(* Title: HOL/Relation.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Stefan Berghofer, TU Muenchen
*)
section \<open>Relations -- as sets of pairs, and binary predicates\<close>
theory Relation
imports Finite_Set
begin
text \<open>A preliminary: classical rules for reasoning on predicates\<close>
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 \<open>Fundamental\<close>
subsubsection \<open>Relations as sets of pairs\<close>
type_synonym 'a rel = "('a \<times> 'a) set"
lemma subrelI: "(\x y. (x, y) \ r \ (x, y) \ s) \ r \ s"
\<comment> \<open>Version of @{thm [source] subsetI} for binary relations\<close>
by auto
lemma lfp_induct2:
"(a, b) \ lfp f \ mono f \
(\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
\<comment> \<open>Version of @{thm [source] lfp_induct} for binary relations\<close>
using lfp_induct_set [of "(a, b)" f "case_prod P"] by auto
subsubsection \<open>Conversions between set and predicate relations\<close>
lemma pred_equals_eq [pred_set_conv]: "(\x. x \ R) = (\x. x \ S) \ R = S"
by (simp add: set_eq_iff fun_eq_iff)
lemma pred_equals_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) = (\x y. (x, y) \ S) \ R = S"
by (simp add: set_eq_iff fun_eq_iff)
lemma pred_subset_eq [pred_set_conv]: "(\x. x \ R) \ (\x. x \ S) \ R \ S"
by (simp add: subset_iff le_fun_def)
lemma pred_subset_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) \ (\x y. (x, y) \ S) \ R \ S"
by (simp add: subset_iff le_fun_def)
lemma bot_empty_eq [pred_set_conv]: "\ = (\x. x \ {})"
by (auto simp add: fun_eq_iff)
lemma bot_empty_eq2 [pred_set_conv]: "\ = (\x y. (x, y) \ {})"
by (auto simp add: fun_eq_iff)
lemma top_empty_eq [pred_set_conv]: "\ = (\x. x \ UNIV)"
by (auto simp add: fun_eq_iff)
lemma top_empty_eq2 [pred_set_conv]: "\ = (\x y. (x, y) \ UNIV)"
by (auto simp add: fun_eq_iff)
lemma inf_Int_eq [pred_set_conv]: "(\x. x \ R) \ (\x. x \ S) = (\x. x \ R \ S)"
by (simp add: inf_fun_def)
lemma inf_Int_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) \ (\x y. (x, y) \ S) = (\x y. (x, y) \ R \ S)"
by (simp add: inf_fun_def)
lemma sup_Un_eq [pred_set_conv]: "(\x. x \ R) \ (\x. x \ S) = (\x. x \ R \ S)"
by (simp add: sup_fun_def)
lemma sup_Un_eq2 [pred_set_conv]: "(\x y. (x, y) \ R) \ (\x y. (x, y) \ S) = (\x y. (x, y) \ R \ S)"
by (simp add: sup_fun_def)
lemma INF_INT_eq [pred_set_conv]: "(\i\S. (\x. x \ r i)) = (\x. x \ (\i\S. r i))"
by (simp add: fun_eq_iff)
lemma INF_INT_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ r i)) = (\x y. (x, y) \ (\i\S. r i))"
by (simp add: fun_eq_iff)
lemma SUP_UN_eq [pred_set_conv]: "(\i\S. (\x. x \ r i)) = (\x. x \ (\i\S. r i))"
by (simp add: fun_eq_iff)
lemma SUP_UN_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ r i)) = (\x y. (x, y) \ (\i\S. r i))"
by (simp add: fun_eq_iff)
lemma Inf_INT_eq [pred_set_conv]: "\S = (\x. x \ (\(Collect ` S)))"
by (simp add: fun_eq_iff)
lemma INF_Int_eq [pred_set_conv]: "(\i\S. (\x. x \ i)) = (\x. x \ \S)"
by (simp add: fun_eq_iff)
lemma Inf_INT_eq2 [pred_set_conv]: "\S = (\x y. (x, y) \ (\(Collect ` case_prod ` S)))"
by (simp add: fun_eq_iff)
lemma INF_Int_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ i)) = (\x y. (x, y) \ \S)"
by (simp add: fun_eq_iff)
lemma Sup_SUP_eq [pred_set_conv]: "\S = (\x. x \ \(Collect ` S))"
by (simp add: fun_eq_iff)
lemma SUP_Sup_eq [pred_set_conv]: "(\i\S. (\x. x \ i)) = (\x. x \ \S)"
by (simp add: fun_eq_iff)
lemma Sup_SUP_eq2 [pred_set_conv]: "\S = (\x y. (x, y) \ (\(Collect ` case_prod ` S)))"
by (simp add: fun_eq_iff)
lemma SUP_Sup_eq2 [pred_set_conv]: "(\i\S. (\x y. (x, y) \ i)) = (\x y. (x, y) \ \S)"
by (simp add: fun_eq_iff)
subsection \<open>Properties of relations\<close>
subsubsection \<open>Reflexivity\<close>
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 :: "('a \ 'a \ bool) \ bool"
