src/HOL/Library/FuncSet.thy
 author wenzelm Tue Nov 07 11:47:57 2006 +0100 (2006-11-07) changeset 21210 c17fd2df4e9e parent 20770 2c583720436e child 21404 eb85850d3eb7 permissions -rw-r--r--
renamed 'const_syntax' to 'notation';
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Florian Kammueller and Lawrence C Paulson
```
```     4 *)
```
```     5
```
```     6 header {* Pi and Function Sets *}
```
```     7
```
```     8 theory FuncSet
```
```     9 imports Main
```
```    10 begin
```
```    11
```
```    12 definition
```
```    13   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set"
```
```    14   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
```
```    15
```
```    16   extensional :: "'a set => ('a => 'b) set"
```
```    17   "extensional A = {f. \<forall>x. x~:A --> f x = arbitrary}"
```
```    18
```
```    19   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)"
```
```    20   "restrict f A = (%x. if x \<in> A then f x else arbitrary)"
```
```    21
```
```    22 abbreviation
```
```    23   funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
```
```    24   "A -> B == Pi A (%_. B)"
```
```    25
```
```    26 notation (xsymbols)
```
```    27   funcset  (infixr "\<rightarrow>" 60)
```
```    28
```
```    29 syntax
```
```    30   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
```
```    31   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    32
```
```    33 syntax (xsymbols)
```
```    34   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    35   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    36
```
```    37 syntax (HTML output)
```
```    38   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    39   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    40
```
```    41 translations
```
```    42   "PI x:A. B" == "CONST Pi A (%x. B)"
```
```    43   "%x:A. f" == "CONST restrict (%x. f) A"
```
```    44
```
```    45 definition
```
```    46   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
```
```    47   "compose A g f = (\<lambda>x\<in>A. g (f x))"
```
```    48
```
```    49
```
```    50 subsection{*Basic Properties of @{term Pi}*}
```
```    51
```
```    52 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    53   by (simp add: Pi_def)
```
```    54
```
```    55 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    56   by (simp add: Pi_def)
```
```    57
```
```    58 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    59   by (simp add: Pi_def)
```
```    60
```
```    61 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    62   by (simp add: Pi_def)
```
```    63
```
```    64 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    65   by (auto simp add: Pi_def)
```
```    66
```
```    67 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    68 apply (simp add: Pi_def, auto)
```
```    69 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    70 picking an element from each non-empty @{term "B x"}*}
```
```    71 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    72 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    73 done
```
```    74
```
```    75 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    76   by (simp add: Pi_def)
```
```    77
```
```    78 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```    79   by (simp add: Pi_def)
```
```    80
```
```    81 text{*Covariance of Pi-sets in their second argument*}
```
```    82 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```    83   by (simp add: Pi_def, blast)
```
```    84
```
```    85 text{*Contravariance of Pi-sets in their first argument*}
```
```    86 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```    87   by (simp add: Pi_def, blast)
```
```    88
```
```    89
```
```    90 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```    91
```
```    92 lemma funcset_compose:
```
```    93     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```    94   by (simp add: Pi_def compose_def restrict_def)
```
```    95
```
```    96 lemma compose_assoc:
```
```    97     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```    98       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```    99   by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
```
```   100
```
```   101 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```   102   by (simp add: compose_def restrict_def)
```
```   103
```
```   104 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```   105   by (auto simp add: image_def compose_eq)
```
```   106
```
```   107
```
```   108 subsection{*Bounded Abstraction: @{term restrict}*}
```
```   109
```
```   110 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   111   by (simp add: Pi_def restrict_def)
```
```   112
```
```   113 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   114   by (simp add: Pi_def restrict_def)
```
```   115
```
```   116 lemma restrict_apply [simp]:
```
```   117     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
```
