author hoelzl Fri Nov 26 10:04:04 2010 +0100 (2010-11-26) changeset 40704 407c6122956f parent 40701 e7aa34600c36 parent 40703 d1fc454d6735 child 40708 739dc2c2ba24
merged
```     1.1 --- a/NEWS	Fri Nov 26 09:15:49 2010 +0100
1.2 +++ b/NEWS	Fri Nov 26 10:04:04 2010 +0100
1.3 @@ -291,9 +291,10 @@
1.4  derive instantiated and simplified equations for inductive predicates,
1.5  similar to inductive_cases.
1.6
1.7 -* "bij f" is now an abbreviation of "bij_betw f UNIV UNIV". surj_on is a
1.8 -generalization of surj, and "surj f" is now a abbreviation of "surj_on f UNIV".
1.9 -The theorems bij_def and surj_def are unchanged.
1.10 +* "bij f" is now an abbreviation of "bij_betw f UNIV UNIV". "surj f" is now an
1.11 +abbreviation of "range f = UIV". The theorems bij_def and surj_def are
1.12 +unchanged.
1.13 +INCOMPATIBILITY.
1.14
1.15  * Function package: .psimps rules are no longer implicitly declared [simp].
1.16  INCOMPATIBILITY.
```
```     2.1 --- a/src/HOL/Finite_Set.thy	Fri Nov 26 09:15:49 2010 +0100
2.2 +++ b/src/HOL/Finite_Set.thy	Fri Nov 26 10:04:04 2010 +0100
2.3 @@ -2179,6 +2179,11 @@
2.4       finite A; finite B |] ==> card A = card B"
2.5  by (auto intro: le_antisym card_inj_on_le)
2.6
2.7 +lemma bij_betw_finite:
2.8 +  assumes "bij_betw f A B"
2.9 +  shows "finite A \<longleftrightarrow> finite B"
2.10 +using assms unfolding bij_betw_def
2.11 +using finite_imageD[of f A] by auto
2.12
2.13  subsubsection {* Pigeonhole Principles *}
2.14
2.15 @@ -2286,7 +2291,7 @@
2.16
2.17  lemma finite_UNIV_surj_inj: fixes f :: "'a \<Rightarrow> 'a"
2.18  shows "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
2.19 -by (blast intro: finite_surj_inj subset_UNIV dest:surj_range)
2.20 +by (blast intro: finite_surj_inj subset_UNIV)
2.21
2.22  lemma finite_UNIV_inj_surj: fixes f :: "'a \<Rightarrow> 'a"
2.23  shows "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
```
```     3.1 --- a/src/HOL/Fun.thy	Fri Nov 26 09:15:49 2010 +0100
3.2 +++ b/src/HOL/Fun.thy	Fri Nov 26 10:04:04 2010 +0100
3.3 @@ -130,24 +130,22 @@
3.5
3.6
3.7 -subsection {* Injectivity, Surjectivity and Bijectivity *}
3.8 +subsection {* Injectivity and Bijectivity *}
3.9
3.10  definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective"
3.11    "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
3.12
3.13 -definition surj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> bool" where -- "surjective"
3.14 -  "surj_on f B \<longleftrightarrow> B \<subseteq> range f"
3.15 -
3.16  definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"
3.17    "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
3.18
3.19 -text{*A common special case: functions injective over the entire domain type.*}
3.20 +text{*A common special case: functions injective, surjective or bijective over
3.21 +the entire domain type.*}
3.22
3.23  abbreviation
3.24    "inj f \<equiv> inj_on f UNIV"
3.25
3.26 -abbreviation
3.27 -  "surj f \<equiv> surj_on f UNIV"
3.28 +abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective"
3.29 +  "surj f \<equiv> (range f = UNIV)"
3.30
3.31  abbreviation
3.32    "bij f \<equiv> bij_betw f UNIV UNIV"
3.33 @@ -171,6 +169,14 @@
3.34  lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
3.35  by (force simp add: inj_on_def)
3.36
3.37 +lemma inj_on_cong:
3.38 +  "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
3.39 +unfolding inj_on_def by auto
3.40 +
3.41 +lemma inj_on_strict_subset:
3.42 +  "\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B"
3.43 +unfolding inj_on_def unfolding image_def by blast
3.44 +
3.45  lemma inj_comp:
3.46    "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
3.48 @@ -187,8 +193,44 @@
3.49  lemma inj_on_id2[simp]: "inj_on (%x. x) A"
3.51
