src/HOL/Quotient.thy
changeset 47096 3ea48c19673e
parent 47094 1a7ad2601cb5
child 47105 e64ffc96a49f
equal deleted inserted replaced
47095:b43ddeea727f 47096:3ea48c19673e
     7 theory Quotient
     7 theory Quotient
     8 imports Plain Hilbert_Choice Equiv_Relations
     8 imports Plain Hilbert_Choice Equiv_Relations
     9 keywords
     9 keywords
    10   "print_quotmaps" "print_quotients" "print_quotconsts" :: diag and
    10   "print_quotmaps" "print_quotients" "print_quotconsts" :: diag and
    11   "quotient_type" :: thy_goal and "/" and
    11   "quotient_type" :: thy_goal and "/" and
       
    12   "setup_lifting" :: thy_decl and
    12   "quotient_definition" :: thy_goal
    13   "quotient_definition" :: thy_goal
    13 uses
    14 uses
    14   ("Tools/Quotient/quotient_info.ML")
    15   ("Tools/Quotient/quotient_info.ML")
    15   ("Tools/Quotient/quotient_type.ML")
    16   ("Tools/Quotient/quotient_type.ML")
    16   ("Tools/Quotient/quotient_def.ML")
    17   ("Tools/Quotient/quotient_def.ML")
   134   assumes a: "Quotient R Abs Rep"
   135   assumes a: "Quotient R Abs Rep"
   135   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
   136   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
   136   using a
   137   using a
   137   unfolding Quotient_def
   138   unfolding Quotient_def
   138   by blast
   139   by blast
       
   140 
       
   141 lemma Quotient_refl1: 
       
   142   assumes a: "Quotient R Abs Rep" 
       
   143   shows "R r s \<Longrightarrow> R r r"
       
   144   using a unfolding Quotient_def 
       
   145   by fast
       
   146 
       
   147 lemma Quotient_refl2: 
       
   148   assumes a: "Quotient R Abs Rep" 
       
   149   shows "R r s \<Longrightarrow> R s s"
       
   150   using a unfolding Quotient_def 
       
   151   by fast
   139 
   152 
   140 lemma Quotient_rel_rep:
   153 lemma Quotient_rel_rep:
   141   assumes a: "Quotient R Abs Rep"
   154   assumes a: "Quotient R Abs Rep"
   142   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
   155   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
   143   using a
   156   using a
   260 
   273 
   261 lemma apply_rsp':
   274 lemma apply_rsp':
   262   assumes a: "(R1 ===> R2) f g" "R1 x y"
   275   assumes a: "(R1 ===> R2) f g" "R1 x y"
   263   shows "R2 (f x) (g y)"
   276   shows "R2 (f x) (g y)"
   264   using a by (auto elim: fun_relE)
   277   using a by (auto elim: fun_relE)
       
   278 
       
   279 lemma apply_rsp'':
       
   280   assumes "Quotient R Abs Rep"
       
   281   and "(R ===> S) f f"
       
   282   shows "S (f (Rep x)) (f (Rep x))"
       
   283 proof -
       
   284   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
       
   285   then show ?thesis using assms(2) by (auto intro: apply_rsp')
       
   286 qed
   265 
   287 
   266 subsection {* lemmas for regularisation of ball and bex *}
   288 subsection {* lemmas for regularisation of ball and bex *}
   267 
   289 
   268 lemma ball_reg_eqv:
   290 lemma ball_reg_eqv:
   269   fixes P :: "'a \<Rightarrow> bool"
   291   fixes P :: "'a \<Rightarrow> bool"
   677       using equivp[simplified part_equivp_def] by metis
   699       using equivp[simplified part_equivp_def] by metis
   678     qed
   700     qed
   679 
   701 
   680 end
   702 end
   681 
   703 
       
   704 subsection {* Quotient composition *}
       
   705 
       
   706 lemma OOO_quotient:
       
   707   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
       
   708   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
       
   709   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
       
   710   fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
       
   711   fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
       
   712   assumes R1: "Quotient R1 Abs1 Rep1"
       
