src/HOL/Fun.thy
author haftmann
Thu, 18 Aug 2011 13:25:17 +0200
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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_Lattice
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uses ("Tools/enriched_type.ML")
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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]: "id ` Y = Y"
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  by (simp add: id_def)
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lemma vimage_id [simp]: "id -` A = A"
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  by (simp add: id_def)
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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 o_apply [simp]: "(f o g) x = f (g x)"
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by (simp add: comp_def)
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lemma o_assoc: "f o (g o h) = f o g o h"
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by (simp add: comp_def)
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lemma id_o [simp]: "id o g = g"
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by (simp add: comp_def)
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lemma o_id [simp]: "f o id = f"
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by (simp add: comp_def)
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lemma o_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 only: comp_def) (fact fun_cong)
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lemma o_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 (erule meta_mp) (fact o_eq_dest) 
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lemma image_compose: "(f o g) ` r = f`(g`r)"
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by (simp add: comp_def, blast)
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lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
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  by auto
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lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
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by (unfold comp_def, blast)
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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"
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unfolding inj_on_def by auto
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lemma inj_on_strict_subset:
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  "\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B"
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hoelzl
parents: 40702
diff changeset
   151
unfolding inj_on_def unfolding image_def by blast
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hoelzl
parents: 40702
diff changeset
   152
38620
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   153
lemma inj_comp:
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   154
  "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   155
  by (simp add: inj_on_def)
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   156
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   157
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
   158
  by (simp add: inj_on_def fun_eq_iff)
38620
b40524b74f77 inj_comp and inj_fun
haftmann
parents: 37767
diff changeset
   159
32988
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   160
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
   161
by (simp add: inj_on_eq_iff)
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   162
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   163
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
   164
  by (simp add: inj_on_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   165
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   166
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
   167
by (simp add: inj_on_def)
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ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   168
40703
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hoelzl
parents: 40702
diff changeset
   169
lemma inj_on_Int: "\<lbrakk>inj_on f A; inj_on f B\<rbrakk> \<Longrightarrow> inj_on f (A \<inter> B)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   170
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
   171
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   172
lemma inj_on_INTER:
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hoelzl
parents: 40702
diff changeset
   173
  "\<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
   174
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
   175
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   176
lemma inj_on_Inter:
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hoelzl
parents: 40702
diff changeset
   177
  "\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)"
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hoelzl
parents: 40702
diff changeset
   178
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
   179
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   180
lemma inj_on_UNION_chain:
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hoelzl
parents: 40702
diff changeset
   181
  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
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hoelzl
parents: 40702
diff changeset
   182
         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
   183
  shows "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
   184
proof(unfold inj_on_def UNION_def, auto)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   185
  fix i j x y
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   186
  assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   187
         and ***: "f x = f y"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   188
  show "x = y"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   189
  proof-
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   190
    {assume "A i \<le> A j"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   191
     with ** have "x \<in> A j" by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   192
     with INJ * ** *** have ?thesis
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   193
     by(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
   194
    }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   195
    moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   196
    {assume "A j \<le> A i"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   197
     with ** have "y \<in> A i" by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   198
     with INJ * ** *** have ?thesis
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hoelzl
parents: 40702
diff changeset
   199
     by(auto simp add: inj_on_def)
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hoelzl
parents: 40702
diff changeset
   200
    }
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   201
    ultimately show ?thesis using  CH * by blast
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hoelzl
parents: 40702
diff changeset
   202
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   203
qed
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hoelzl
parents: 40702
diff changeset
   204
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
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   205
lemma surj_id: "surj id"
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hoelzl
parents: 40602
diff changeset
   206
by simp
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   207
39101
606432dd1896 Revert bij_betw changes to simp set (Problem in afp/Ordinals_and_Cardinals)
hoelzl
parents: 39076
diff changeset
   208
lemma bij_id[simp]: "bij id"
39076
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hoelzl
parents: 39075
diff changeset
   209
by (simp add: bij_betw_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   210
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   211
lemma inj_onI:
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paulson
parents: 12460
diff changeset
   212
    "(!! 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
   213
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   214
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   215
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
   216
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   217
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   218
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
   219
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   220
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   221
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
   222
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   223
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   224
lemma comp_inj_on:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   225
     "[| 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
   226
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   227
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   228
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   229
apply(simp add:inj_on_def image_def)
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nipkow
parents: 15140
diff changeset
   230
apply blast
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   231
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   232
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   233
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
   234
  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
   235
apply(unfold inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   236
apply blast
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   237
done
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   238
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
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
   240
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   241
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   242
lemma inj_singleton: "inj (%s. {s})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   243
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   244
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   245
lemma inj_on_empty[iff]: "inj_on f {}"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   246
by(simp add: inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   247
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   248