3.52 -lemma surj_id[simp]: "surj_on id A"
3.54 +lemma inj_on_Int: "\<lbrakk>inj_on f A; inj_on f B\<rbrakk> \<Longrightarrow> inj_on f (A \<inter> B)"
3.55 +unfolding inj_on_def by blast
3.56 +
3.57 +lemma inj_on_INTER:
3.58 +  "\<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)"
3.59 +unfolding inj_on_def by blast
3.60 +
3.61 +lemma inj_on_Inter:
3.62 +  "\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)"
3.63 +unfolding inj_on_def by blast
3.64 +
3.65 +lemma inj_on_UNION_chain:
3.66 +  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
3.67 +         INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
3.68 +  shows "inj_on f (\<Union> i \<in> I. A i)"
3.69 +proof(unfold inj_on_def UNION_def, auto)
3.70 +  fix i j x y
3.71 +  assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
3.72 +         and ***: "f x = f y"
3.73 +  show "x = y"
3.74 +  proof-
3.75 +    {assume "A i \<le> A j"
3.76 +     with ** have "x \<in> A j" by auto
3.77 +     with INJ * ** *** have ?thesis
3.78 +     by(auto simp add: inj_on_def)
3.79 +    }
3.80 +    moreover
3.81 +    {assume "A j \<le> A i"
3.82 +     with ** have "y \<in> A i" by auto
3.83 +     with INJ * ** *** have ?thesis
3.84 +     by(auto simp add: inj_on_def)
3.85 +    }
3.86 +    ultimately show ?thesis using  CH * by blast
3.87 +  qed
3.88 +qed
3.89 +
3.90 +lemma surj_id: "surj id"
3.91 +by simp
3.92
3.93  lemma bij_id[simp]: "bij id"
3.95 @@ -251,20 +293,19 @@
3.96  apply (blast)
3.97  done
3.98
3.99 -lemma surj_onI: "(\<And>x. x \<in> B \<Longrightarrow> g (f x) = x) \<Longrightarrow> surj_on g B"
3.100 -  by (simp add: surj_on_def) (blast intro: sym)
3.101 +lemma comp_inj_on_iff:
3.102 +  "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
3.103 +by(auto simp add: comp_inj_on inj_on_def)
3.104
3.105 -lemma surj_onD: "surj_on f B \<Longrightarrow> y \<in> B \<Longrightarrow> \<exists>x. y = f x"
3.106 -  by (auto simp: surj_on_def)
3.107 -
3.108 -lemma surj_on_range_iff: "surj_on f B \<longleftrightarrow> (\<exists>A. f ` A = B)"
3.109 -  unfolding surj_on_def by (auto intro!: exI[of _ "f -` B"])
3.110 +lemma inj_on_imageI2:
3.111 +  "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
3.112 +by(auto simp add: comp_inj_on inj_on_def)
3.113
3.114  lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
3.115 -  by (simp add: surj_on_def subset_eq image_iff)
3.116 +  by auto
3.117
3.118 -lemma surjI: "(\<And> x. g (f x) = x) \<Longrightarrow> surj g"
3.119 -  by (blast intro: surj_onI)
3.120 +lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
3.121 +  using *[symmetric] by auto
3.122
3.123  lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
3.125 @@ -278,17 +319,25 @@
3.126  apply (drule_tac x = x in spec, blast)
3.127  done
3.128
3.129 -lemma surj_range: "surj f \<Longrightarrow> range f = UNIV"
3.130 -  by (auto simp add: surj_on_def)
3.131 +lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
3.132 +  unfolding bij_betw_def by auto
3.133 +
3.134 +lemma bij_betw_empty1:
3.135 +  assumes "bij_betw f {} A"
3.136 +  shows "A = {}"
3.137 +using assms unfolding bij_betw_def by blast
3.138
3.139 -lemma surj_range_iff: "surj f \<longleftrightarrow> range f = UNIV"
3.140 -  unfolding surj_on_def by auto
3.141 +lemma bij_betw_empty2:
3.142 +  assumes "bij_betw f A {}"
3.143 +  shows "A = {}"
3.144 +using assms unfolding bij_betw_def by blast
3.145
3.146 -lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
3.147 -  unfolding bij_betw_def surj_range_iff by auto
3.148 +lemma inj_on_imp_bij_betw:
3.149 +  "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
3.150 +unfolding bij_betw_def by simp
3.151
3.152  lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
3.153 -  unfolding surj_range_iff bij_betw_def ..
