src/HOL/Fun.thy
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(*  Title:      HOL/Fun.thy
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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*)
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header {* Notions about functions *}
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theory Fun
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imports Complete_Lattices
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keywords "enriched_type" :: thy_goal
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begin
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lemma apply_inverse:
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  "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
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  by auto
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subsection {* The Identity Function @{text id} *}
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definition id :: "'a \<Rightarrow> 'a" where
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  "id = (\<lambda>x. x)"
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lemma id_apply [simp]: "id x = x"
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  by (simp add: id_def)
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lemma image_id [simp]: "image id = id"
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  by (simp add: id_def fun_eq_iff)
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lemma vimage_id [simp]: "vimage id = id"
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  by (simp add: id_def fun_eq_iff)
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subsection {* The Composition Operator @{text "f \<circ> g"} *}
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definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) where
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  "f o g = (\<lambda>x. f (g x))"
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notation (xsymbols)
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  comp  (infixl "\<circ>" 55)
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notation (HTML output)
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  comp  (infixl "\<circ>" 55)
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lemma comp_apply [simp]: "(f o g) x = f (g x)"
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  by (simp add: comp_def)
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lemma comp_assoc: "(f o g) o h = f o (g o h)"
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  by (simp add: fun_eq_iff)
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lemma id_comp [simp]: "id o g = g"
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  by (simp add: fun_eq_iff)
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lemma comp_id [simp]: "f o id = f"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_dest:
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  "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
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  by (simp add: fun_eq_iff)
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lemma comp_eq_elim:
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  "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
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  by (simp add: fun_eq_iff) 
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lemma image_comp:
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  "(f o g) ` r = f ` (g ` r)"
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  by auto
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lemma vimage_comp:
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  "(g \<circ> f) -` x = f -` (g -` x)"
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  by auto
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lemma INF_comp:
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  "INFI A (g \<circ> f) = INFI (f ` A) g"
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  by (simp add: INF_def image_comp)
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lemma SUP_comp:
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  "SUPR A (g \<circ> f) = SUPR (f ` A) g"
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  by (simp add: SUP_def image_comp)
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subsection {* The Forward Composition Operator @{text fcomp} *}
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definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
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  "f \<circ>> g = (\<lambda>x. g (f x))"
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lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
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  by (simp add: fcomp_def)
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lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
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  by (simp add: fcomp_def)
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lemma id_fcomp [simp]: "id \<circ>> g = g"
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  by (simp add: fcomp_def)
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lemma fcomp_id [simp]: "f \<circ>> id = f"
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  by (simp add: fcomp_def)
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code_const fcomp
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  (Eval infixl 1 "#>")
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no_notation fcomp (infixl "\<circ>>" 60)
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subsection {* Mapping functions *}
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definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
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  "map_fun f g h = g \<circ> h \<circ> f"
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lemma map_fun_apply [simp]:
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  "map_fun f g h x = g (h (f x))"
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  by (simp add: map_fun_def)
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subsection {* Injectivity and Bijectivity *}
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definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective"
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  "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
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definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"
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  "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
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text{*A common special case: functions injective, surjective or bijective over
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the entire domain type.*}
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abbreviation
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  "inj f \<equiv> inj_on f UNIV"
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abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective"
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  "surj f \<equiv> (range f = UNIV)"
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abbreviation
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  "bij f \<equiv> bij_betw f UNIV UNIV"
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text{* The negated case: *}
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translations
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"\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"
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lemma injI:
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  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
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  shows "inj f"
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  using assms unfolding inj_on_def by auto
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theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
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  by (unfold inj_on_def, blast)
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lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
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by (simp add: inj_on_def)
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lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
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by (force simp add: inj_on_def)
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lemma inj_on_cong:
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  "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   154
unfolding inj_on_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   155
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   156
lemma inj_on_strict_subset:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   157
  "\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   158
unfolding inj_on_def unfolding image_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   159
38620
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   160
lemma inj_comp:
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   161
  "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   162
  by (simp add: inj_on_def)
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   163
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   164
lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39213
diff changeset
   165
  by (simp add: inj_on_def fun_eq_iff)
38620
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   166
32988
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   167
lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   168
by (simp add: inj_on_eq_iff)
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   169
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   170
lemma inj_on_id[simp]: "inj_on id A"
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   171
  by (simp add: inj_on_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   172
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   173
lemma inj_on_id2[simp]: "inj_on (%x. x) A"
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   174
by (simp add: inj_on_def)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   175
46586
abbec6fa25c8 generalizing inj_on_Int
bulwahn
parents: 46420
diff changeset
   176
lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)"
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   177