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
   249
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   250
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   251
lemma inj_on_Un:
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nipkow
parents: 14565
diff changeset
   252
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   253
  (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
   254
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   255
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   256
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   257
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   258
lemma inj_on_insert[iff]:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   259
  "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
   260
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   261
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   262
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   263
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   264
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   265
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   266
apply (blast)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   267
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   268
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   269
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
   270
  "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
   271
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
   272
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   273
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
   274
  "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
   275
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
   276
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   277
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
   278
  by auto
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   279
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   280
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
   281
  using *[symmetric] by auto
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   282
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   283
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
   284
  by (simp add: surj_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   285
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   286
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
   287
  by (simp add: surj_def, blast)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   288
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   289
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   290
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   291
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   292
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   293
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   294
39074
211e4f6aad63 bij <--> bij_betw
hoelzl
parents: 38620
diff changeset
   295
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
   296
  unfolding bij_betw_def by auto
39074
211e4f6aad63 bij <--> bij_betw
hoelzl
parents: 38620
diff changeset
   297
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   298
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
   299
  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
   300
  shows "A = {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   301
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
   302
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   303
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
   304
  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
   305
  shows "A = {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   306
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
   307
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   308
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
   309
  "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
   310
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
   311
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   312
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
   313
  unfolding bij_betw_def ..
39074
211e4f6aad63 bij <--> bij_betw
hoelzl
parents: 38620
diff changeset
   314
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   315
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   316
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   317
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   318
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   319
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   320
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   321
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   322
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   323
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   324
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
   325
by (simp add: bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   326
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   327
lemma bij_betw_trans:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   328
  "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
   329
by(auto simp add:bij_betw_def comp_inj_on)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   330
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   331
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
   332
  by (rule bij_betw_trans)
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40602
diff changeset
   333
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   334
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
   335
  "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
   336
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
   337
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   338
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
   339
  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
   340
  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
   341
using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   342
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
   343
  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
   344
  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
   345
  using IM
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   346
  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
   347
    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
   348
    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
   349
  next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   350
    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
   351
    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
   352
    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
   353
    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
   354
    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
   355
    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
   356
    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
   357
  qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   358
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   359
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   360
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
   361
proof -
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   362
  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
   363
    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
   364
  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
   365
  { 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
   366
    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
   367
    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
   368
    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
   369
  } note g = this
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   370
  have "inj_on ?g B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   371
  proof(rule inj_onI)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   372
    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
   373
    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
   374
    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
   375
    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
   376
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   377
  moreover have "?g ` B = A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   378
  proof(auto simp:image_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   379
    fix b assume "b:B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   380
    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
   381
    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
   382
  next
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   383
    fix a assume "a:A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   384
    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
   385
    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
   386
    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
   387
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   388
  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
   389
qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   390
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   391
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
   392
  "(\<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
   393
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
   394
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   395
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
   396
  "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
   397
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
   398
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   399
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
   400
  "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
   401
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
   402
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   403
lemma bij_betw_combine:
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   404
  assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   405
  shows "bij_betw f (A \<union> C) (B \<union> D)"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   406
  using assms unfolding bij_betw_def inj_on_Un image_Un by auto
