src/HOL/Fun.thy
author haftmann
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(*  Title:      HOL/Fun.thy
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    ID:         $Id$
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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 Set
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begin
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text{*As a simplification rule, it replaces all function equalities by
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  first-order equalities.*}
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lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
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apply (rule iffI)
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apply (simp (no_asm_simp))
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apply (rule ext)
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apply (simp (no_asm_simp))
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done
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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
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  id :: "'a \<Rightarrow> 'a"
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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_ident [simp]: "(%x. x) ` Y = Y"
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by blast
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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_ident [simp]: "(%x. x) -` Y = Y"
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by blast
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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
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  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
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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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text{*compatibility*}
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lemmas o_def = comp_def
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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 image_compose: "(f o g) ` r = f`(g`r)"
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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 "f \<circ>> g"} *}
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definition
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  fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
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where
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  "f o> g = (\<lambda>x. g (f x))"
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notation (xsymbols)
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  fcomp  (infixl "\<circ>>" 60)
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notation (HTML output)
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  fcomp  (infixl "\<circ>>" 60)
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lemma fcomp_apply:  "(f o> g) x = g (f x)"
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  by (simp add: fcomp_def)
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lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
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  by (simp add: fcomp_def)
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lemma id_fcomp [simp]: "id o> g = g"
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  by (simp add: fcomp_def)
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lemma fcomp_id [simp]: "f o> id = f"
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  by (simp add: fcomp_def)
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subsection {* Injectivity and Surjectivity *}
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constdefs
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  inj_on :: "['a => 'b, 'a set] => bool"  -- "injective"
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  "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
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text{*A common special case: functions injective over the entire domain type.*}
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abbreviation
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  "inj f == inj_on f UNIV"
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definition
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  bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
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  "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
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constdefs
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  surj :: "('a => 'b) => bool"                   (*surjective*)
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  "surj f == ! y. ? x. y=f(x)"
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  bij :: "('a => 'b) => bool"                    (*bijective*)
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  "bij f == inj f & surj f"
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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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text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
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lemma datatype_injI:
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    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
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by (simp add: inj_on_def)
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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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(*Useful with the simplifier*)
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lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
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by (force simp add: inj_on_def)
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lemma inj_on_id[simp]: "inj_on id A"
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  by (simp add: inj_on_def) 
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lemma inj_on_id2[simp]: "inj_on (%x. x) A"
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by (simp add: inj_on_def) 
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lemma surj_id[simp]: "surj id"
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by (simp add: surj_def) 
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lemma bij_id[simp]: "bij id"
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by (simp add: bij_def inj_on_id surj_id) 
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lemma inj_onI:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   166
    "(!! 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
   167
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   168
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   169
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
   170
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   171
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   172
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
   173
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   174
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   175
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
   176
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   177
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   178
lemma comp_inj_on:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   179
     "[| 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
   180
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   181
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   182
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   183
apply(simp add:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   184
apply blast
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   185
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   186
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   187
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
   188
  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
   189
apply(unfold inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   190
apply blast
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   191
done
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   192
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   193
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
   194
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   195
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   196
lemma inj_singleton: "inj (%s. {s})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   197
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   198
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   199
lemma inj_on_empty[iff]: "inj_on f {}"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   200
by(simp add: inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   201
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   202
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
   203
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   204
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   205
lemma inj_on_Un:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   206
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   207
  (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
   208
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   209
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   210
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   211
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   212
lemma inj_on_insert[iff]:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   213
  "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
   214
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   215
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   216
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   217
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   218
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   219
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   220
apply (blast)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   221
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   222
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   223
lemma surjI: "(!! x. g(f x) = x) ==> surj g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   224
apply (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   225
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   226
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   227
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   228
lemma surj_range: "surj f ==> range f = UNIV"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   229
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   230
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   231
lemma surjD: "surj f ==> EX x. y = f x"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   232
by (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   233
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   234
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   235
by (simp add: surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   236
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   237
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   238
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   240
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   241
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   242
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   243
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   244
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   245
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   246
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   247
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   248
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   249
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   250
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   251
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   252
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
   253
by (simp add: bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   254
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   255
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
   256
proof -
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   257
  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
   258
    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
   259
  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
   260
  { 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
   261
    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
   262
    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
   263
    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
   264
  } note g = this
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   265
  have "inj_on ?g B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   266
  proof(rule inj_onI)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   267
    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
   268
    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
   269
    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
   270
    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
   271
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   272
  moreover have "?g ` B = A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   273
  proof(auto simp:image_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   274
    fix b assume "b:B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   275
    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
   276
    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
   277
  next
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   278
    fix a assume "a:A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   279
    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
   280
    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
   281
    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
   282
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   283
  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
   284
qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   285
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   286
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   287
by (simp add: surj_range)
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 inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   290
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   291
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   292
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   293
apply (unfold surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   294
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   295
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   296
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   297
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   298
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   299
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   300
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   301
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   302
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   303
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   304
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   305
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   306
   "[| 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
   307
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   308
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   309
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   310
lemma inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   311
   "[| 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
   312
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   313
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   314
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   315
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
   316
by (simp add: inj_on_def, blast)
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 image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   319