unfolding inj_on_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   178
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   179
lemma inj_on_INTER:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   180
  "\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)\<rbrakk> \<Longrightarrow> inj_on f (\<Inter> i \<in> I. A i)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   181
unfolding inj_on_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   182
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   183
lemma inj_on_Inter:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   184
  "\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   185
unfolding inj_on_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   186
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   187
lemma inj_on_UNION_chain:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   188
  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   189
         INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   190
  shows "inj_on f (\<Union> i \<in> I. A i)"
49905
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   191
proof -
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   192
  {
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   193
    fix i j x y
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   194
    assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   195
      and ***: "f x = f y"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   196
    have "x = y"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   197
    proof -
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   198
      {
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   199
        assume "A i \<le> A j"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   200
        with ** have "x \<in> A j" by auto
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   201
        with INJ * ** *** have ?thesis
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   202
        by(auto simp add: inj_on_def)
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   203
      }
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   204
      moreover
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   205
      {
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   206
        assume "A j \<le> A i"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   207
        with ** have "y \<in> A i" by auto
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   208
        with INJ * ** *** have ?thesis
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   209
        by(auto simp add: inj_on_def)
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   210
      }
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   211
      ultimately show ?thesis using CH * by blast
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   212
    qed
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   213
  }
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   214
  then show ?thesis by (unfold inj_on_def UNION_eq) auto
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   215
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   216
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   217
lemma surj_id: "surj id"
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   218
by simp
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   219
39101
606432dd1896 Revert bij_betw changes to simp set (Problem in afp/Ordinals_and_Cardinals)
hoelzl
parents: 39076
diff changeset
   220
lemma bij_id[simp]: "bij id"
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   221
by (simp add: bij_betw_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   222
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   223
lemma inj_onI:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   224
    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   225
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   226
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   227
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   228
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   229
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   230
lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   231
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   232
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   233
lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   234
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   235
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   236
lemma comp_inj_on:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   237
     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   238
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   240
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   241
apply(simp add:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   242
apply blast
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   243
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   244
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   245
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   246
  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   247
apply(unfold inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   248
apply blast
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   249
done
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   250
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   251
lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   252
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   253
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   254
lemma inj_singleton: "inj (%s. {s})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   255
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   256
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   257
lemma inj_on_empty[iff]: "inj_on f {}"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   258
by(simp add: inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   259
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   260
lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   261
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   262
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   263
lemma inj_on_Un:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   264
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   265
  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   266
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   267
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   268
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   269
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   270
lemma inj_on_insert[iff]:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   271
  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   272
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   273
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   274
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   275
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   276
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   277
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   278
apply (blast)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   279
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   280
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   281
lemma comp_inj_on_iff:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   282
  "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   283
by(auto simp add: comp_inj_on inj_on_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   284
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   285
lemma inj_on_imageI2:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   286
  "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   287
by(auto simp add: comp_inj_on inj_on_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   288
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   289
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   290
  by auto
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   291
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   292
lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   293
  using *[symmetric] by auto
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   294
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   295
lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   296
  by (simp add: surj_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   297
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   298
lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   299
  by (simp add: surj_def, blast)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   300
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   301
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   302
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   303
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   304
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   305
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   306
39074
211e4f6aad63 bij <--> bij_betw
hoelzl
parents: 38620
diff changeset
   307
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   308
  unfolding bij_betw_def by auto
39074
211e4f6aad63 bij <--> bij_betw
hoelzl
parents: 38620
diff changeset
   309
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   310
lemma bij_betw_empty1:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   311
  assumes "bij_betw f {} A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   312
  shows "A = {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   313
using assms unfolding bij_betw_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   314
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   315
lemma bij_betw_empty2:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   316
  assumes "bij_betw f A {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   317
  shows "A = {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   318
using assms unfolding bij_betw_def by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   319
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   320
lemma inj_on_imp_bij_betw:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   321
  "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   322
unfolding bij_betw_def by simp
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   323
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   324
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   325
  unfolding bij_betw_def ..
