r ` I) = (⨆i∈I. s OO r i)"
  by (fact relcomp_UNION_distrib [to_pred])
    
lemma relcomp_UNION_distrib2: "⋃(r ` I) O s = (⋃i∈I. r i O s) "
  by auto

lemma relcompp_SUP_distrib2: "⨆(r ` I) OO s = (⨆i∈I. r i OO s)"
  by (fact relcomp_UNION_distrib2 [to_pred])
    
lemma single_valued_relcomp: "single_valued r ⟹ single_valued s ⟹ single_valued (r O s)"
  unfolding single_valued_def by blast

lemma relcomp_unfold: "r O s = {(x, z). ∃y. (x, y) ∈ r ∧ (y, z) ∈ s}"
  by (auto simp add: set_eq_iff)

lemma relcompp_apply: "(R OO S) a c ⟷ (∃b. R a b ∧ S b c)"
  unfolding relcomp_unfold [to_pred] ..

lemma eq_OO: "(=) OO R = R"
  by blast

lemma OO_eq: "R OO (=) = R"
  by blast


subsubsection ‹Converse›

inductive_set converse :: "('a × 'b) set ⇒ ('b × 'a) set"  ("(_¯)" [1000] 999)
  for r :: "('a × 'b) set"
  where "(a, b) ∈ r ⟹ (b, a) ∈ r¯"

notation conversep  ("(_¯¯)" [1000] 1000)

notation (ASCII)
  converse  ("(_^-1)" [1000] 999) and
  conversep ("(_^--1)" [1000] 1000)

lemma converseI [sym]: "(a, b) ∈ r ⟹ (b, a) ∈ r¯"
  by (fact converse.intros)

lemma conversepI (* CANDIDATE [sym] *): "r a b ⟹ r¯¯ b a"
  by (fact conversep.intros)

lemma converseD [sym]: "(a, b) ∈ r¯ ⟹ (b, a) ∈ r"
  by (erule converse.cases) iprover

lemma conversepD (* CANDIDATE [sym] *): "r¯¯ b a ⟹ r a b"
  by (fact converseD [to_pred])

lemma converseE [elim!]: "yx ∈ r¯ ⟹ (⋀x y. yx = (y, x) ⟹ (x, y) ∈ r ⟹ P) ⟹ P"
  ― ‹More general than ‹converseD›, as it ``splits'' the member of the relation.›
  apply (cases yx)
  apply simp
  apply (erule converse.cases)
  apply iprover
  done

lemmas conversepE [elim!] = conversep.cases

lemma converse_iff [iff]: "(a, b) ∈ r¯ ⟷ (b, a) ∈ r"
  by (auto intro: converseI)

lemma conversep_iff [iff]: "r¯¯ a b = r b a"
  by (fact converse_iff [to_pred])

lemma converse_converse [simp]: "(r¯)¯ = r"
  by (simp add: set_eq_iff)

lemma conversep_conversep [simp]: "(r¯¯)¯¯ = r"
  by (fact converse_converse [to_pred])

lemma converse_empty[simp]: "{}¯ = {}"
  by auto

lemma converse_UNIV[simp]: "UNIV¯ = UNIV"
  by auto

lemma converse_relcomp: "(r O s)¯ = s¯ O r¯"
  by blast

lemma converse_relcompp: "(r OO s)¯¯ = s¯¯ OO r¯¯"
  by (iprover intro: order_antisym conversepI relcomppI elim: relcomppE dest: conversepD)

lemma converse_Int: "(r ∩ s)¯ = r¯ ∩ s¯"
  by blast

lemma converse_meet: "(r ⊓ s)¯¯ = r¯¯ ⊓ s¯¯"
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)

lemma converse_Un: "(r ∪ s)¯ = r¯ ∪ s¯"
  by blast

lemma converse_join: "(r ⊔ s)¯¯ = r¯¯ ⊔ s¯¯"
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)

lemma converse_INTER: "(⋂(r ` S))¯ = (⋂x∈S. (r x)¯)"
  by fast

lemma converse_UNION: "(⋃(r ` S))¯ = (⋃x∈S. (r x)¯)"
  by blast

lemma converse_mono[simp]: "r¯ ⊆ s ¯ ⟷ r ⊆ s"
  by auto

lemma conversep_mono[simp]: "r¯¯ ≤ s ¯¯ ⟷ r ≤ s"
  by (fact converse_mono[to_pred])

