tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/Algebra/Congruence.thy
Author: Clemens Ballarin, started 3 January 2008
Copyright: Clemens Ballarin
*)
theory Congruence
imports Main
begin
section {* Objects *}
subsection {* Structure with Carrier Set. *}
record 'a partial_object =
carrier :: "'a set"
subsection {* Structure with Carrier and Equivalence Relation @{text eq} *}
record 'a eq_object = "'a partial_object" +
eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".=\<index>" 50)
definition
elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<in>\<index>" 50)
where "x .\<in>\<^bsub>S\<^esub> A \<longleftrightarrow> (\<exists>y \<in> A. x .=\<^bsub>S\<^esub> y)"
definition
set_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.=}\<index>" 50)
where "A {.=}\<^bsub>S\<^esub> B \<longleftrightarrow> ((\<forall>x \<in> A. x .\<in>\<^bsub>S\<^esub> B) \<and> (\<forall>x \<in> B. x .\<in>\<^bsub>S\<^esub> A))"
definition
eq_class_of :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set" ("class'_of\<index>")
where "class_of\<^bsub>S\<^esub> x = {y \<in> carrier S. x .=\<^bsub>S\<^esub> y}"
definition
eq_closure_of :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set" ("closure'_of\<index>")
where "closure_of\<^bsub>S\<^esub> A = {y \<in> carrier S. y .\<in>\<^bsub>S\<^esub> A}"
definition
eq_is_closed :: "_ \<Rightarrow> 'a set \<Rightarrow> bool" ("is'_closed\<index>")
where "is_closed\<^bsub>S\<^esub> A \<longleftrightarrow> A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A"
abbreviation
not_eq :: "_ \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl ".\<noteq>\<index>" 50)
where "x .\<noteq>\<^bsub>S\<^esub> y == ~(x .=\<^bsub>S\<^esub> y)"
abbreviation
not_elem :: "_ \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl ".\<notin>\<index>" 50)
where "x .\<notin>\<^bsub>S\<^esub> A == ~(x .\<in>\<^bsub>S\<^esub> A)"
abbreviation
set_not_eq :: "_ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "{.\<noteq>}\<index>" 50)
where "A {.\<noteq>}\<^bsub>S\<^esub> B == ~(A {.=}\<^bsub>S\<^esub> B)"
locale equivalence =
fixes S (structure)
assumes refl [simp, intro]: "x \<in> carrier S \<Longrightarrow> x .= x"
and sym [sym]: "\<lbrakk> x .= y; x \<in> carrier S; y \<in> carrier S \<rbrakk> \<Longrightarrow> y .= x"
and trans [trans]:
"\<lbrakk> x .= y; y .= z; x \<in> carrier S; y \<in> carrier S; z \<in> carrier S \<rbrakk> \<Longrightarrow> x .= z"
(* Lemmas by Stephan Hohe *)
lemma elemI:
fixes R (structure)
assumes "a' \<in> A" and "a .= a'"
shows "a .\<in> A"
unfolding elem_def
using assms
by fast
lemma (in equivalence) elem_exact:
assumes "a \<in> carrier S" and "a \<in> A"
shows "a .\<in> A"
using assms
by (fast intro: elemI)
lemma elemE:
fixes S (structure)
assumes "a .\<in> A"
and "\<And>a'. \<lbrakk>a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
shows "P"
using assms
unfolding elem_def
by fast
lemma (in equivalence) elem_cong_l [trans]:
assumes cong: "a' .= a"
and a: "a .\<in> A"
and carr: "a \<in> carrier S" "a' \<in> carrier S"
and Acarr: "A \<subseteq> carrier S"
shows "a' .\<in> A"
using a
apply (elim elemE, intro elemI)
proof assumption
fix b
assume bA: "b \<in> A"
note [simp] = carr bA[THEN subsetD[OF Acarr]]
note cong
also assume "a .= b"
finally show "a' .= b" by simp
qed
lemma (in equivalence) elem_subsetD:
assumes "A \<subseteq> B"
and aA: "a .\<in> A"
shows "a .\<in> B"
using assms
by (fast intro: elemI elim: elemE dest: subsetD)
lemma (in equivalence) mem_imp_elem [simp, intro]:
"[| x \<in> A; x \<in> carrier S |] ==> x .\<in> A"
unfolding elem_def by blast
lemma set_eqI:
fixes R (structure)
assumes ltr: "\<And>a. a \<in> A \<Longrightarrow> a .\<in> B"
and rtl: "\<And>b. b \<in> B \<Longrightarrow> b .\<in> A"
shows "A {.=} B"
unfolding set_eq_def
by (fast intro: ltr rtl)