39074
211e4f6aad63 bij <--> bij_betw
hoelzl
parents: 38620
diff changeset
   326
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   327
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   328
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   329
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   330
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   331
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   332
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   333
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   334
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   335
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   336
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   337
by (simp add: bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   338
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   339
lemma bij_betw_trans:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   340
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   341
by(auto simp add:bij_betw_def comp_inj_on)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   342
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   343
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   344
  by (rule bij_betw_trans)
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   345
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   346
lemma bij_betw_comp_iff:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   347
  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   348
by(auto simp add: bij_betw_def inj_on_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   349
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   350
lemma bij_betw_comp_iff2:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   351
  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   352
  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   353
using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   354
proof(auto simp add: bij_betw_comp_iff)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   355
  assume *: "bij_betw (f' \<circ> f) A A''"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   356
  thus "bij_betw f A A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   357
  using IM
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   358
  proof(auto simp add: bij_betw_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   359
    assume "inj_on (f' \<circ> f) A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   360
    thus "inj_on f A" using inj_on_imageI2 by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   361
  next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   362
    fix a' assume **: "a' \<in> A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   363
    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   364
    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   365
    unfolding bij_betw_def by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   366
    hence "f a \<in> A'" using IM by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   367
    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   368
    thus "a' \<in> f ` A" using 1 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   369
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   370
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   371
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   372
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   373
proof -
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   374
  have i: "inj_on f A" and s: "f ` A = B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   375
    using assms by(auto simp:bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   376
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   377
  { fix a b assume P: "?P b a"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   378
    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   379
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   380
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   381
  } note g = this
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   382
  have "inj_on ?g B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   383
  proof(rule inj_onI)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   384
    fix x y assume "x:B" "y:B" "?g x = ?g y"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   385
    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   386
    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   387
    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   388
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   389
  moreover have "?g ` B = A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   390
  proof(auto simp:image_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   391
    fix b assume "b:B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   392
    with s obtain a where P: "?P b a" unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   393
    thus "?g b \<in> A" using g[OF P] by auto
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   394
  next
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   395
    fix a assume "a:A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   396
    then obtain b where P: "?P b a" using s unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   397
    then have "b:B" using s unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   398
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   399
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   400
  ultimately show ?thesis by(auto simp:bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   401
qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   402
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   403
lemma bij_betw_cong:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   404
  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   405
unfolding bij_betw_def inj_on_def by force
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   406
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   407
lemma bij_betw_id[intro, simp]:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   408
  "bij_betw id A A"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   409
unfolding bij_betw_def id_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   410
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   411
lemma bij_betw_id_iff:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   412
  "bij_betw id A B \<longleftrightarrow> A = B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   413
by(auto simp add: bij_betw_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   414
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   415
lemma bij_betw_combine:
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   416
  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   417
  shows "bij_betw f (A \<union> C) (B \<union> D)"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   418
  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   419
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   420
lemma bij_betw_UNION_chain:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   421
  assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   422
         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   423
  shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
49905
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   424
proof (unfold bij_betw_def, auto)
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   425
  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   426
  using BIJ bij_betw_def[of f] by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   427
  thus "inj_on f (\<Union> i \<in> I. A i)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   428
  using CH inj_on_UNION_chain[of I A f] by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   429
next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   430
  fix i x
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   431
  assume *: "i \<in> I" "x \<in> A i"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   432
  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   433
  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   434
next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   435
  fix i x'
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   436
  assume *: "i \<in> I" "x' \<in> A' i"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   437
  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
49905
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   438