3.154 +  unfolding bij_betw_def ..
3.155
3.156  lemma bijI: "[| inj f; surj f |] ==> bij f"
3.158 @@ -302,16 +351,39 @@
3.159  lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
3.161
3.162 -lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> surj_on f B"
3.163 -by (auto simp: bij_betw_def surj_on_range_iff)
3.164 -
3.165 -lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
3.166 -by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
3.167 -
3.168  lemma bij_betw_trans:
3.169    "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
3.171
3.172 +lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
3.173 +  by (rule bij_betw_trans)
3.174 +
3.175 +lemma bij_betw_comp_iff:
3.176 +  "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
3.177 +by(auto simp add: bij_betw_def inj_on_def)
3.178 +
3.179 +lemma bij_betw_comp_iff2:
3.180 +  assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
3.181 +  shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
3.182 +using assms
3.184 +  assume *: "bij_betw (f' \<circ> f) A A''"
3.185 +  thus "bij_betw f A A'"
3.186 +  using IM
3.187 +  proof(auto simp add: bij_betw_def)
3.188 +    assume "inj_on (f' \<circ> f) A"
3.189 +    thus "inj_on f A" using inj_on_imageI2 by blast
3.190 +  next
3.191 +    fix a' assume **: "a' \<in> A'"
3.192 +    hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
3.193 +    then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
3.194 +    unfolding bij_betw_def by force
3.195 +    hence "f a \<in> A'" using IM by auto
3.196 +    hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
3.197 +    thus "a' \<in> f ` A" using 1 by auto
3.198 +  qed
3.199 +qed
3.200 +
3.201  lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
3.202  proof -
3.203    have i: "inj_on f A" and s: "f ` A = B"
3.204 @@ -343,21 +415,81 @@
3.205    ultimately show ?thesis by(auto simp:bij_betw_def)
3.206  qed
3.207
3.208 +lemma bij_betw_cong:
3.209 +  "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
3.210 +unfolding bij_betw_def inj_on_def by force
3.211 +
3.212 +lemma bij_betw_id[intro, simp]:
3.213 +  "bij_betw id A A"
3.214 +unfolding bij_betw_def id_def by auto
3.215 +
3.216 +lemma bij_betw_id_iff:
3.217 +  "bij_betw id A B \<longleftrightarrow> A = B"
3.219 +
3.220  lemma bij_betw_combine:
3.221    assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
3.222    shows "bij_betw f (A \<union> C) (B \<union> D)"
3.223    using assms unfolding bij_betw_def inj_on_Un image_Un by auto
3.224
3.225 +lemma bij_betw_UNION_chain:
3.226 +  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
3.227 +         BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
3.228 +  shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
3.229 +proof(unfold bij_betw_def, auto simp add: image_def)
3.230 +  have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
3.231 +  using BIJ bij_betw_def[of f] by auto
3.232 +  thus "inj_on f (\<Union> i \<in> I. A i)"
3.233 +  using CH inj_on_UNION_chain[of I A f] by auto
3.234 +next
3.235 +  fix i x
3.236 +  assume *: "i \<in> I" "x \<in> A i"
3.237 +  hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
3.238 +  thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
3.239 +next
3.240 +  fix i x'
3.241 +  assume *: "i \<in> I" "x' \<in> A' i"
3.242 +  hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
3.243 +  thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
3.244 +  using * by blast
3.245 +qed
3.246 +
3.247 +lemma bij_betw_Disj_Un:
3.248 +  assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and
3.249 +          B1: "bij_betw f A A'" and B2: "bij_betw f B B'"
3.250 +  shows "bij_betw f (A \<union> B) (A' \<union> B')"
3.251 +proof-
3.252 +  have 1: "inj_on f A \<and> inj_on f B"
3.253 +  using B1 B2 by (auto simp add: bij_betw_def)
