src/HOL/Algebra/Bij.thy
 author haftmann Mon Mar 01 13:40:23 2010 +0100 (2010-03-01) changeset 35416 d8d7d1b785af parent 33057 764547b68538 child 35848 5443079512ea permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
 wenzelm@14706 ` 1` ```(* Title: HOL/Algebra/Bij.thy ``` paulson@13945 ` 2` ``` Author: Florian Kammueller, with new proofs by L C Paulson ``` paulson@13945 ` 3` ```*) ``` paulson@13945 ` 4` ballarin@20318 ` 5` ```theory Bij imports Group begin ``` paulson@13945 ` 6` ballarin@20318 ` 7` ballarin@27717 ` 8` ```section {* Bijections of a Set, Permutation and Automorphism Groups *} ``` paulson@13945 ` 9` haftmann@35416 ` 10` ```definition Bij :: "'a set \ ('a \ 'a) set" where ``` paulson@13945 ` 11` ``` --{*Only extensional functions, since otherwise we get too many.*} ``` paulson@14963 ` 12` ``` "Bij S \ extensional S \ {f. bij_betw f S S}" ``` paulson@13945 ` 13` haftmann@35416 ` 14` ```definition BijGroup :: "'a set \ ('a \ 'a) monoid" where ``` paulson@14963 ` 15` ``` "BijGroup S \ ``` paulson@14963 ` 16` ``` \carrier = Bij S, ``` paulson@14963 ` 17` ``` mult = \g \ Bij S. \f \ Bij S. compose S g f, ``` paulson@14963 ` 18` ``` one = \x \ S. x\" ``` paulson@13945 ` 19` paulson@13945 ` 20` paulson@13945 ` 21` ```declare Id_compose [simp] compose_Id [simp] ``` paulson@13945 ` 22` paulson@14963 ` 23` ```lemma Bij_imp_extensional: "f \ Bij S \ f \ extensional S" ``` wenzelm@14666 ` 24` ``` by (simp add: Bij_def) ``` paulson@13945 ` 25` paulson@14963 ` 26` ```lemma Bij_imp_funcset: "f \ Bij S \ f \ S \ S" ``` paulson@14853 ` 27` ``` by (auto simp add: Bij_def bij_betw_imp_funcset) ``` paulson@13945 ` 28` paulson@13945 ` 29` wenzelm@14666 ` 30` ```subsection {*Bijections Form a Group *} ``` paulson@13945 ` 31` nipkow@33057 ` 32` ```lemma restrict_inv_into_Bij: "f \ Bij S \ (\x \ S. (inv_into S f) x) \ Bij S" ``` nipkow@33057 ` 33` ``` by (simp add: Bij_def bij_betw_inv_into) ``` paulson@13945 ` 34` paulson@13945 ` 35` ```lemma id_Bij: "(\x\S. x) \ Bij S " ``` paulson@14853 ` 36` ``` by (auto simp add: Bij_def bij_betw_def inj_on_def) ``` paulson@13945 ` 37` paulson@14963 ` 38` ```lemma compose_Bij: "\x \ Bij S; y \ Bij S\ \ compose S x y \ Bij S" ``` paulson@14853 ` 39` ``` by (auto simp add: Bij_def bij_betw_compose) ``` paulson@13945 ` 40` paulson@13945 ` 41` ```lemma Bij_compose_restrict_eq: ``` nipkow@33057 ` 42` ``` "f \ Bij S \ compose S (restrict (inv_into S f) S) f = (\x\S. x)" ``` nipkow@33057 ` 43` ``` by (simp add: Bij_def compose_inv_into_id) ``` paulson@13945 ` 44` paulson@13945 ` 45` ```theorem group_BijGroup: "group (BijGroup S)" ``` wenzelm@14666 ` 46` ```apply (simp add: BijGroup_def) ``` paulson@13945 ` 47` ```apply (rule groupI) ``` paulson@13945 ` 48` ``` apply (simp add: compose_Bij) ``` paulson@13945 ` 49` ``` apply (simp add: id_Bij) ``` paulson@13945 ` 50` ``` apply (simp add: compose_Bij) ``` nipkow@31754 ` 51` ``` apply (blast intro: compose_assoc [symmetric] dest: Bij_imp_funcset) ``` paulson@13945 ` 52` ``` apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp) ``` nipkow@33057 ` 53` ```apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij) ``` paulson@13945 ` 54` ```done ``` paulson@13945 ` 55` paulson@13945 ` 56` paulson@13945 ` 57` ```subsection{*Automorphisms Form a Group*} ``` paulson@13945 ` 58` nipkow@33057 ` 59` ```lemma Bij_inv_into_mem: "\ f \ Bij S; x \ S\ \ inv_into S f x \ S" ``` nipkow@33057 ` 60` ```by (simp add: Bij_def bij_betw_def inv_into_into) ``` paulson@13945 ` 61` nipkow@33057 ` 62` ```lemma Bij_inv_into_lemma: ``` paulson@14963 ` 63` ``` assumes eq: "\x y. \x \ S; y \ S\ \ h(g x