lemma converse_inject[simp]: "r¯ = s ¯ ⟷ r = s"
  by auto

lemma conversep_inject[simp]: "r¯¯ = s ¯¯ ⟷ r = s"
  by (fact converse_inject[to_pred])

lemma converse_subset_swap: "r ⊆ s ¯ ⟷ r ¯ ⊆ s"
  by auto

lemma conversep_le_swap: "r ≤ s ¯¯ ⟷ r ¯¯ ≤ s"
  by (fact converse_subset_swap[to_pred])

lemma converse_Id [simp]: "Id¯ = Id"
  by blast

lemma converse_Id_on [simp]: "(Id_on A)¯ = Id_on A"
  by blast

lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
  by (auto simp: refl_on_def)

lemma sym_converse [simp]: "sym (converse r) = sym r"
  unfolding sym_def by blast

lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
  unfolding antisym_def by blast

lemma trans_converse [simp]: "trans (converse r) = trans r"
  unfolding trans_def by blast

lemma sym_conv_converse_eq: "sym r ⟷ r¯ = r"
  unfolding sym_def by fast

lemma sym_Un_converse: "sym (r ∪ r¯)"
  unfolding sym_def by blast

lemma sym_Int_converse: "sym (r ∩ r¯)"
  unfolding sym_def by blast

lemma total_on_converse [simp]: "total_on A (r¯) = total_on A r"
  by (auto simp: total_on_def)

lemma finite_converse [iff]: "finite (r¯) = finite r"
unfolding converse_def conversep_iff using [[simproc add: finite_Collect]]
by (auto elim: finite_imageD simp: inj_on_def)

lemma card_inverse[simp]: "card (R¯) = card R"
proof -
  have *: "⋀R. prod.swap ` R = R¯" by auto
  {
    assume "¬finite R"
    hence ?thesis
      by auto
  } moreover {
    assume "finite R"
    with card_image_le[of R prod.swap] card_image_le[of "R¯" prod.swap]
    have ?thesis by (auto simp: *)
  } ultimately show ?thesis by blast
qed  

lemma conversep_noteq [simp]: "(≠)¯¯ = (≠)"
  by (auto simp add: fun_eq_iff)

lemma conversep_eq [simp]: "(=)¯¯ = (=)"
  by (auto simp add: fun_eq_iff)

lemma converse_unfold [code]: "r¯ = {(y, x). (x, y) ∈ r}"
  by (simp add: set_eq_iff)


subsubsection ‹Domain, range and field›

inductive_set Domain :: "('a × 'b) set ⇒ 'a set" for r :: "('a × 'b) set"
  where DomainI [intro]: "(a, b) ∈ r ⟹ a ∈ Domain r"

lemmas DomainPI = Domainp.DomainI

inductive_cases DomainE [elim!]: "a ∈ Domain r"
inductive_cases DomainpE [elim!]: "Domainp r a"

inductive_set Range :: "('a × 'b) set ⇒ 'b set" for r :: "('a × 'b) set"
  where RangeI [intro]: "(a, b) ∈ r ⟹ b ∈ Range r"

lemmas RangePI = Rangep.RangeI

inductive_cases RangeE [elim!]: "b ∈ Range r"
inductive_cases RangepE [elim!]: "Rangep r b"

definition Field :: "'a rel ⇒ 'a set"
  where "Field r = Domain r ∪ Range r"

lemma FieldI1: "(i, j) ∈ R ⟹ i ∈ Field R"
  unfolding Field_def by blast

lemma FieldI2: "(i, j) ∈ R ⟹ j ∈ Field R"
  unfolding Field_def by auto

lemma Domain_fst [code]: "Domain r = fst ` r"
  by force

lemma Range_snd [code]: "Range r = snd ` r"
  by force

lemma fst_eq_Domain: "fst ` R = Domain R"
  by force

lemma snd_eq_Range: "snd ` R = Range R"
  by force

lemma range_fst [simp]: "range fst = UNIV"
  by (auto simp: fst_eq_Domain)

lemma range_snd [simp]: "range snd = UNIV"
  by (auto simp: snd_eq_Range)

lemma Domain_empty [simp]: "Domain {} = {}"
  by auto

lemma Range_empty [simp]: "Range {} = {}"
  by auto

lemma Field_empty [simp]: "Field {} = {}"
  by (simp add: Field_def)