3.254 +  have 2: "f`A = A' \<and> f`B = B'"
3.255 +  using B1 B2 by (auto simp add: bij_betw_def)
3.256 +  hence "f`(A - B) \<inter> f`(B - A) = {}"
3.257 +  using DISJ DISJ' by blast
3.258 +  hence "inj_on f (A \<union> B)"
3.259 +  using 1 by (auto simp add: inj_on_Un)
3.260 +  (*  *)
3.261 +  moreover
3.262 +  have "f`(A \<union> B) = A' \<union> B'"
3.263 +  using 2 by auto
3.264 +  ultimately show ?thesis
3.265 +  unfolding bij_betw_def by auto
3.266 +qed
3.267 +
3.268 +lemma bij_betw_subset:
3.269 +  assumes BIJ: "bij_betw f A A'" and
3.270 +          SUB: "B \<le> A" and IM: "f ` B = B'"
3.271 +  shows "bij_betw f B B'"
3.272 +using assms
3.273 +by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
3.274 +
3.275  lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
3.277 +by simp
3.278
3.279  lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
3.280  by (simp add: inj_on_def, blast)
3.281
3.282  lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
3.283 -apply (unfold surj_def)
3.284 -apply (blast intro: sym)
3.285 -done
3.286 +by (blast intro: sym)
3.287
3.288  lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
3.289  by (unfold inj_on_def, blast)
3.290 @@ -410,7 +542,7 @@
3.291  done
3.292
3.293  lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
3.294 -by (auto simp add: surj_def)
3.295 +by auto
3.296
3.297  lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
3.298  by (auto simp add: inj_on_def)
3.299 @@ -559,10 +691,10 @@
3.300  qed
3.301
3.302  lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
3.303 -  unfolding surj_range_iff by simp
3.304 +  by simp
3.305
3.306  lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
3.307 -  unfolding surj_range_iff by simp
3.308 +  by simp
3.309
3.310  lemma bij_betw_swap_iff [simp]:
3.311    "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
3.312 @@ -635,6 +767,17 @@
3.313    shows "the_inv f (f x) = x" using assms UNIV_I
3.314    by (rule the_inv_into_f_f)
3.315
3.316 +subsection {* Cantor's Paradox *}
3.317 +
3.319 +  "\<not>(\<exists>f. f ` A = Pow A)"
3.320 +proof clarify
3.321 +  fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
3.322 +  let ?X = "{a \<in> A. a \<notin> f a}"
3.323 +  have "?X \<in> Pow A" unfolding Pow_def by auto
3.324 +  with * obtain x where "x \<in> A \<and> f x = ?X" by blast
3.325 +  thus False by best
3.326 +qed
3.327
3.328  subsection {* Proof tool setup *}
3.329
```
```     4.1 --- a/src/HOL/Hilbert_Choice.thy	Fri Nov 26 09:15:49 2010 +0100
4.2 +++ b/src/HOL/Hilbert_Choice.thy	Fri Nov 26 10:04:04 2010 +0100
4.3 @@ -151,10 +151,10 @@
4.4  by(fastsimp simp: image_def)
4.5
4.6  lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
4.7 -by (blast intro: surjI inv_into_f_f)
4.8 +by (blast intro!: surjI inv_into_f_f)
4.9
4.10  lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
4.11 -by (simp add: f_inv_into_f surj_range)
4.13
4.14  lemma inv_into_injective:
4.15    assumes eq: "inv_into A f x = inv_into A f y"
4.16 @@ -173,12 +173,13 @@
4.17  by (auto simp add: bij_betw_def inj_on_inv_into)
4.18
4.19  lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
4.20 -by (simp add: inj_on_inv_into surj_range)
4.22
4.23  lemma surj_iff: "(surj f) = (f o inv f = id)"
4.24 -apply (simp add: o_def fun_eq_iff)
4.25 -apply (blast intro: surjI surj_f_inv_f)
4.26 -done
4.27 +by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
4.28 +
4.29 +lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
4.30 +  unfolding surj_iff by (simp add: o_def fun_eq_iff)
4.31
4.32  lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
4.33  apply (rule ext)
4.34 @@ -260,6 +261,208 @@
4.35    ultimately show "finite (UNIV :: 'a set)" by simp
4.36  qed
4.37
4.38 +lemma image_inv_into_cancel:
4.39 +  assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