  then have "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   439
    using * by blast
a81f95693c68 simp results for simplification results of Inf/Sup expressions on bool;
haftmann
parents: 49739
diff changeset
   440
  then show "x' \<in> f ` (\<Union>x\<in>I. A x)" by (simp add: image_def)
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   441
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   442
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   443
lemma bij_betw_subset:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   444
  assumes BIJ: "bij_betw f A A'" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   445
          SUB: "B \<le> A" and IM: "f ` B = B'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   446
  shows "bij_betw f B B'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   447
using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   448
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   449
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   450
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   451
by simp
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   452
42903
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   453
lemma surj_vimage_empty:
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   454
  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   455
  using surj_image_vimage_eq[OF `surj f`, of A]
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44860
diff changeset
   456
  by (intro iffI) fastforce+
42903
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   457
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   458
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   459
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   460
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   461
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   462
by (blast intro: sym)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   463
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   464
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   465
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   466
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   467
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   468
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   469
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   470
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   471
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   472
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   473
by(blast dest: inj_onD)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   474
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   475
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   476
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   477
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   478
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   479
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   480
lemma inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   481
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   482
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   483
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   484
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   485
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   486
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   487
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   488
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   489
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   490
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   491
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   492
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   493
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   494
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   495
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   496
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   497
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   498
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   499
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   500
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   501
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   502
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   503
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   504
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   505
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   506
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   507
(*Compare with image_INT: no use of inj_on, and if f is surjective then
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   508
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   509
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   510
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   511
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   512
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   513
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   514
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   515
by auto
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   516
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   517
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   518
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   519
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   520
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   521
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   522
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   523
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   524
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   525
41657
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   526
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   527
  -- {* The inverse image of a singleton under an injective function
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   528
         is included in a singleton. *}
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   529
  apply (auto simp add: inj_on_def)
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   530
  apply (blast intro: the_equality [symmetric])
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   531
  done
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   532
43991
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   533
lemma inj_on_vimage_singleton:
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   534
  "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   535
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   536
35584
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   537
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35416
diff changeset
   538
  by (auto intro!: inj_onI)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   539
35584
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   540
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   541
  by (auto intro!: inj_onI dest: strict_mono_eq)
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   542
41657
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   543
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   544
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   545
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   546
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   547
  "fun_upd f a b == % x. if x=a then b else f x"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   548
41229
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 40969
diff changeset
   549
nonterminal updbinds and updbind
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 40969
diff changeset
   550
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   551
syntax
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   552
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   553
  ""         :: "updbind => updbinds"             ("_")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   554
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   555
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   556
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   557
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   558
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   559
  "f(x:=y)" == "CONST fun_upd f x y"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   560
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   561
(* Hint: to define the sum of two functions (or maps), use sum_case.