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   407
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   408
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
   409
  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
   410
         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
   411
  shows "bij_betw f (\<Union> i \<in> I. A i) (\<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
   412
proof(unfold bij_betw_def, auto simp add: image_def)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   413
  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
   414
  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
   415
  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
   416
  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
   417
next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   418
  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
   419
  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
   420
  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
   421
  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
   422
next
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   423
  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
   424
  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
   425
  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   426
  thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   427
  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
   428
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   429
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   430
lemma bij_betw_Disj_Un:
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   431
  assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   432
          B1: "bij_betw f A A'" and B2: "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
   433
  shows "bij_betw f (A \<union> B) (A' \<union> B')"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   434
proof-
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   435
  have 1: "inj_on f A \<and> inj_on f B"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   436
  using B1 B2 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
   437
  have 2: "f`A = A' \<and> 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
   438
  using B1 B2 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
   439
  hence "f`(A - B) \<inter> f`(B - A) = {}"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   440
  using DISJ DISJ' by blast
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   441
  hence "inj_on f (A \<union> B)"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   442
  using 1 by (auto simp add: inj_on_Un)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   443
  (*  *)
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   444
  moreover
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   445
  have "f`(A \<union> B) = A' \<union> B'"
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   446
  using 2 by auto
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   447
  ultimately show ?thesis
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   448
  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
   449
qed
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   450
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   451
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
   452
  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
   453
          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
   454
  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
   455
using assms
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   456
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
   457
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   458
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
   459
by simp
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   460
42903
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   461
lemma surj_vimage_empty:
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   462
  assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   463
  using surj_image_vimage_eq[OF `surj f`, of A]
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   464
  by (intro iffI) fastsimp+
ec9eb1fbfcb8 add surj_vimage_empty
hoelzl
parents: 42238
diff changeset
   465
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   466
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   467
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   468
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   469
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
   470
by (blast intro: sym)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   471
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   472
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   473
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   474
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   475
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   476
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   477
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
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
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   480
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
   481
by(blast dest: inj_onD)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   482
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   483
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   484
   "[| 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
   485
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   486
done
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 inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   489
   "[| 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
   490
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   491
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   492
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   493
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
   494
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   495
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   496
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   497
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   498
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   499
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   500
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   501
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   502
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   503
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   504
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   505
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   506
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   507
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   508
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   509
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   510
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   511
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   512
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   513
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   514
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   515
(*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
   516
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   517
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
   518
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   519
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   520
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   521
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   522
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
   523
by auto
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   524
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   525
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   526
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   527
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   528
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   529
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   530
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   531
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
   532
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   533
41657
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   534
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   535
  -- {* The inverse image of a singleton under an injective function
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   536
         is included in a singleton. *}
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   537
  apply (auto simp add: inj_on_def)
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   538
  apply (blast intro: the_equality [symmetric])
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   539
  done
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   540
43991
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   541
lemma inj_on_vimage_singleton:
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   542
  "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
   543
  by (auto simp add: inj_on_def intro: the_equality [symmetric])
f4a7697011c5 finite vimage on arbitrary domains
hoelzl
parents: 43874
diff changeset
   544
35584
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   545
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
   546
  by (auto intro!: inj_onI)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   547
35584
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   548
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
   549
  by (auto intro!: inj_onI dest: strict_mono_eq)
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   550
41657
89451110ba8e moved theorem
haftmann
parents: 41505
diff changeset
   551
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   552
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   553
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   554
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   555
  "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
   556
41229
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 40969
diff changeset
   557
nonterminal updbinds and updbind
d797baa3d57c replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents: 40969
diff changeset
   558