4.40 +  shows "f `((inv_into A f)`B') = B'"
4.41 +  using assms
4.42 +proof (auto simp add: f_inv_into_f)
4.43 +  let ?f' = "(inv_into A f)"
4.44 +  fix a' assume *: "a' \<in> B'"
4.45 +  then have "a' \<in> A'" using SUB by auto
4.46 +  then have "a' = f (?f' a')"
4.47 +    using SURJ by (auto simp add: f_inv_into_f)
4.48 +  then show "a' \<in> f ` (?f' ` B')" using * by blast
4.49 +qed
4.50 +
4.51 +lemma inv_into_inv_into_eq:
4.52 +  assumes "bij_betw f A A'" "a \<in> A"
4.53 +  shows "inv_into A' (inv_into A f) a = f a"
4.54 +proof -
4.55 +  let ?f' = "inv_into A f"   let ?f'' = "inv_into A' ?f'"
4.56 +  have 1: "bij_betw ?f' A' A" using assms
4.57 +  by (auto simp add: bij_betw_inv_into)
4.58 +  obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
4.59 +    using 1 `a \<in> A` unfolding bij_betw_def by force
4.60 +  hence "?f'' a = a'"
4.61 +    using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
4.62 +  moreover have "f a = a'" using assms 2 3
4.63 +    by (auto simp add: inv_into_f_f bij_betw_def)
4.64 +  ultimately show "?f'' a = f a" by simp
4.65 +qed
4.66 +
4.67 +lemma inj_on_iff_surj:
4.68 +  assumes "A \<noteq> {}"
4.69 +  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
4.70 +proof safe
4.71 +  fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
4.72 +  let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"  let ?csi = "\<lambda>a. a \<in> A"
4.73 +  let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
4.74 +  have "?g ` A' = A"
4.75 +  proof
4.76 +    show "?g ` A' \<le> A"
4.77 +    proof clarify
4.78 +      fix a' assume *: "a' \<in> A'"
4.79 +      show "?g a' \<in> A"
4.80 +      proof cases
4.81 +        assume Case1: "a' \<in> f ` A"
4.82 +        then obtain a where "?phi a' a" by blast
4.83 +        hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
4.84 +        with Case1 show ?thesis by auto
4.85 +      next
4.86 +        assume Case2: "a' \<notin> f ` A"
4.87 +        hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
4.88 +        with Case2 show ?thesis by auto
4.89 +      qed
4.90 +    qed
4.91 +  next
4.92 +    show "A \<le> ?g ` A'"
4.93 +    proof-
4.94 +      {fix a assume *: "a \<in> A"
4.95 +       let ?b = "SOME aa. ?phi (f a) aa"
4.96 +       have "?phi (f a) a" using * by auto
4.97 +       hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
4.98 +       hence "?g(f a) = ?b" using * by auto
4.99 +       moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
4.100 +       ultimately have "?g(f a) = a" by simp
4.101 +       with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
4.102 +      }
4.103 +      thus ?thesis by force
4.104 +    qed
4.105 +  qed
4.106 +  thus "\<exists>g. g ` A' = A" by blast
4.107 +next
4.108 +  fix g  let ?f = "inv_into A' g"
4.109 +  have "inj_on ?f (g ` A')"
4.110 +    by (auto simp add: inj_on_inv_into)
4.111 +  moreover
4.112 +  {fix a' assume *: "a' \<in> A'"
4.113 +   let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
4.114 +   have "?phi a'" using * by auto
4.115 +   hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
4.116 +   hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
4.117 +  }
4.118 +  ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
4.119 +qed
4.120 +
4.121 +lemma Ex_inj_on_UNION_Sigma:
4.122 +  "\<exists>f. (inj_on f (\<Union> i \<in> I. A i) \<and> f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i))"
4.123 +proof
4.124 +  let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
4.125 +  let ?sm = "\<lambda> a. SOME i. ?phi a i"
4.126 +  let ?f = "\<lambda>a. (?sm a, a)"
4.127 +  have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
4.128 +  moreover
4.129 +  { { fix i a assume "i \<in> I" and "a \<in> A i"
4.130 +      hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
4.131 +    }