lemma Domain_empty_iff: "Domain r = {} ⟷ r = {}"
  by auto

lemma Range_empty_iff: "Range r = {} ⟷ r = {}"
  by auto

lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
  by blast

lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
  by blast

lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} ∪ Field r"
  by (auto simp add: Field_def)

lemma Domain_iff: "a ∈ Domain r ⟷ (∃y. (a, y) ∈ r)"
  by blast

lemma Range_iff: "a ∈ Range r ⟷ (∃y. (y, a) ∈ r)"
  by blast

lemma Domain_Id [simp]: "Domain Id = UNIV"
  by blast

lemma Range_Id [simp]: "Range Id = UNIV"
  by blast

lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
  by blast

lemma Range_Id_on [simp]: "Range (Id_on A) = A"
  by blast

lemma Domain_Un_eq: "Domain (A ∪ B) = Domain A ∪ Domain B"
  by blast

lemma Range_Un_eq: "Range (A ∪ B) = Range A ∪ Range B"
  by blast

lemma Field_Un [simp]: "Field (r ∪ s) = Field r ∪ Field s"
  by (auto simp: Field_def)

lemma Domain_Int_subset: "Domain (A ∩ B) ⊆ Domain A ∩ Domain B"
  by blast

lemma Range_Int_subset: "Range (A ∩ B) ⊆ Range A ∩ Range B"
  by blast

lemma Domain_Diff_subset: "Domain A - Domain B ⊆ Domain (A - B)"
  by blast

lemma Range_Diff_subset: "Range A - Range B ⊆ Range (A - B)"
  by blast

lemma Domain_Union: "Domain (⋃S) = (⋃A∈S. Domain A)"
  by blast

lemma Range_Union: "Range (⋃S) = (⋃A∈S. Range A)"
  by blast

lemma Field_Union [simp]: "Field (⋃R) = ⋃(Field ` R)"
  by (auto simp: Field_def)

lemma Domain_converse [simp]: "Domain (r¯) = Range r"
  by auto

lemma Range_converse [simp]: "Range (r¯) = Domain r"
  by blast

lemma Field_converse [simp]: "Field (r¯) = Field r"
  by (auto simp: Field_def)

lemma Domain_Collect_case_prod [simp]: "Domain {(x, y). P x y} = {x. ∃y. P x y}"
  by auto

lemma Range_Collect_case_prod [simp]: "Range {(x, y). P x y} = {y. ∃x. P x y}"
  by auto

lemma finite_Domain: "finite r ⟹ finite (Domain r)"
  by (induct set: finite) auto

lemma finite_Range: "finite r ⟹ finite (Range r)"
  by (induct set: finite) auto

lemma finite_Field: "finite r ⟹ finite (Field r)"
  by (simp add: Field_def finite_Domain finite_Range)

lemma Domain_mono: "r ⊆ s ⟹ Domain r ⊆ Domain s"
  by blast

lemma Range_mono: "r ⊆ s ⟹ Range r ⊆ Range s"
  by blast

lemma mono_Field: "r ⊆ s ⟹ Field r ⊆ Field s"
  by (auto simp: Field_def Domain_def Range_def)

lemma Domain_unfold: "Domain r = {x. ∃y. (x, y) ∈ r}"
  by blast

lemma Field_square [simp]: "Field (x × x) = x"
  unfolding Field_def by blast


subsubsection ‹Image of a set under a relation›

definition Image :: "('a × 'b) set ⇒ 'a set ⇒ 'b set"  (infixr "``" 90)
  where "r `` s = {y. ∃x∈s. (x, y) ∈ r}"

lemma Image_iff: "b ∈ r``A ⟷ (∃x∈A. (x, b) ∈ r)"
  by (simp add: Image_def)

lemma Image_singleton: "r``{a} = {b. (a, b) ∈ r}"
  by (simp add: Image_def)

lemma Image_singleton_iff [iff]: "b ∈ r``{a} ⟷ (a, b) ∈ r"
  by (rule Image_iff [THEN trans]) simp

lemma ImageI [intro]: "(a, b) ∈ r ⟹ a ∈ A ⟹ b ∈ r``A"
  unfolding Image_def by blast