   713   assumes R2: "Quotient R2 Abs2 Rep2"
       
   714   assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
       
   715   assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
       
   716   shows "Quotient (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
       
   717 apply (rule QuotientI)
       
   718    apply (simp add: o_def Quotient_abs_rep [OF R2] Quotient_abs_rep [OF R1])
       
   719   apply simp
       
   720   apply (rule_tac b="Rep1 (Rep2 a)" in pred_compI)
       
   721    apply (rule Quotient_rep_reflp [OF R1])
       
   722   apply (rule_tac b="Rep1 (Rep2 a)" in pred_compI [rotated])
       
   723    apply (rule Quotient_rep_reflp [OF R1])
       
   724   apply (rule Rep1)
       
   725   apply (rule Quotient_rep_reflp [OF R2])
       
   726  apply safe
       
   727     apply (rename_tac x y)
       
   728     apply (drule Abs1)
       
   729       apply (erule Quotient_refl2 [OF R1])
       
   730      apply (erule Quotient_refl1 [OF R1])
       
   731     apply (drule Quotient_refl1 [OF R2], drule Rep1)
       
   732     apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
       
   733      apply (rule_tac b="Rep1 (Abs1 x)" in pred_compI, assumption)
       
   734      apply (erule pred_compI)
       
   735      apply (erule Quotient_symp [OF R1, THEN sympD])
       
   736     apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
       
   737     apply (rule conjI, erule Quotient_refl1 [OF R1])
       
   738     apply (rule conjI, rule Quotient_rep_reflp [OF R1])
       
   739     apply (subst Quotient_abs_rep [OF R1])
       
   740     apply (erule Quotient_rel_abs [OF R1])
       
   741    apply (rename_tac x y)
       
   742    apply (drule Abs1)
       
   743      apply (erule Quotient_refl2 [OF R1])
       
   744     apply (erule Quotient_refl1 [OF R1])
       
   745    apply (drule Quotient_refl2 [OF R2], drule Rep1)
       
   746    apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
       
   747     apply (rule_tac b="Rep1 (Abs1 y)" in pred_compI, assumption)
       
   748     apply (erule pred_compI)
       
   749     apply (erule Quotient_symp [OF R1, THEN sympD])
       
   750    apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
       
   751    apply (rule conjI, erule Quotient_refl2 [OF R1])
       
   752    apply (rule conjI, rule Quotient_rep_reflp [OF R1])
       
   753    apply (subst Quotient_abs_rep [OF R1])
       
   754    apply (erule Quotient_rel_abs [OF R1, THEN sym])
       
   755   apply simp
       
   756   apply (rule Quotient_rel_abs [OF R2])
       
   757   apply (rule Quotient_rel_abs [OF R1, THEN ssubst], assumption)
       
   758   apply (rule Quotient_rel_abs [OF R1, THEN subst], assumption)
       
   759   apply (erule Abs1)
       
   760    apply (erule Quotient_refl2 [OF R1])
       
   761   apply (erule Quotient_refl1 [OF R1])
       
   762  apply (rename_tac a b c d)
       
   763  apply simp
       
   764  apply (rule_tac b="Rep1 (Abs1 r)" in pred_compI)
       
   765   apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
       
   766   apply (rule conjI, erule Quotient_refl1 [OF R1])
       
   767   apply (simp add: Quotient_abs_rep [OF R1] Quotient_rep_reflp [OF R1])
       
   768  apply (rule_tac b="Rep1 (Abs1 s)" in pred_compI [rotated])
       
   769   apply (rule Quotient_rel[symmetric, OF R1, THEN iffD2])
       
   770   apply (simp add: Quotient_abs_rep [OF R1] Quotient_rep_reflp [OF R1])
       
   771   apply (erule Quotient_refl2 [OF R1])
       
   772  apply (rule Rep1)
       
   773  apply (drule Abs1)
       
   774    apply (erule Quotient_refl2 [OF R1])
       