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   562
         A nice infix syntax could be defined (in Datatype.thy or below) by
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   563
notation
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   564
  sum_case  (infixr "'(+')"80)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   565
*)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   566
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   567
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   568
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   569
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   570
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   571
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   572
45603
d2d9ef16ccaf explicit is better than implicit;
wenzelm
parents: 45174
diff changeset
   573
lemma fun_upd_idem: "f x = y ==> f(x:=y) = f"
d2d9ef16ccaf explicit is better than implicit;
wenzelm
parents: 45174
diff changeset
   574
  by (simp only: fun_upd_idem_iff)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   575
45603
d2d9ef16ccaf explicit is better than implicit;
wenzelm
parents: 45174
diff changeset
   576
lemma fun_upd_triv [iff]: "f(x := f x) = f"
d2d9ef16ccaf explicit is better than implicit;
wenzelm
parents: 45174
diff changeset
   577
  by (simp only: fun_upd_idem)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   578
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   579
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   580
by (simp add: fun_upd_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   581
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   582
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   583
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   584
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   585
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   586
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   587
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   588
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   589
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   590
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39213
diff changeset
   591
by (simp add: fun_eq_iff)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   592
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   593
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   594
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   595
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   596
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44860
diff changeset
   597
by (fastforce simp:inj_on_def image_def)
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   598
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   599
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   600
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   601
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   602
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30301
diff changeset
   603
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
44921
58eef4843641 tuned proofs
huffman
parents: 44890
diff changeset
   604
  by auto
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30301
diff changeset
   605
44744
bdf8eb8f126b added new lemmas
nipkow
parents: 44277
diff changeset
   606
lemma UNION_fun_upd:
bdf8eb8f126b added new lemmas
nipkow
parents: 44277
diff changeset
   607
  "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
bdf8eb8f126b added new lemmas
nipkow
parents: 44277
diff changeset
   608
by (auto split: if_splits)
bdf8eb8f126b added new lemmas
nipkow
parents: 44277
diff changeset
   609
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   610
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   611
subsection {* @{text override_on} *}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   612
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   613
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   614
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   615
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   616
lemma override_on_emptyset[simp]: "override_on f g {} = f"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   617
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   618
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   619
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   620
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   621
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   622
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   623
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   624
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   625
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   626
subsection {* @{text swap} *}
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   627
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   628
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" where
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   629
  "swap a b f = f (a := f b, b:= f a)"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   630
34101
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   631
lemma swap_self [simp]: "swap a a f = f"
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   632
by (simp add: swap_def)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   633
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   634
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   635
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   636
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   637
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   638
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   639
34145
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   640
lemma swap_triple:
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   641
  assumes "a \<noteq> c" and "b \<noteq> c"
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   642
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39213
diff changeset
   643
  using assms by (simp add: fun_eq_iff swap_def)
34145
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   644
34101
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   645
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   646
by (rule ext, simp add: fun_upd_def swap_def)
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   647
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   648
lemma swap_image_eq [simp]:
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   649
  assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   650
proof -
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   651
  have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   652
    using assms by (auto simp: image_iff swap_def)
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   653
  then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   654
  with subset[of f] show ?thesis by auto
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   655
qed
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   656
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   657
lemma inj_on_imp_inj_on_swap:
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   658
  "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   659
  by (simp add: inj_on_def swap_def, blast)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   660
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   661
lemma inj_on_swap_iff [simp]:
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   662
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   663
proof
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   664
  assume "inj_on (swap a b f) A"
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   665
  with A have "inj_on (swap a b (swap a b f)) A"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   666
    by (iprover intro: inj_on_imp_inj_on_swap)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   667
  thus "inj_on f A" by simp
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   668
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   669
  assume "inj_on f A"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   670
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   671
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   672
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   673
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   674
  by simp
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   675
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   676
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   677
  by simp
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   678
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   679
lemma bij_betw_swap_iff [simp]:
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   680
  "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   681
  by (auto simp: bij_betw_def)
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   682
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   683
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   684
  by simp
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   685
36176
3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents: 35584
diff changeset
   686
hide_const (open) swap
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   687
31949
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   688
subsection {* Inversion of injective functions *}
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   689
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   690
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   691
  "the_inv_into A f == %x. THE y. y : A & f y = x"
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   692
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   693
lemma the_inv_into_f_f:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   694
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   695
apply (simp add: the_inv_into_def inj_on_def)
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   696
apply blast
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   697
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   698
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   699