lemma ImageE [elim!]: "b ∈ r `` A ⟹ (⋀x. (x, b) ∈ r ⟹ x ∈ A ⟹ P) ⟹ P"
  unfolding Image_def by (iprover elim!: CollectE bexE)

lemma rev_ImageI: "a ∈ A ⟹ (a, b) ∈ r ⟹ b ∈ r `` A"
  ― ‹This version's more effective when we already have the required ‹a››
  by blast

lemma Image_empty1 [simp]: "{} `` X = {}"
by auto

lemma Image_empty2 [simp]: "R``{} = {}"
by blast

lemma Image_Id [simp]: "Id `` A = A"
  by blast

lemma Image_Id_on [simp]: "Id_on A `` B = A ∩ B"
  by blast

lemma Image_Int_subset: "R `` (A ∩ B) ⊆ R `` A ∩ R `` B"
  by blast

lemma Image_Int_eq: "single_valued (converse R) ⟹ R `` (A ∩ B) = R `` A ∩ R `` B"
  by (auto simp: single_valued_def)

lemma Image_Un: "R `` (A ∪ B) = R `` A ∪ R `` B"
  by blast

lemma Un_Image: "(R ∪ S) `` A = R `` A ∪ S `` A"
  by blast

lemma Image_subset: "r ⊆ A × B ⟹ r``C ⊆ B"
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)

lemma Image_eq_UN: "r``B = (⋃y∈ B. r``{y})"
  ― ‹NOT suitable for rewriting›
  by blast

lemma Image_mono: "r' ⊆ r ⟹ A' ⊆ A ⟹ (r' `` A') ⊆ (r `` A)"
  by blast

lemma Image_UN: "r `` (⋃(B ` A)) = (⋃x∈A. r `` (B x))"
  by blast

lemma UN_Image: "(⋃i∈I. X i) `` S = (⋃i∈I. X i `` S)"
  by auto

lemma Image_INT_subset: "(r `` (⋂(B ` A))) ⊆ (⋂x∈A. r `` (B x))"
  by blast

text ‹Converse inclusion requires some assumptions›
lemma Image_INT_eq:
  assumes "single_valued (r¯)"
    and "A ≠ {}"
  shows "r `` (⋂(B ` A)) = (⋂x∈A. r `` B x)"
proof(rule equalityI, rule Image_INT_subset)
  show "(⋂x∈A. r `` B x) ⊆ r `` ⋂ (B ` A)"
  proof
    fix x
    assume "x ∈ (⋂x∈A. r `` B x)"
    then show "x ∈ r `` ⋂ (B ` A)"
      using assms unfolding single_valued_def by simp blast
  qed
qed

lemma Image_subset_eq: "r``A ⊆ B ⟷ A ⊆ - ((r¯) `` (- B))"
  by blast

lemma Image_Collect_case_prod [simp]: "{(x, y). P x y} `` A = {y. ∃x∈A. P x y}"
  by auto

lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (⋃x∈X ∩ A. B x)"
  by auto

lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)"
  by auto

lemma finite_Image[simp]: assumes "finite R" shows "finite (R `` A)"
by(rule finite_subset[OF _ finite_Range[OF assms]]) auto


subsubsection ‹Inverse image›

definition inv_image :: "'b rel ⇒ ('a ⇒ 'b) ⇒ 'a rel"
  where "inv_image r f = {(x, y). (f x, f y) ∈ r}"

definition inv_imagep :: "('b ⇒ 'b ⇒ bool) ⇒ ('a ⇒ 'b) ⇒ 'a ⇒ 'a ⇒ bool"
  where "inv_imagep r f = (λx y. r (f x) (f y))"

lemma [pred_set_conv]: "inv_imagep (λx y. (x, y) ∈ r) f = (λx y. (x, y) ∈ inv_image r f)"
  by (simp add: inv_image_def inv_imagep_def)

lemma sym_inv_image: "sym r ⟹ sym (inv_image r f)"
  unfolding sym_def inv_image_def by blast

lemma trans_inv_image: "trans r ⟹ trans (inv_image r f)"
  unfolding trans_def inv_image_def
  by (simp (no_asm)) blast

lemma total_inv_image: "⟦inj f; total r⟧ ⟹ total (inv_image r f)"
  unfolding inv_image_def total_on_def by (auto simp: inj_eq)