   775   apply (erule Quotient_refl1 [OF R1])
       
   776  apply (drule Abs1)
       
   777   apply (erule Quotient_refl2 [OF R1])
       
   778  apply (erule Quotient_refl1 [OF R1])
       
   779  apply (drule Quotient_rel_abs [OF R1])
       
   780  apply (drule Quotient_rel_abs [OF R1])
       
   781  apply (drule Quotient_rel_abs [OF R1])
       
   782  apply (drule Quotient_rel_abs [OF R1])
       
   783  apply simp
       
   784  apply (rule Quotient_rel[symmetric, OF R2, THEN iffD2])
       
   785  apply simp
       
   786 done
       
   787 
       
   788 lemma OOO_eq_quotient:
       
   789   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
       
   790   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
       
   791   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
       
   792   assumes R1: "Quotient R1 Abs1 Rep1"
       
   793   assumes R2: "Quotient op= Abs2 Rep2"
       
   794   shows "Quotient (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
       
   795 using assms
       
   796 by (rule OOO_quotient) auto
       
   797 
       
   798 subsection {* Invariant *}
       
   799 
       
   800 definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
       
   801   where "invariant R = (\<lambda>x y. R x \<and> x = y)"
       
   802 
       
   803 lemma invariant_to_eq:
       
   804   assumes "invariant P x y"
       
   805   shows "x = y"
       
   806 using assms by (simp add: invariant_def)
       
   807 
       
   808 lemma fun_rel_eq_invariant:
       
   809   shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
       
   810 by (auto simp add: invariant_def fun_rel_def)
       
   811 
       
   812 lemma invariant_same_args:
       
   813   shows "invariant P x x \<equiv> P x"
       
   814 using assms by (auto simp add: invariant_def)
       
   815 
       
   816 lemma copy_type_to_Quotient:
       
   817   assumes "type_definition Rep Abs UNIV"
       
   818   shows "Quotient (op =) Abs Rep"
       
   819 proof -
       
   820   interpret type_definition Rep Abs UNIV by fact
       
   821   from Abs_inject Rep_inverse show ?thesis by (auto intro!: QuotientI)
       
   822 qed
       
   823 
       
   824 lemma copy_type_to_equivp:
       
   825   fixes Abs :: "'a \<Rightarrow> 'b"
       
   826   and Rep :: "'b \<Rightarrow> 'a"
       
   827   assumes "type_definition Rep Abs (UNIV::'a set)"
       
   828   shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
       
   829 by (rule identity_equivp)
       
   830 
       
   831 lemma invariant_type_to_Quotient:
       
   832   assumes "type_definition Rep Abs {x. P x}"
       
   833   shows "Quotient (invariant P) Abs Rep"
       
   834 proof -
       
   835   interpret type_definition Rep Abs "{x. P x}" by fact
       
   836   from Rep Abs_inject Rep_inverse show ?thesis by (auto intro!: QuotientI simp: invariant_def)
       
   837 qed
       
   838 
       
   839 lemma invariant_type_to_part_equivp:
       
   840   assumes "type_definition Rep Abs {x. P x}"
       
   841   shows "part_equivp (invariant P)"
       
   842 proof (intro part_equivpI)
       
   843   interpret type_definition Rep Abs "{x. P x}" by fact
       
   844   show "\<exists>x. invariant P x x" using Rep by (auto simp: invariant_def)
       
   845 next
       
   846   show "symp (invariant P)" by (auto intro: sympI simp: invariant_def)
       
   847 next
       
   848   show "transp (invariant P)" by (auto intro: transpI simp: invariant_def)
       
   849 qed
       
   850 
   682 subsection {* ML setup *}
   851 subsection {* ML setup *}
   683 
   852 
   684 text {* Auxiliary data for the quotient package *}
   853 text {* Auxiliary data for the quotient package *}
   685 
   854 
   686 use "Tools/Quotient/quotient_info.ML"
   855 use "Tools/Quotient/quotient_info.ML"