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   559
syntax
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   560
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   561
  ""         :: "updbind => updbinds"             ("_")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   562
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   563
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   564
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   565
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   566
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   567
  "f(x:=y)" == "CONST fun_upd f x y"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   568
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   569
(* 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
   570
         A nice infix syntax could be defined (in Datatype.thy or below) by
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   571
notation
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   572
  sum_case  (infixr "'(+')"80)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   573
*)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   574
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   575
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   576
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   577
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   578
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   579
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   580
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   581
(* f x = y ==> f(x:=y) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   582
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   583
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   584
(* f(x := f x) = f *)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   585
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   586
declare fun_upd_triv [iff]
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   587
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   588
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
   589
by (simp add: fun_upd_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   590
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   591
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   592
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   593
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   594
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   595
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   596
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   597
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   598
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   599
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
   600
by (simp add: fun_eq_iff)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   601
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   602
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
   603
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   604
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   605
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   606
by (fastsimp simp:inj_on_def image_def)
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   607
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   608
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   609
     "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
   610
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   611
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30301
diff changeset
   612
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   613
by (auto intro: ext)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30301
diff changeset
   614
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   615
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   616
subsection {* @{text override_on} *}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   617
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   618
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
   619
  "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
   620
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   621
lemma override_on_emptyset[simp]: "override_on f g {} = f"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   622
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   623
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   624
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
   625
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   626
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   627
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
   628
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   629
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   630
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   631
subsection {* @{text swap} *}
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   632
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   633
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
   634
  "swap a b f = f (a := f b, b:= f a)"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   635
34101
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   636
lemma swap_self [simp]: "swap a a f = f"
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   637
by (simp add: swap_def)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   638
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   639
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   640
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   641
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   642
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   643
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   644
34145
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   645
lemma swap_triple:
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   646
  assumes "a \<noteq> c" and "b \<noteq> c"
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   647
  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
   648
  using assms by (simp add: fun_eq_iff swap_def)
34145
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   649
34101
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   650
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
   651
by (rule ext, simp add: fun_upd_def swap_def)
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   652
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   653
lemma swap_image_eq [simp]:
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   654
  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
   655
proof -
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   656
  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
   657
    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
   658
  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
   659
  with subset[of f] show ?thesis by auto
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   660
qed
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   661
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   662
lemma inj_on_imp_inj_on_swap:
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   663
  "\<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
   664
  by (simp add: inj_on_def swap_def, blast)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   665
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   666
lemma inj_on_swap_iff [simp]:
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   667
  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
   668
proof
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   669
  assume "inj_on (swap a b f) A"
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   670
  with A have "inj_on (swap a b (swap a b f)) A"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   671
    by (iprover intro: inj_on_imp_inj_on_swap)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   672
  thus "inj_on f A" by simp
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   673
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   674
  assume "inj_on f A"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   675
  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
   676
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   677
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   678
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
   679
  by simp
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   680
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   681
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
   682
  by simp
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   683
39076
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   684
lemma bij_betw_swap_iff [simp]:
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   685
  "\<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
   686
  by (auto simp: bij_betw_def)
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   687
b3a9b6734663 Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents: 39075
diff changeset
   688
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
   689
  by simp
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 39074
diff changeset
   690
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
   691
hide_const (open) swap
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   692
31949
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   693
subsection {* Inversion of injective functions *}
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   694
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   695
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
   696
  "the_inv_into A f == %x. THE y. y : A & f y = x"
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   697
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   698
lemma the_inv_into_f_f:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   699
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   700
apply (simp add: the_inv_into_def inj_on_def)
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   701
apply blast
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   702
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   703
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   704
lemma f_the_inv_into_f:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   705
  "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
