src/HOL/BNF_Composition.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 62777 596baa1a3251 child 67091 1393c2340eec permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     1 (*  Title:      HOL/BNF_Composition.thy
```
```     2     Author:     Dmitriy Traytel, TU Muenchen
```
```     3     Author:     Jasmin Blanchette, TU Muenchen
```
```     4     Copyright   2012, 2013, 2014
```
```     5
```
```     6 Composition of bounded natural functors.
```
```     7 *)
```
```     8
```
```     9 section \<open>Composition of Bounded Natural Functors\<close>
```
```    10
```
```    11 theory BNF_Composition
```
```    12 imports BNF_Def
```
```    13 keywords
```
```    14   "copy_bnf" :: thy_decl and
```
```    15   "lift_bnf" :: thy_goal
```
```    16 begin
```
```    17
```
```    18 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X"
```
```    19   by simp
```
```    20
```
```    21 lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"
```
```    22   by (rule ext) simp
```
```    23
```
```    24 lemma Union_natural: "Union o image (image f) = image f o Union"
```
```    25   by (rule ext) (auto simp only: comp_apply)
```
```    26
```
```    27 lemma in_Union_o_assoc: "x \<in> (Union o gset o gmap) A \<Longrightarrow> x \<in> (Union o (gset o gmap)) A"
```
```    28   by (unfold comp_assoc)
```
```    29
```
```    30 lemma comp_single_set_bd:
```
```    31   assumes fbd_Card_order: "Card_order fbd" and
```
```    32     fset_bd: "\<And>x. |fset x| \<le>o fbd" and
```
```    33     gset_bd: "\<And>x. |gset x| \<le>o gbd"
```
```    34   shows "|\<Union>(fset ` gset x)| \<le>o gbd *c fbd"
```
```    35   apply simp
```
```    36   apply (rule ordLeq_transitive)
```
```    37   apply (rule card_of_UNION_Sigma)
```
```    38   apply (subst SIGMA_CSUM)
```
```    39   apply (rule ordLeq_transitive)
```
```    40   apply (rule card_of_Csum_Times')
```
```    41   apply (rule fbd_Card_order)
```
```    42   apply (rule ballI)
```
```    43   apply (rule fset_bd)
```
```    44   apply (rule ordLeq_transitive)
```
```    45   apply (rule cprod_mono1)
```
```    46   apply (rule gset_bd)
```
```    47   apply (rule ordIso_imp_ordLeq)
```
```    48   apply (rule ordIso_refl)
```
```    49   apply (rule Card_order_cprod)
```
```    50   done
```
```    51
```
```    52 lemma csum_dup: "cinfinite r \<Longrightarrow> Card_order r \<Longrightarrow> p +c p' =o r +c r \<Longrightarrow> p +c p' =o r"
```
```    53   apply (erule ordIso_transitive)
```
```    54   apply (frule csum_absorb2')
```
```    55   apply (erule ordLeq_refl)
```
```    56   by simp
```
```    57
```
```    58 lemma cprod_dup: "cinfinite r \<Longrightarrow> Card_order r \<Longrightarrow> p *c p' =o r *c r \<Longrightarrow> p *c p' =o r"
```
```    59   apply (erule ordIso_transitive)
```
```    60   apply (rule cprod_infinite)
```
```    61   by simp
```
```    62
```
```    63 lemma Union_image_insert: "\<Union>(f ` insert a B) = f a \<union> \<Union>(f ` B)"
```
```    64   by simp
```
```    65
```
```    66 lemma Union_image_empty: "A \<union> \<Union>(f ` {}) = A"
```
```    67   by simp
```
```    68
```
```    69 lemma image_o_collect: "collect ((\<lambda>f. image g o f) ` F) = image g o collect F"
```
```    70   by (rule ext) (auto simp add: collect_def)
```
```    71
```