y) = g (h x) (h y)" ``` paulson@14963 ` 64` ``` shows "\h \ Bij S; g \ S \ S \ S; x \ S; y \ S\ ``` nipkow@33057 ` 65` ``` \ inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)" ``` paulson@14853 ` 66` ```apply (simp add: Bij_def bij_betw_def) ``` paulson@14853 ` 67` ```apply (subgoal_tac "\x'\S. \y'\S. x = h x' & y = h y'", clarify) ``` nipkow@32988 ` 68` ``` apply (simp add: eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem], blast) ``` paulson@13945 ` 69` ```done ``` paulson@13945 ` 70` paulson@14963 ` 71` haftmann@35416 ` 72` ```definition auto :: "('a, 'b) monoid_scheme \ ('a \ 'a) set" where ``` paulson@14963 ` 73` ``` "auto G \ hom G G \ Bij (carrier G)" ``` paulson@13945 ` 74` haftmann@35416 ` 75` ```definition AutoGroup :: "('a, 'c) monoid_scheme \ ('a \ 'a) monoid" where ``` paulson@14963 ` 76` ``` "AutoGroup G \ BijGroup (carrier G) \carrier := auto G\" ``` paulson@13945 ` 77` paulson@14963 ` 78` ```lemma (in group) id_in_auto: "(\x \ carrier G. x) \ auto G" ``` wenzelm@14666 ` 79` ``` by (simp add: auto_def hom_def restrictI group.axioms id_Bij) ``` paulson@13945 ` 80` paulson@14963 ` 81` ```lemma (in group) mult_funcset: "mult G \ carrier G \ carrier G \ carrier G" ``` paulson@13945 ` 82` ``` by (simp add: Pi_I group.axioms) ``` paulson@13945 ` 83` nipkow@33057 ` 84` ```lemma (in group) restrict_inv_into_hom: ``` paulson@14963 ` 85` ``` "\h \ hom G G; h \ Bij (carrier G)\ ``` nipkow@33057 ` 86` ``` \ restrict (inv_into (carrier G) h) (carrier G) \ hom G G" ``` nipkow@33057 ` 87` ``` by (simp add: hom_def Bij_inv_into_mem restrictI mult_funcset ``` nipkow@33057 ` 88` ``` group.axioms Bij_inv_into_lemma) ``` paulson@13945 ` 89` paulson@13945 ` 90` ```lemma inv_BijGroup: ``` nipkow@33057 ` 91` ``` "f \ Bij S \ m_inv (BijGroup S) f = (\x \ S. (inv_into S f) x)" ``` paulson@13945 ` 92` ```apply (rule group.inv_equality) ``` paulson@13945 ` 93` ```apply (rule group_BijGroup) ``` nipkow@33057 ` 94` ```apply (simp_all add:BijGroup_def restrict_inv_into_Bij Bij_compose_restrict_eq) ``` paulson@13945 ` 95` ```done ``` paulson@13945 ` 96` paulson@14963 ` 97` ```lemma (in group) subgroup_auto: ``` paulson@14963 ` 98` ``` "subgroup (auto G) (BijGroup (carrier G))" ``` paulson@14963 ` 99` ```proof (rule subgroup.intro) ``` paulson@14963 ` 100` ``` show "auto G \ carrier (BijGroup (carrier G))" ``` paulson@14963 ` 101` ``` by (force simp add: auto_def BijGroup_def) ``` paulson@14963 ` 102` ```next ``` paulson@14963 ` 103` ``` fix x y ``` paulson@14963 ` 104` ``` assume "x \ auto G" "y \ auto G" ``` paulson@14963 ` 105` ``` thus "x \\<^bsub>BijGroup (carrier G)\<^esub> y \ auto G" ``` paulson@14963 ` 106` ``` by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset ``` paulson@14963 ` 107` ``` group.hom_compose compose_Bij) ``` paulson@14963 ` 108` ```next ``` paulson@14963 ` 109` ``` show "\\<^bsub>BijGroup (carrier G)\<^esub> \ auto G" by (simp add: BijGroup_def id_in_auto) ``` paulson@14963 ` 110` ```next ``` paulson@14963 ` 111` ``` fix x ``` paulson@14963 ` 112` ``` assume "x \ auto G" ``` paulson@14963 ` 113` ``` thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \ auto G" ``` paulson@14963 ` 114` ``` by (simp del: restrict_apply ``` nipkow@33057 ` 115` ``` add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom) ``` paulson@14963 ` 116` ```qed ``` paulson@13945 ` 117` paulson@14963 ` 118` ```theorem (in group) AutoGroup: "group (AutoGroup G)" ``` paulson@14963 ` 119` ```by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto ``` paulson@14963 ` 120` ``` group_BijGroup) ``` paulson@13945 ` 121` paulson@13945 ` 122` ```end ```