Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 4,167 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 |
(* Title: IsomorphismClass
Author: Eugene W. Stark <stark@cs.stonybrook.edu>, 2019
Maintainer: Eugene W. Stark <stark@cs.stonybrook.edu>
*)
section "Isomorphism Classes"
text \<open>
The following is a small theory that facilitates working with isomorphism classes of objects
in a category.
\<close>
theory IsomorphismClass
imports Category3.EpiMonoIso Category3.NaturalTransformation
begin
context category
begin
notation isomorphic (infix "\<cong>" 50)
definition iso_class ("\<lbrakk>_\<rbrakk>")
where "iso_class f \<equiv> {f'. f \<cong> f'}"
definition is_iso_class
where "is_iso_class F \<equiv> \<exists>f. f \<in> F \<and> F = iso_class f"
definition iso_class_rep
where "iso_class_rep F \<equiv> SOME f. f \<in> F"
lemmas isomorphic_transitive [trans]
lemmas naturally_isomorphic_transitive [trans]
lemma inv_in_homI [intro]:
assumes "iso f" and "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
shows "\<guillemotleft>inv f : b \<rightarrow> a\<guillemotright>"
using assms inv_is_inverse seqE inverse_arrowsE
by (metis ide_compE in_homE in_homI)
lemma iso_class_is_nonempty:
assumes "is_iso_class F"
shows "F \<noteq> {}"
using assms is_iso_class_def iso_class_def by auto
lemma iso_class_memb_is_ide:
assumes "is_iso_class F" and "f \<in> F"
shows "ide f"
using assms is_iso_class_def iso_class_def isomorphic_def by auto
lemma ide_in_iso_class:
assumes "ide f"
shows "f \<in> \<lbrakk>f\<rbrakk>"
using assms iso_class_def is_iso_class_def isomorphic_reflexive by simp
lemma rep_in_iso_class:
assumes "is_iso_class F"
shows "iso_class_rep F \<in> F"
using assms is_iso_class_def iso_class_rep_def someI_ex [of "\<lambda>f. f \<in> F"]
ide_in_iso_class
by metis
lemma is_iso_classI:
assumes "ide f"
shows "is_iso_class \<lbrakk>f\<rbrakk>"
using assms iso_class_def is_iso_class_def isomorphic_reflexive by blast
lemma rep_iso_class:
assumes "ide f"
shows "iso_class_rep \<lbrakk>f\<rbrakk> \<cong> f"
using assms is_iso_classI rep_in_iso_class iso_class_def isomorphic_symmetric
by blast
lemma iso_class_elems_isomorphic:
assumes "is_iso_class F" and "f \<in> F" and "f' \<in> F"
shows "f \<cong> f'"
using assms iso_class_def
by (metis is_iso_class_def isomorphic_symmetric isomorphic_transitive mem_Collect_eq)
lemma iso_class_eqI [intro]:
assumes "f \<cong> g"
shows "\<lbrakk>f\<rbrakk> = \<lbrakk>g\<rbrakk>"
proof -
have f: "ide f"
using assms isomorphic_def by auto
have g: "ide g"
using assms isomorphic_def by auto
have "f \<in> \<lbrakk>g\<rbrakk>"
using assms iso_class_def isomorphic_symmetric by simp
moreover have "g \<in> \<lbrakk>g\<rbrakk>"
using assms g iso_class_def isomorphic_reflexive isomorphic_def by simp
ultimately have "\<And>h. (h \<in> \<lbrakk>f\<rbrakk>) = (h \<in> \<lbrakk>g\<rbrakk>)"
using assms g iso_class_def [of f] iso_class_def [of g]
isomorphic_reflexive isomorphic_symmetric isomorphic_transitive
by blast
thus ?thesis by auto
qed
lemma iso_class_eq:
assumes "is_iso_class F" and "is_iso_class G" and "F \<inter> G \<noteq> {}"
shows "F = G"
proof -
obtain h where h: "h \<in> F \<and> h \<in> G"
using assms by auto
show ?thesis
using assms h
by (metis is_iso_class_def iso_class_elems_isomorphic iso_class_eqI)
qed
lemma iso_class_rep [simp]:
assumes "is_iso_class F"
shows "\<lbrakk>iso_class_rep F\<rbrakk> = F"
proof (intro iso_class_eq)
show "is_iso_class \<lbrakk>iso_class_rep F\<rbrakk>"
using assms is_iso_classI iso_class_memb_is_ide rep_in_iso_class by blast
show "is_iso_class F"
using assms by simp
show "\<lbrakk>iso_class_rep F\<rbrakk> \<inter> F \<noteq> {}"
using assms rep_in_iso_class
by (meson disjoint_iff_not_equal ide_in_iso_class iso_class_memb_is_ide)
qed
end
end
|