4.132 +    hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
4.133 +  }
4.134 +  ultimately
4.135 +  show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
4.136 +  by auto
4.137 +qed
4.138 +
4.139 +subsection {* The Cantor-Bernstein Theorem *}
4.140 +
4.141 +lemma Cantor_Bernstein_aux:
4.142 +  shows "\<exists>A' h. A' \<le> A \<and>
4.143 +                (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
4.144 +                (\<forall>a \<in> A'. h a = f a) \<and>
4.145 +                (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
4.146 +proof-
4.147 +  obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
4.148 +  have 0: "mono H" unfolding mono_def H_def by blast
4.149 +  then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
4.150 +  hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
4.151 +  hence 3: "A' \<le> A" by blast
4.152 +  have 4: "\<forall>a \<in> A'.  a \<notin> g`(B - f ` A')"
4.153 +  using 2 by blast
4.154 +  have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
4.155 +  using 2 by blast
4.156 +  (*  *)
4.157 +  obtain h where h_def:
4.158 +  "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
4.159 +  hence "\<forall>a \<in> A'. h a = f a" by auto
4.160 +  moreover
4.161 +  have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
4.162 +  proof
4.163 +    fix a assume *: "a \<in> A - A'"
4.164 +    let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
4.165 +    have "h a = (SOME b. ?phi b)" using h_def * by auto
4.166 +    moreover have "\<exists>b. ?phi b" using 5 *  by auto
4.167 +    ultimately show  "?phi (h a)" using someI_ex[of ?phi] by auto
4.168 +  qed
4.169 +  ultimately show ?thesis using 3 4 by blast
4.170 +qed
4.171 +
4.172 +theorem Cantor_Bernstein:
4.173 +  assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
4.174 +          INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
4.175 +  shows "\<exists>h. bij_betw h A B"
4.176 +proof-
4.177 +  obtain A' and h where 0: "A' \<le> A" and
4.178 +  1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
4.179 +  2: "\<forall>a \<in> A'. h a = f a" and
4.180 +  3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
4.181 +  using Cantor_Bernstein_aux[of A g B f] by blast
4.182 +  have "inj_on h A"
4.183 +  proof (intro inj_onI)
4.184 +    fix a1 a2
4.185 +    assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
4.186 +    show "a1 = a2"
4.187 +    proof(cases "a1 \<in> A'")
4.188 +      assume Case1: "a1 \<in> A'"
4.189 +      show ?thesis
4.190 +      proof(cases "a2 \<in> A'")
4.191 +        assume Case11: "a2 \<in> A'"
4.192 +        hence "f a1 = f a2" using Case1 2 6 by auto
4.193 +        thus ?thesis using INJ1 Case1 Case11 0
4.194 +        unfolding inj_on_def by blast
4.195 +      next
4.196 +        assume Case12: "a2 \<notin> A'"
4.197 +        hence False using 3 5 2 6 Case1 by force
4.198 +        thus ?thesis by simp
4.199 +      qed
4.200 +    next
4.201 +    assume Case2: "a1 \<notin> A'"
4.202 +      show ?thesis
4.203 +      proof(cases "a2 \<in> A'")
4.204 +        assume Case21: "a2 \<in> A'"
4.205 +        hence False using 3 4 2 6 Case2 by auto
4.206 +        thus ?thesis by simp
4.207 +      next
4.208 +        assume Case22: "a2 \<notin> A'"
4.209 +        hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
4.210 +        thus ?thesis using 6 by simp
4.211 +      qed
4.212 +    qed
4.213 +  qed
4.214 +  (*  *)
4.215 +  moreover
4.216 +  have "h ` A = B"
4.217 +  proof safe
4.218 +    fix a assume "a \<in> A"
4.219 +    thus "h a \<in> B" using SUB1 2 3 by (case_tac "a \<in> A'", auto)
4.220 +  next
4.221 +    fix b assume *: "b \<in> B"
4.222 +    show "b \<in> h ` A"
4.223 +    proof(cases "b \<in> f ` A'")
4.224 +      assume Case1: "b \<in> f ` A'"
4.225 +      then obtain a where "a \<in> A' \<and> b = f a" by blast