where "reflp r \ (\x. r x x)"
lemma reflp_refl_eq [pred_set_conv]: "reflp (\x 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 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 reflpI [intro?]: "(\x. r x x) \ reflp r"
by (auto intro: refl_onI simp add: reflp_def)
lemma reflpE:
assumes "reflp r"
obtains "r x x"
using assms by (auto dest: refl_onD simp add: reflp_def)
lemma reflpD [dest?]:
assumes "reflp r"
shows "r x x"
using assms by (auto elim: reflpE)
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_inf: "reflp r \ reflp s \ reflp (r \ s)"
by (auto intro: reflpI elim: reflpE)
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_sup: "reflp r \ reflp s \ reflp (r \ s)"
by (auto intro: reflpI elim: reflpE)
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 refl_on_UNION: "\x\S. refl_on (A x) (r x) \ refl_on (\(A ` S)) (\(r ` S))"
unfolding refl_on_def by blast
lemma refl_on_empty [simp]: "refl_on {} {}"
by (simp add: refl_on_def)
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_mono: "reflp R \ (\x y. R x y \ Q x y) \ reflp Q"
by (auto intro: reflpI dest: reflpD)
subsubsection \<open>Irreflexivity\<close>
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 (\a 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)
subsubsection \<open>Asymmetry\<close>
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 (\a 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 asym_iff: "asym R \ (\x y. (x,y) \ R \ (y,x) \ R)"
by (blast intro: asymI dest: asymD)
subsubsection \<open>Symmetry\<close>
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 (\x 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 \<open>Antisymmetry\<close>
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 (\x 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 \<open>Transitivity\<close>
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 (\x 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. \(y2, z) \ r. y1 = y2 \ (x, z) \ r)"
by (auto simp add: trans_def)
lemma transp_trans: "transp r \ trans {(x, y). r x y}"
by (simp add: trans_def transp_def)
lemma transp_equality [simp]: "transp (=)"
by (auto intro: transpI)
lemma trans_empty [simp]: "trans {}"
by (blast intro: transI)
lemma transp_empty [simp]: "transp (\x y. False)"
using trans_empty[to_pred] by (simp add: bot_fun_def)
lemma trans_singleton [simp]: "trans {(a, a)}"
by (blast intro: transI)
lemma transp_singleton [simp]: "transp (\x y. x = a \ y = a)"
by (simp add: transp_def)
context preorder
begin
lemma transp_le[simp]: "transp (\)"
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 \<open>Totality\<close>
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"
lemma total_on_empty [simp]: "total_on {} r"
by (simp add: total_on_def)
lemma total_on_singleton [simp]: "total_on {x} {(x, x)}"
unfolding total_on_def by blast
subsubsection \<open>Single valued relations\<close>
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 (\x 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. \\<^sub>\\<^sub>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 \<open>Relation operations\<close>
subsubsection \<open>The identity relation\<close>
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"
\<comment> \<open>A strange result, since \<open>Id\<close> is also symmetric.\<close>
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 \<open>Diagonal: identity over a set\<close>
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"
\<comment> \<open>The general elimination rule.\<close>
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 (\(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 \<open>Composition\<close>
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 \<open>
For historic reasons, the elimination rules are not wholly corresponding.