by (simp add: inj_on_def, blast)
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 inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   322
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   323
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   324
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   325
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   326
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   327
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   328
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   329
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   330
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   331
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   332
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   333
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   334
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   335
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   336
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   337
(*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
   338
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   339
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
   340
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   341
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   342
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   343
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   344
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   345
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   346
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   347
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   348
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   349
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   350
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   351
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   352
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   353
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
   354
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   355
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   356
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   357
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   358
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   359
constdefs
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   360
  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   361
  "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
   362
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   363
nonterminals
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   364
  updbinds updbind
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   365
syntax
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   366
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   367
  ""         :: "updbind => updbinds"             ("_")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   368
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   369
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   370
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   371
translations
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   372
  "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   373
  "f(x:=y)"                     == "fun_upd f x y"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   374
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   375
(* 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
   376
         A nice infix syntax could be defined (in Datatype.thy or below) by
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   377
consts
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   378
  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   379
translations
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   380
 "fun_sum" == sum_case
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   381
*)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   382
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   383
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   384
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   385
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   386
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   387
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   388
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   389
(* f x = y ==> f(x:=y) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   390
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   391
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   392
(* f(x := f x) = f *)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   393
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   394
declare fun_upd_triv [iff]
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   395
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   396
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
   397
by (simp add: fun_upd_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   398
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   399
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   400
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   401
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   402
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   403
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   404
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   405
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   406
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   407
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   408
by (simp add: expand_fun_eq)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   409
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   410
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
   411
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   412
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   413
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   414
by(fastsimp simp:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   415
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   416
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   417
     "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
   418
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   419
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   420
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   421
subsection {* @{text override_on} *}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   422
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   423
definition
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   424
  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   425
where
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   426
  "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
   427
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   428
lemma override_on_emptyset[simp]: "override_on f g {} = f"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   429
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   430
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   431
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
   432
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   433
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   434
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
   435
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   436
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   437
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   438
subsection {* @{text swap} *}
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   439
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   440
definition
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   441
  swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   442
where
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   443
  "swap a b f = f (a := f b, b:= f a)"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   444
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   445
lemma swap_self: "swap a a f = f"
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   446
by (simp add: swap_def)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   447
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   448
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   449
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   450
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   451
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   452
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   453
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   454
lemma inj_on_imp_inj_on_swap:
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   455
  "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   456
by (simp add: inj_on_def swap_def, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   457
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   458
lemma inj_on_swap_iff [simp]:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   459
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   460
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   461
  assume "inj_on (swap a b f) A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   462
  with A have "inj_on (swap a b (swap a b f)) A" 
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   463
    by (iprover intro: inj_on_imp_inj_on_swap) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   464
  thus "inj_on f A" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   465
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   466
  assume "inj_on f A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   467
  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
   468
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   469
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   470
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   471
apply (simp add: surj_def swap_def, clarify)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   472
apply (rule_tac P = "y = f b" in case_split_thm, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   473
apply (rule_tac P = "y = f a" in case_split_thm, auto)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   474
  --{*We don't yet have @{text case_tac}*}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   475
done
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   476
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   477
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   478
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   479
  assume "surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   480
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   481
  thus "surj f" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   482
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   483
  assume "surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   484
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   485
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   486
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   487
lemma bij_swap_iff: "bij (swap a b f) = bij f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   488
by (simp add: bij_def)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   489
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   490
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   491
subsection {* Proof tool setup *} 
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   492
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   493
text {* simplifies terms of the form
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   494
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   495
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   496
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   497
let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   498
  fun gen_fun_upd NONE T _ _ = NONE
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   499
    | 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
   500
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   501
  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
   502
    let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   503
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   504
            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
   505
        | find t = NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   506
    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
   507
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   508
  fun proc ss ct =
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   509
    let
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   510
      val ctxt = Simplifier.the_context ss
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   511
      val t = Thm.term_of ct
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   512
    in
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   513
      case find_double t of
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   514
        (T, NONE) => NONE
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   515
      | (T, SOME rhs) =>
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   516
          SOME (Goal.prove ctxt [] [] (Term.equals T $ t $ rhs)
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   517
            (fn _ =>
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   518
              rtac eq_reflection 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   519
              rtac ext 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   520
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   521
    end
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   522
in proc end
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   523
*}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   524
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   525
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   526
subsection {* Code generator setup *}
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   527
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   528
types_code
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   529
  "fun"  ("(_ ->/ _)")
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   530
attach (term_of) {*
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   531
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
   532
*}
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   533
attach (test) {*
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   534
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
   535
  let
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   536
    val tab = ref [];
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   537
    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
   538
      (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
   539
  in
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   540
    (fn x =>
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   541
       case AList.lookup op = (!tab) x of
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   542
         NONE =>
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   543
           let val p as (y, _) = bG i
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   544
           in (tab := (x, p) :: !tab; y) end
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   545
       | SOME (y, _) => y,
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   546
     fn () => Basics.fold mk_upd (!tab) (Const ("arbitrary", aT --> bT)))
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   547
  end;
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   548
*}
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   549
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   550
code_const "op \<circ>"
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   551
  (SML infixl 5 "o")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   552
  (Haskell infixr 9 ".")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   553
21906
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   554
code_const "id"
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   555
  (Haskell "id")
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   556
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   557
end