```    72 lemma conj_subset_def: "A \<subseteq> {x. P x \<and> Q x} = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
```
```    73   by blast
```
```    74
```
```    75 lemma UN_image_subset: "\<Union>(f ` g x) \<subseteq> X = (g x \<subseteq> {x. f x \<subseteq> X})"
```
```    76   by blast
```
```    77
```
```    78 lemma comp_set_bd_Union_o_collect: "|\<Union>\<Union>((\<lambda>f. f x) ` X)| \<le>o hbd \<Longrightarrow> |(Union \<circ> collect X) x| \<le>o hbd"
```
```    79   by (unfold comp_apply collect_def) simp
```
```    80
```
```    81 lemma Collect_inj: "Collect P = Collect Q \<Longrightarrow> P = Q"
```
```    82   by blast
```
```    83
```
```    84 lemma Grp_fst_snd: "(Grp (Collect (case_prod R)) fst)^--1 OO Grp (Collect (case_prod R)) snd = R"
```
```    85   unfolding Grp_def fun_eq_iff relcompp.simps by auto
```
```    86
```
```    87 lemma OO_Grp_cong: "A = B \<Longrightarrow> (Grp A f)^--1 OO Grp A g = (Grp B f)^--1 OO Grp B g"
```
```    88   by (rule arg_cong)
```
```    89
```
```    90 lemma vimage2p_relcompp_mono: "R OO S \<le> T \<Longrightarrow>
```
```    91   vimage2p f g R OO vimage2p g h S \<le> vimage2p f h T"
```
```    92   unfolding vimage2p_def by auto
```
```    93
```
```    94 lemma type_copy_map_cong0: "M (g x) = N (h x) \<Longrightarrow> (f o M o g) x = (f o N o h) x"
```
```    95   by auto
```
```    96
```
```    97 lemma type_copy_set_bd: "(\<And>y. |S y| \<le>o bd) \<Longrightarrow> |(S o Rep) x| \<le>o bd"
```
```    98   by auto
```
```    99
```
```   100 lemma vimage2p_cong: "R = S \<Longrightarrow> vimage2p f g R = vimage2p f g S"
```
```   101   by simp
```
```   102
```
```   103 lemma Ball_comp_iff: "(\<lambda>x. Ball (A x) f) o g = (\<lambda>x. Ball ((A o g) x) f)"
```
```   104   unfolding o_def by auto
```
```   105
```
```   106 lemma conj_comp_iff: "(\<lambda>x. P x \<and> Q x) o g = (\<lambda>x. (P o g) x \<and> (Q o g) x)"
```
```   107   unfolding o_def by auto
```
```   108
```
```   109 context
```
```   110   fixes Rep Abs
```
```   111   assumes type_copy: "type_definition Rep Abs UNIV"
```
```   112 begin
```
```   113
```
```   114 lemma type_copy_map_id0: "M = id \<Longrightarrow> Abs o M o Rep = id"
```
```   115   using type_definition.Rep_inverse[OF type_copy] by auto
```
```   116
```
```   117 lemma type_copy_map_comp0: "M = M1 o M2 \<Longrightarrow> f o M o g = (f o M1 o Rep) o (Abs o M2 o g)"
```
```   118   using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto
```
```   119
```
```   120 lemma type_copy_set_map0: "S o M = image f o S' \<Longrightarrow> (S o Rep) o (Abs o M o g) = image f o (S' o g)"
```
```   121   using type_definition.Abs_inverse[OF type_copy UNIV_I] by (auto simp: o_def fun_eq_iff)
```
```   122
```
```   123 lemma type_copy_wit: "x \<in> (S o Rep) (Abs y) \<Longrightarrow> x \<in> S y"
```
```   124   using type_definition.Abs_inverse[OF type_copy UNIV_I] by auto
```
```   125
```
```   126 lemma type_copy_vimage2p_Grp_Rep: "vimage2p f Rep (Grp (Collect P) h) =
```
```   127     Grp (Collect (\<lambda>x. P (f x))) (Abs o h o f)"
```
```   128   unfolding vimage2p_def Grp_def fun_eq_iff
```
```   129   by (auto simp: type_definition.Abs_inverse[OF type_copy UNIV_I]
```
```   130    type_definition.Rep_inverse[OF type_copy] dest: sym)
```
```   131
```