Feel free to consolidate this.
\<close>
inductive_cases relcompEpair: "(a, c) \ r O s"
inductive_cases relcomppE [elim!]: "(r OO s) a c"
lemma relcompE [elim!]: "xz \ r O s \
(\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s \<Longrightarrow> P) \<Longrightarrow> 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: "\(r ` I) OO s = (\i\I. r i OO s)"
by (fact relcomp_UNION_distrib2 [to_pred])
lemma single_valued_relcomp: "single_valued r \ single_valued s \ single_valued (r O s)"
unfolding single_valued_def by blast
lemma relcomp_unfold: "r O s = {(x, z). \y. (x, y) \ r \ (y, z) \ s}"
by (auto simp add: set_eq_iff)
lemma relcompp_apply: "(R OO S) a c \ (\b. R a b \ S b c)"
unfolding relcomp_unfold [to_pred] ..
lemma eq_OO: "(=) OO R = R"
by blast
lemma OO_eq: "R OO (=) = R"
by blast
subsubsection \<open>Converse\<close>
inductive_set converse :: "('a \ 'b) set \ ('b \ 'a) set" ("(_\)" [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 \<Longrightarrow> r\<inverse>\<inverse> 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\<inverse>\<inverse> b a \<Longrightarrow> r a b"
by (fact converseD [to_pred])
lemma converseE [elim!]: "yx \ r\ \ (\x y. yx = (y, x) \ (x, y) \ r \ P) \ P"
\<comment> \<open>More general than \<open>converseD\<close>, as it ``splits'' the member of the relation.\<close>
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 \<open>Domain, range and field\<close>
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 \<open>Image of a set under a relation\<close>
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"
\<comment> \<open>This version's more effective when we already have the required \<open>a\<close>\<close>
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})"
\<comment> \<open>NOT suitable for rewriting\<close>
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 `` (\(B ` A))) \ (\x\A. r `` (B x))"
by blast
text \<open>Converse inclusion requires some assumptions\<close>
lemma Image_INT_eq: "single_valued (r\) \ A \ {} \ r `` (\(B ` A)) = (\x\A. r `` B x)"
apply (rule equalityI)
apply (rule Image_INT_subset)
apply (auto simp add: single_valued_def)
apply blast
done
lemma Image_subset_eq: "r``A \ B \ A \ - ((r\) `` (- 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 \<open>Inverse image\<close>
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 = (\x y. r (f x) (f y))"
lemma [pred_set_conv]: "inv_imagep (\x y. (x, y) \ r) f = (\x 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 \<open>Powerset\<close>
definition Powp :: "('a \ bool) \ 'a set \ bool"
where "Powp A = (\B. \x \ B. A x)"
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\x. x \ A) = (\x. x \ Pow A)"
by (auto simp add: Powp_def fun_eq_iff)
lemmas Powp_mono [mono] = Pow_mono [to_pred]
subsubsection \<open>Expressing relation operations via \<^const>\<open>Finite_Set.fold\<close>\<close>
lemma Id_on_fold:
assumes "finite A"
shows "Id_on A = Finite_Set.fold (\x. Set.insert (Pair x x)) {} A"
proof -
interpret comp_fun_commute "\x. 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 (\(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 (\(x,y) A. if x \ S then Set.insert y A else A) {} R"
proof -
interpret comp_fun_commute "(\(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 (\(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
proof -
interpret comp_fun_commute "\(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 (\(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 \<open>finite S\<close> 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 (\<lambda>(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 (\(x,y) A.
Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
proof -
have *: "\a b A.
Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> 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
(\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
using assms
by (induct R)
(auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold
cong: if_cong)
end
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