lemma f_the_inv_into_f:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   700
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   701
apply (simp add: the_inv_into_def)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   702
apply (rule the1I2)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   703
 apply(blast dest: inj_onD)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   704
apply blast
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   705
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   706
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   707
lemma the_inv_into_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   708
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   709
apply (simp add: the_inv_into_def)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   710
apply (rule the1I2)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   711
 apply(blast dest: inj_onD)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   712
apply blast
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   713
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   714
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   715
lemma the_inv_into_onto[simp]:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   716
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   717
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   718
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   719
lemma the_inv_into_f_eq:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   720
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   721
  apply (erule subst)
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   722
  apply (erule the_inv_into_f_f, assumption)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   723
  done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   724
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   725
lemma the_inv_into_comp:
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   726
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   727
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   728
apply (rule the_inv_into_f_eq)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   729
  apply (fast intro: comp_inj_on)
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   730
 apply (simp add: f_the_inv_into_f the_inv_into_into)
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   731
apply (simp add: the_inv_into_into)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   732
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   733
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   734
lemma inj_on_the_inv_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   735
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   736
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   737
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   738
lemma bij_betw_the_inv_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   739
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   740
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   741
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   742
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   743
  "the_inv f \<equiv> the_inv_into UNIV f"
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   744
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   745
lemma the_inv_f_f:
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   746
  assumes "inj f"
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   747
  shows "the_inv f (f x) = x" using assms UNIV_I
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   748
  by (rule the_inv_into_f_f)
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   749
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   750
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   751
subsection {* Cantor's Paradox *}
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   752
42238
d53dccb38dd1 added "no_atp" to Cantor's paradox
blanchet
parents: 41657
diff changeset
   753
lemma Cantors_paradox [no_atp]:
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   754
  "\<not>(\<exists>f. f ` A = Pow A)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   755
proof clarify
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   756
  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   757
  let ?X = "{a \<in> A. a \<notin> f a}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   758
  have "?X \<in> Pow A" unfolding Pow_def by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   759
  with * obtain x where "x \<in> A \<and> f x = ?X" by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   760
  thus False by best
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   761
qed
31949
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   762
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   763
subsection {* Setup *} 
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   764
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   765
subsubsection {* Proof tools *}
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   766
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   767
text {* simplifies terms of the form
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   768
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   769
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   770
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   771
let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   772
  fun gen_fun_upd NONE T _ _ = NONE
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   773
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   774
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   775
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   776
    let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   777
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   778
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   779
        | find t = NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   780
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   781
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   782
  fun proc ss ct =
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   783
    let
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   784
      val ctxt = Simplifier.the_context ss
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   785
      val t = Thm.term_of ct
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   786
    in
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   787
      case find_double t of
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   788
        (T, NONE) => NONE
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   789
      | (T, SOME rhs) =>
27330
1af2598b5f7d Logic.all/mk_equals/mk_implies;
wenzelm
parents: 27188
diff changeset
   790
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   791
            (fn _ =>
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   792
              rtac eq_reflection 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   793
              rtac ext 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   794
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   795
    end
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   796
in proc end
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   797
*}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   798
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   799
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   800
subsubsection {* Code generator *}
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   801
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   802
code_const "op \<circ>"
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   803
  (SML infixl 5 "o")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   804
  (Haskell infixr 9 ".")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   805
21906
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   806
code_const "id"
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   807
  (Haskell "id")
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   808
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   809
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   810
subsubsection {* Functorial structure of types *}
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   811
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47579
diff changeset
   812
ML_file "Tools/enriched_type.ML"
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   813
47488
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 46950
diff changeset
   814
enriched_type map_fun: map_fun
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 46950
diff changeset
   815
  by (simp_all add: fun_eq_iff)
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 46950
diff changeset
   816
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 46950
diff changeset
   817
enriched_type vimage
49739
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   818
  by (simp_all add: fun_eq_iff vimage_comp)
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   819
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   820
text {* Legacy theorem names *}
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   821
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   822
lemmas o_def = comp_def
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   823
lemmas o_apply = comp_apply
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   824
lemmas o_assoc = comp_assoc [symmetric]
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   825
lemmas id_o = id_comp
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   826
lemmas o_id = comp_id
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   827
lemmas o_eq_dest = comp_eq_dest
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   828
lemmas o_eq_elim = comp_eq_elim
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   829
lemmas image_compose = image_comp
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 48891
diff changeset
   830
lemmas vimage_compose = vimage_comp
47488
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 46950
diff changeset
   831
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   832
end
47488
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 46950
diff changeset
   833