```   132 lemma type_copy_vimage2p_Grp_Abs:
```
```   133   "\<And>h. vimage2p g Abs (Grp (Collect P) h) = Grp (Collect (\<lambda>x. P (g x))) (Rep o h o g)"
```
```   134   unfolding vimage2p_def Grp_def fun_eq_iff
```
```   135   by (auto simp: type_definition.Abs_inverse[OF type_copy UNIV_I]
```
```   136    type_definition.Rep_inverse[OF type_copy] dest: sym)
```
```   137
```
```   138 lemma type_copy_ex_RepI: "(\<exists>b. F b) = (\<exists>b. F (Rep b))"
```
```   139 proof safe
```
```   140   fix b assume "F b"
```
```   141   show "\<exists>b'. F (Rep b')"
```
```   142   proof (rule exI)
```
```   143     from \<open>F b\<close> show "F (Rep (Abs b))" using type_definition.Abs_inverse[OF type_copy] by auto
```
```   144   qed
```
```   145 qed blast
```
```   146
```
```   147 lemma vimage2p_relcompp_converse:
```
```   148   "vimage2p f g (R^--1 OO S) = (vimage2p Rep f R)^--1 OO vimage2p Rep g S"
```
```   149   unfolding vimage2p_def relcompp.simps conversep.simps fun_eq_iff image_def
```
```   150   by (auto simp: type_copy_ex_RepI)
```
```   151
```
```   152 end
```
```   153
```
```   154 bnf DEADID: 'a
```
```   155   map: "id :: 'a \<Rightarrow> 'a"
```
```   156   bd: natLeq
```
```   157   rel: "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
```
```   158   by (auto simp add: natLeq_card_order natLeq_cinfinite)
```
```   159
```
```   160 definition id_bnf :: "'a \<Rightarrow> 'a" where
```
```   161   "id_bnf \<equiv> (\<lambda>x. x)"
```
```   162
```
```   163 lemma id_bnf_apply: "id_bnf x = x"
```
```   164   unfolding id_bnf_def by simp
```
```   165
```
```   166 bnf ID: 'a
```
```   167   map: "id_bnf :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   168   sets: "\<lambda>x. {x}"
```
```   169   bd: natLeq
```
```   170   rel: "id_bnf :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
```
```   171   pred: "id_bnf :: ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"
```
```   172   unfolding id_bnf_def
```
```   173   apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)
```
```   174   apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
```
```   175   apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]
```
```   176   done
```
```   177
```
```   178 lemma type_definition_id_bnf_UNIV: "type_definition id_bnf id_bnf UNIV"
```
```   179   unfolding id_bnf_def by unfold_locales auto
```
```   180
```
```   181 ML_file "Tools/BNF/bnf_comp_tactics.ML"
```
```   182 ML_file "Tools/BNF/bnf_comp.ML"
```
```   183 ML_file "Tools/BNF/bnf_lift.ML"
```
```   184
```
```   185 hide_fact
```
```   186   DEADID.inj_map DEADID.inj_map_strong DEADID.map_comp DEADID.map_cong DEADID.map_cong0
```
```   187   DEADID.map_cong_simp DEADID.map_id DEADID.map_id0 DEADID.map_ident DEADID.map_transfer
```
```   188   DEADID.rel_Grp DEADID.rel_compp DEADID.rel_compp_Grp DEADID.rel_conversep DEADID.rel_eq
```
```   189   DEADID.rel_flip DEADID.rel_map DEADID.rel_mono DEADID.rel_transfer
```
```   190   ID.inj_map ID.inj_map_strong ID.map_comp ID.map_cong ID.map_cong0 ID.map_cong_simp ID.map_id
```
```   191   ID.map_id0 ID.map_ident ID.map_transfer ID.rel_Grp ID.rel_compp ID.rel_compp_Grp ID.rel_conversep
```
```   192   ID.rel_eq ID.rel_flip ID.rel_map ID.rel_mono ID.rel_transfer ID.set_map ID.set_transfer
```
```   193
```
```   194 end
```