```   118   by (simp add: restrict_def)
```
```   119
```
```   120 lemma restrict_ext:
```
```   121     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   122   by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
```
```   123
```
```   124 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   125   by (simp add: inj_on_def restrict_def)
```
```   126
```
```   127 lemma Id_compose:
```
```   128     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   129   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   130
```
```   131 lemma compose_Id:
```
```   132     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   133   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   134
```
```   135 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   136   by (auto simp add: restrict_def)
```
```   137
```
```   138
```
```   139 subsection{*Bijections Between Sets*}
```
```   140
```
```   141 text{*The basic definition could be moved to @{text "Fun.thy"}, but most of
```
```   142 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
```
```   143
```
```   144 definition
```
```   145   bij_betw :: "['a => 'b, 'a set, 'b set] => bool"         -- {* bijective *}
```
```   146   "bij_betw f A B = (inj_on f A & f ` A = B)"
```
```   147
```
```   148 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   149   by (simp add: bij_betw_def)
```
```   150
```
```   151 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   152   by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
```
```   153
```
```   154 lemma bij_betw_Inv: "bij_betw f A B \<Longrightarrow> bij_betw (Inv A f) B A"
```
```   155   apply (auto simp add: bij_betw_def inj_on_Inv Inv_mem)
```
```   156   apply (simp add: image_compose [symmetric] o_def)
```
```   157   apply (simp add: image_def Inv_f_f)
```
```   158   done
```
```   159
```
```   160 lemma inj_on_compose:
```
```   161     "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
```
```   162   by (auto simp add: bij_betw_def inj_on_def compose_eq)
```
```   163
```
```   164 lemma bij_betw_compose:
```
```   165     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   166   apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   167   apply (auto simp add: compose_def image_def)
```
```   168   done
```
```   169
```
```   170 lemma bij_betw_restrict_eq [simp]:
```
```   171      "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   172   by (simp add: bij_betw_def)
```
```   173
```
```   174
```
```   175 subsection{*Extensionality*}
```
```   176
```
```   177 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
```
```   178   by (simp add: extensional_def)
```
```   179
```
```   180 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   181   by (simp add: restrict_def extensional_def)
```
```   182
```
```   183 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   184   by (simp add: compose_def)
```
```   185
```
```   186 lemma extensionalityI:
```
```   187     "[| f \<in> extensional A; g \<in> extensional A;
```
```   188       !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   189   by (force simp add: expand_fun_eq extensional_def)
```
```   190
```
```   191 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
```
```   192   by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
```
```   193
```
```   194 lemma compose_Inv_id:
```
```   195     "bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
```
```   196   apply (simp add: bij_betw_def compose_def)
```
```   197   apply (rule restrict_ext, auto)
```
```   198   apply (erule subst)
```
```   199   apply (simp add: Inv_f_f)
```
```   200   done
```
```   201
```
```   202 lemma compose_id_Inv:
```
```   203     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
```
```   204   apply (simp add: compose_def)
```
```   205   apply (rule restrict_ext)
```
```   206   apply (simp add: f_Inv_f)
```
```   207   done
```
```   208
```
```   209
```
```   210 subsection{*Cardinality*}
```
```   211
```
```   212 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
```
```   213   apply (rule card_inj_on_le)
```
```   214     apply (auto simp add: Pi_def)
```
```   215   done
```
```   216
```
```   217 lemma card_bij:
```
```   218      "[|f \<in> A\<rightarrow>B; inj_on f A;
```
```   219         g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   220   by (blast intro: card_inj order_antisym)
```
```   221
```
```   222
```
```   223 (*The following declarations generate polymorphic Skolem functions for
```
```   224   these theorems. Eventually they should become redundant, once this
```
```   225   is done automatically.*)
```
```   226
```
```   227 declare FuncSet.Pi_I [skolem]
```
```   228 declare FuncSet.Pi_mono [skolem]
```
```   229 declare FuncSet.extensionalityI [skolem]
```
```   230 declare FuncSet.funcsetI [skolem]
```
```   231 declare FuncSet.restrictI [skolem]
```
```   232 declare FuncSet.restrict_in_funcset [skolem]
```
```   233
```
```   234 end
```