lemma set_eqI2:
fixes R (structure)
assumes ltr: "\<And>a b. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a .= b"
and rtl: "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. b .= a"
shows "A {.=} B"
by (intro set_eqI, unfold elem_def) (fast intro: ltr rtl)+
lemma set_eqD1:
fixes R (structure)
assumes AA': "A {.=} A'"
and "a \<in> A"
shows "\<exists>a'\<in>A'. a .= a'"
using assms
unfolding set_eq_def elem_def
by fast
lemma set_eqD2:
fixes R (structure)
assumes AA': "A {.=} A'"
and "a' \<in> A'"
shows "\<exists>a\<in>A. a' .= a"
using assms
unfolding set_eq_def elem_def
by fast
lemma set_eqE:
fixes R (structure)
assumes AB: "A {.=} B"
and r: "\<lbrakk>\<forall>a\<in>A. a .\<in> B; \<forall>b\<in>B. b .\<in> A\<rbrakk> \<Longrightarrow> P"
shows "P"
using AB
unfolding set_eq_def
by (blast dest: r)
lemma set_eqE2:
fixes R (structure)
assumes AB: "A {.=} B"
and r: "\<lbrakk>\<forall>a\<in>A. (\<exists>b\<in>B. a .= b); \<forall>b\<in>B. (\<exists>a\<in>A. b .= a)\<rbrakk> \<Longrightarrow> P"
shows "P"
using AB
unfolding set_eq_def elem_def
by (blast dest: r)
lemma set_eqE':
fixes R (structure)
assumes AB: "A {.=} B"
and aA: "a \<in> A" and bB: "b \<in> B"
and r: "\<And>a' b'. \<lbrakk>a' \<in> A; b .= a'; b' \<in> B; a .= b'\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
from AB aA
have "\<exists>b'\<in>B. a .= b'" by (rule set_eqD1)
from this obtain b'
where b': "b' \<in> B" "a .= b'" by auto
from AB bB
have "\<exists>a'\<in>A. b .= a'" by (rule set_eqD2)
from this obtain a'
where a': "a' \<in> A" "b .= a'" by auto
from a' b'
show "P" by (rule r)
qed
lemma (in equivalence) eq_elem_cong_r [trans]:
assumes a: "a .\<in> A"
and cong: "A {.=} A'"
and carr: "a \<in> carrier S"
and Carr: "A \<subseteq> carrier S" "A' \<subseteq> carrier S"
shows "a .\<in> A'"
using a cong
proof (elim elemE set_eqE)
fix b
assume bA: "b \<in> A"
and inA': "\<forall>b\<in>A. b .\<in> A'"
note [simp] = carr Carr Carr[THEN subsetD] bA
assume "a .= b"
also from bA inA'
have "b .\<in> A'" by fast
finally
show "a .\<in> A'" by simp
qed
lemma (in equivalence) set_eq_sym [sym]:
assumes "A {.=} B"
and "A \<subseteq> carrier S" "B \<subseteq> carrier S"
shows "B {.=} A"
using assms
unfolding set_eq_def elem_def
by fast
(* FIXME: the following two required in Isabelle 2008, not Isabelle 2007 *)
(* alternatively, could declare lemmas [trans] = ssubst [where 'a = "'a set"] *)
lemma (in equivalence) equal_set_eq_trans [trans]:
assumes AB: "A = B" and BC: "B {.=} C"
shows "A {.=} C"
using AB BC by simp
lemma (in equivalence) set_eq_equal_trans [trans]:
assumes AB: "A {.=} B" and BC: "B = C"
shows "A {.=} C"
using AB BC by simp
lemma (in equivalence) set_eq_trans [trans]:
assumes AB: "A {.=} B" and BC: "B {.=} C"
and carr: "A \<subseteq> carrier S" "B \<subseteq> carrier S" "C \<subseteq> carrier S"
shows "A {.=} C"
proof (intro set_eqI)
fix a
assume aA: "a \<in> A"
with carr have "a \<in> carrier S" by fast
note [simp] = carr this
from aA
have "a .\<in> A" by (simp add: elem_exact)
also note AB
also note BC
finally
show "a .\<in> C" by simp
next
fix c
assume cC: "c \<in> C"
with carr have "c \<in> carrier S" by fast
note [simp] = carr this
from cC
have "c .\<in> C" by (simp add: elem_exact)
also note BC[symmetric]
also note AB[symmetric]
finally
show "c .\<in> A" by simp
qed
(* FIXME: generalise for insert *)
(*
lemma (in equivalence) set_eq_insert:
assumes x: "x .= x'"
and carr: "x \<in> carrier S" "x' \<in> carrier S" "A \<subseteq> carrier S"
shows "insert x A {.=} insert x' A"
unfolding set_eq_def elem_def
apply rule
apply rule
apply (case_tac "xa = x")
using x apply fast
apply (subgoal_tac "xa \<in> A") prefer 2 apply fast
apply (rule_tac x=xa in bexI)
using carr apply (rule_tac refl) apply auto [1]
apply safe
*)
lemma (in equivalence) set_eq_pairI:
assumes xx': "x .= x'"
and carr: "x \<in> carrier S" "x' \<in> carrier S" "y \<in> carrier S"
shows "{x, y} {.=} {x', y}"
unfolding set_eq_def elem_def
proof safe
have "x' \<in> {x', y}" by fast
with xx' show "\<exists>b\<in>{x', y}. x .= b" by fast
next
have "y \<in> {x', y}" by fast
with carr show "\<exists>b\<in>{x', y}. y .= b" by fast
next
have "x \<in> {x, y}" by fast
with xx'[symmetric] carr
show "\<exists>a\<in>{x, y}. x' .= a" by fast
next
have "y \<in> {x, y}" by fast
with carr show "\<exists>a\<in>{x, y}. y .= a" by fast
qed
lemma (in equivalence) is_closedI:
assumes closed: "!!x y. [| x .= y; x \<in> A; y \<in> carrier S |] ==> y \<in> A"
and S: "A \<subseteq> carrier S"
shows "is_closed A"
unfolding eq_is_closed_def eq_closure_of_def elem_def
using S
by (blast dest: closed sym)
lemma (in equivalence) closure_of_eq:
"[| x .= x'; A \<subseteq> carrier S; x \<in> closure_of A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> closure_of A"
unfolding eq_closure_of_def elem_def
by (blast intro: trans sym)
lemma (in equivalence) is_closed_eq [dest]:
"[| x .= x'; x \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x' \<in> A"
unfolding eq_is_closed_def
using closure_of_eq [where A = A]
by simp
lemma (in equivalence) is_closed_eq_rev [dest]:
"[| x .= x'; x' \<in> A; is_closed A; x \<in> carrier S; x' \<in> carrier S |] ==> x \<in> A"
by (drule sym) (simp_all add: is_closed_eq)
lemma closure_of_closed [simp, intro]:
fixes S (structure)
shows "closure_of A \<subseteq> carrier S"
unfolding eq_closure_of_def
by fast
lemma closure_of_memI:
fixes S (structure)
assumes "a .\<in> A"
and "a \<in> carrier S"
shows "a \<in> closure_of A"
unfolding eq_closure_of_def
using assms
by fast
lemma closure_ofI2:
fixes S (structure)
assumes "a .= a'"
and "a' \<in> A"
and "a \<in> carrier S"
shows "a \<in> closure_of A"
unfolding eq_closure_of_def elem_def
using assms
by fast
lemma closure_of_memE:
fixes S (structure)
assumes p: "a \<in> closure_of A"
and r: "\<lbrakk>a \<in> carrier S; a .\<in> A\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
from p
have acarr: "a \<in> carrier S"
and "a .\<in> A"
by (simp add: eq_closure_of_def)+
thus "P" by (rule r)
qed
lemma closure_ofE2:
fixes S (structure)
assumes p: "a \<in> closure_of A"
and r: "\<And>a'. \<lbrakk>a \<in> carrier S; a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
from p have acarr: "a \<in> carrier S" by (simp add: eq_closure_of_def)
from p have "\<exists>a'\<in>A. a .= a'" by (simp add: eq_closure_of_def elem_def)
from this obtain a'
where "a' \<in> A" and "a .= a'" by auto
from acarr and this
show "P" by (rule r)
qed
(*
lemma (in equivalence) classes_consistent:
assumes Acarr: "A \<subseteq> carrier S"
shows "is_closed (closure_of A)"
apply (blast intro: elemI elim elemE)
using assms
apply (intro is_closedI closure_of_memI, simp)
apply (elim elemE closure_of_memE)
proof -
fix x a' a''
assume carr: "x \<in> carrier S" "a' \<in> carrier S"
assume a''A: "a'' \<in> A"
with Acarr have "a'' \<in> carrier S" by fast
note [simp] = carr this Acarr
assume "x .= a'"
also assume "a' .= a''"
also from a''A
have "a'' .\<in> A" by (simp add: elem_exact)
finally show "x .\<in> A" by simp
qed
*)
(*
lemma (in equivalence) classes_small:
assumes "is_closed B"
and "A \<subseteq> B"
shows "closure_of A \<subseteq> B"
using assms
by (blast dest: is_closedD2 elem_subsetD elim: closure_of_memE)
lemma (in equivalence) classes_eq:
assumes "A \<subseteq> carrier S"
shows "A {.=} closure_of A"
using assms
by (blast intro: set_eqI elem_exact closure_of_memI elim: closure_of_memE)
lemma (in equivalence) complete_classes:
assumes c: "is_closed A"
shows "A = closure_of A"
using assms
by (blast intro: closure_of_memI elem_exact dest: is_closedD1 is_closedD2 closure_of_memE)
*)
end