   706
apply (simp add: the_inv_into_def)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   707
apply (rule the1I2)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   708
 apply(blast dest: inj_onD)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   709
apply blast
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   710
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   711
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   712
lemma the_inv_into_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   713
  "[| 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
   714
apply (simp add: the_inv_into_def)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   715
apply (rule the1I2)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   716
 apply(blast dest: inj_onD)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   717
apply blast
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   718
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   719
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   720
lemma the_inv_into_onto[simp]:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   721
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   722
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   723
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   724
lemma the_inv_into_f_eq:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   725
  "[| 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
   726
  apply (erule subst)
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   727
  apply (erule the_inv_into_f_f, assumption)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   728
  done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   729
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   730
lemma the_inv_into_comp:
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   731
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   732
  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
   733
apply (rule the_inv_into_f_eq)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   734
  apply (fast intro: comp_inj_on)
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   735
 apply (simp add: f_the_inv_into_f the_inv_into_into)
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   736
apply (simp add: the_inv_into_into)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   737
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   738
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   739
lemma inj_on_the_inv_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   740
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   741
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   742
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   743
lemma bij_betw_the_inv_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   744
  "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
   745
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
   746
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   747
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   748
  "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
   749
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   750
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
   751
  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
   752
  shows "the_inv f (f x) = x" using assms UNIV_I
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   753
  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
   754
44277
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   755
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   756
text{*compatibility*}
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   757
lemmas o_def = comp_def
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   758
bcb696533579 moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents: 43991
diff changeset
   759
40703
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   760
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
   761
42238
d53dccb38dd1 added "no_atp" to Cantor's paradox
blanchet
parents: 41657
diff changeset
   762
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
   763
  "\<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
   764
proof clarify
d1fc454d6735 Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents: 40702
diff changeset
   765
  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
   766
  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
   767
  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
   768
  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
   769
  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
   770
qed
31949
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   771
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   772
subsection {* Setup *} 
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   773
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   774
subsubsection {* Proof tools *}
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   775
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   776
text {* simplifies terms of the form
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   777
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   778
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   779
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   780
let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   781
  fun gen_fun_upd NONE T _ _ = NONE
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   782
    | 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
   783
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   784
  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
   785
    let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   786
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   787
            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
   788
        | find t = NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   789
    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
   790
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   791
  fun proc ss ct =
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   792
    let
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   793
      val ctxt = Simplifier.the_context ss
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   794
      val t = Thm.term_of ct
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   795
    in
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   796
      case find_double t of
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   797
        (T, NONE) => NONE
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   798
      | (T, SOME rhs) =>
27330
1af2598b5f7d Logic.all/mk_equals/mk_implies;
wenzelm
parents: 27188
diff changeset
   799
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   800
            (fn _ =>
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   801
              rtac eq_reflection 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   802
              rtac ext 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   803
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   804
    end
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   805
in proc end
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   806
*}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   807
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   808
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   809
subsubsection {* Code generator *}
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   810
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   811
types_code
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   812
  "fun"  ("(_ ->/ _)")
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   813
attach (term_of) {*
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   814
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   815
*}
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   816
attach (test) {*
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   817
fun gen_fun_type aF aT bG bT i =
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   818
  let
32740
9dd0a2f83429 explicit indication of Unsynchronized.ref;
wenzelm
parents: 32554
diff changeset
   819
    val tab = Unsynchronized.ref [];
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   820
    fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   821
      (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   822
  in
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   823
    (fn x =>
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   824
       case AList.lookup op = (!tab) x of
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   825
         NONE =>
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   826
           let val p as (y, _) = bG i
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   827
           in (tab := (x, p) :: !tab; y) end
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   828
       | SOME (y, _) => y,
28711
60e51a045755 Replaced arbitrary by undefined.
berghofe
parents: 28562
diff changeset
   829
     fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   830
  end;
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   831
*}
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   832
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   833
code_const "op \<circ>"
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   834
  (SML infixl 5 "o")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   835
  (Haskell infixr 9 ".")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   836
21906
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   837
code_const "id"
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   838
  (Haskell "id")
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   839
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   840
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   841
subsubsection {* Functorial structure of types *}
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   842
41505
6d19301074cf "enriched_type" replaces less specific "type_lifting"
haftmann
parents: 41229
diff changeset
   843
use "Tools/enriched_type.ML"
40969
fb2d3ccda5a7 moved bootstrap of type_lifting to Fun
haftmann
parents: 40968
diff changeset
   844
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   845
end