lemma asym_inv_image: "asym R ⟹ asym (inv_image R f)"
  by (simp add: inv_image_def asym_iff)

lemma in_inv_image[simp]: "(x, y) ∈ inv_image r f ⟷ (f x, f y) ∈ r"
  by (auto simp: inv_image_def)

lemma converse_inv_image[simp]: "(inv_image R f)¯ = inv_image (R¯) f"
  unfolding inv_image_def converse_unfold by auto

lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
  by (simp add: inv_imagep_def)


subsubsection ‹Powerset›

definition Powp :: "('a ⇒ bool) ⇒ 'a set ⇒ bool"
  where "Powp A = (λB. ∀x ∈ B. A x)"

lemma Powp_Pow_eq [pred_set_conv]: "Powp (λx. x ∈ A) = (λx. x ∈ Pow A)"
  by (auto simp add: Powp_def fun_eq_iff)

lemmas Powp_mono [mono] = Pow_mono [to_pred]


subsubsection ‹Expressing relation operations via \<^const>‹Finite_Set.fold››

lemma Id_on_fold:
  assumes "finite A"
  shows "Id_on A = Finite_Set.fold (λx. Set.insert (Pair x x)) {} A"
proof -
  interpret comp_fun_commute "λx. Set.insert (Pair x x)"
    by standard auto
  from assms show ?thesis
    unfolding Id_on_def by (induct A) simp_all
qed

lemma comp_fun_commute_Image_fold:
  "comp_fun_commute (λ(x,y) A. if x ∈ S then Set.insert y A else A)"
proof -
  interpret comp_fun_idem Set.insert
    by (fact comp_fun_idem_insert)
  show ?thesis
    by standard (auto simp: fun_eq_iff comp_fun_commute split: prod.split)
qed

lemma Image_fold:
  assumes "finite R"
  shows "R `` S = Finite_Set.fold (λ(x,y) A. if x ∈ S then Set.insert y A else A) {} R"
proof -
  interpret comp_fun_commute "(λ(x,y) A. if x ∈ S then Set.insert y A else A)"
    by (rule comp_fun_commute_Image_fold)
  have *: "⋀x F. Set.insert x F `` S = (if fst x ∈ S then Set.insert (snd x) (F `` S) else (F `` S))"
    by (force intro: rev_ImageI)
  show ?thesis
    using assms by (induct R) (auto simp: *)
qed

lemma insert_relcomp_union_fold:
  assumes "finite S"
  shows "{x} O S ∪ X = Finite_Set.fold (λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
proof -
  interpret comp_fun_commute "λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
  proof -
    interpret comp_fun_idem Set.insert
      by (fact comp_fun_idem_insert)
    show "comp_fun_commute (λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
      by standard (auto simp add: fun_eq_iff split: prod.split)
  qed
  have *: "{x} O S = {(x', z). x' = fst x ∧ (snd x, z) ∈ S}"
    by (auto simp: relcomp_unfold intro!: exI)
  show ?thesis
    unfolding * using ‹finite S› by (induct S) (auto split: prod.split)
qed

lemma insert_relcomp_fold:
  assumes "finite S"
  shows "Set.insert x R O S =
    Finite_Set.fold (λ(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S"
proof -
  have "Set.insert x R O S = ({x} O S) ∪ (R O S)"
    by auto
  then show ?thesis
    by (auto simp: insert_relcomp_union_fold [OF assms])
qed

lemma comp_fun_commute_relcomp_fold:
  assumes "finite S"
  shows "comp_fun_commute (λ(x,y) A.
    Finite_Set.fold (λ(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
proof -
  have *: "⋀a b A.
    Finite_Set.fold (λ(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S ∪ A"
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
  show ?thesis
    by standard (auto simp: *)
qed

lemma relcomp_fold:
  assumes "finite R" "finite S"
  shows "R O S = Finite_Set.fold
    (λ(x,y) A. Finite_Set.fold (λ(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
proof -
  interpret commute_relcomp_fold: comp_fun_commute
    "(λ(x, y) A. Finite_Set.fold (λ(w, z) A'. if y = w then insert (x, z) A' else A') A S)"
    by (fact comp_fun_commute_relcomp_fold[OF ‹finite S›])
  from assms show ?thesis
    by (induct R) (auto simp: comp_fun_commute_relcomp_fold insert_relcomp_fold cong: if_cong)
qed

end