4.226 +      thus ?thesis using 2 0 by force
4.227 +    next
4.228 +      assume Case2: "b \<notin> f ` A'"
4.229 +      hence "g b \<notin> A'" using 1 * by auto
4.230 +      hence 4: "g b \<in> A - A'" using * SUB2 by auto
4.231 +      hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
4.232 +      using 3 by auto
4.233 +      hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
4.234 +      thus ?thesis using 4 by force
4.235 +    qed
4.236 +  qed
4.237 +  (*  *)
4.238 +  ultimately show ?thesis unfolding bij_betw_def by auto
4.239 +qed
4.240
4.241  subsection {*Other Consequences of Hilbert's Epsilon*}
4.242
```
```     5.1 --- a/src/HOL/Lattice/Orders.thy	Fri Nov 26 09:15:49 2010 +0100
5.2 +++ b/src/HOL/Lattice/Orders.thy	Fri Nov 26 10:04:04 2010 +0100
5.3 @@ -118,8 +118,8 @@
5.4    qed
5.5  qed
5.6
5.7 -lemma range_dual [simp]: "dual ` UNIV = UNIV"
5.8 -proof (rule surj_range)
5.9 +lemma range_dual [simp]: "surj dual"
5.10 +proof -
5.11    have "\<And>x'. dual (undual x') = x'" by simp
5.12    thus "surj dual" by (rule surjI)
5.13  qed
```
```     6.1 --- a/src/HOL/Library/ContNotDenum.thy	Fri Nov 26 09:15:49 2010 +0100
6.2 +++ b/src/HOL/Library/ContNotDenum.thy	Fri Nov 26 10:04:04 2010 +0100
6.3 @@ -565,8 +565,7 @@
6.4    shows "\<not> (\<exists>f::nat\<Rightarrow>real. surj f)"
6.6    assume "\<exists>f::nat\<Rightarrow>real. surj f"
6.7 -  then obtain f::"nat\<Rightarrow>real" where "surj f" by auto
6.8 -  hence rangeF: "range f = UNIV" by (rule surj_range)
6.9 +  then obtain f::"nat\<Rightarrow>real" where rangeF: "surj f" by auto
6.10    -- "We now produce a real number x that is not in the range of f, using the properties of newInt. "
6.11    have "\<exists>x. x \<in> (\<Inter>n. newInt n f)" using newInt_notempty by blast
6.12    moreover have "\<forall>n. f n \<notin> (\<Inter>n. newInt n f)" by (rule newInt_inter)
```
```     7.1 --- a/src/HOL/Library/Countable.thy	Fri Nov 26 09:15:49 2010 +0100
7.2 +++ b/src/HOL/Library/Countable.thy	Fri Nov 26 10:04:04 2010 +0100
7.3 @@ -165,7 +165,7 @@
7.4  qed
7.5
7.6  lemma Rats_eq_range_nat_to_rat_surj: "\<rat> = range nat_to_rat_surj"
7.7 -by (simp add: Rats_def surj_nat_to_rat_surj surj_range)
7.8 +by (simp add: Rats_def surj_nat_to_rat_surj)
7.9
7.10  context field_char_0
7.11  begin
```
```     8.1 --- a/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Nov 26 09:15:49 2010 +0100
8.2 +++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Fri Nov 26 10:04:04 2010 +0100
8.3 @@ -948,14 +948,12 @@
8.4     assumes lf: "linear f" and gf: "f o g = id"
8.5     shows "linear g"
8.6   proof-
8.7 -   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def fun_eq_iff)
8.8 -     by metis
8.9 +   from gf have fi: "surj f" by (auto simp add: surj_def o_def id_def) metis
8.10     from linear_surjective_isomorphism[OF lf fi]
8.11     obtain h:: "'a => 'a" where
8.12       h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
8.13     have "h = g" apply (rule ext) using gf h(2,3)
8.14 -     apply (simp add: o_def id_def fun_eq_iff)
8.15 -     by metis
8.16 +     by (simp add: o_def id_def fun_eq_iff) metis
8.17     with h(1) show ?thesis by blast
8.18   qed
8.19
```
```     9.1 --- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Fri Nov 26 09:15:49 2010 +0100
9.2 +++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Fri Nov 26 10:04:04 2010 +0100
9.3 @@ -2843,7 +2843,7 @@
9.4      h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. h x = inv f x" by blast
9.5    from h(2)
9.6    have th: "\<forall>i<DIM('b). (f o h) (basis i) = id (basis i)"
9.7 -    using sf by(auto simp add: surj_iff o_def fun_eq_iff)
9.8 +    using sf by(auto simp add: surj_iff_all)
9.9    from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
9.10    have "f o h = id" .
9.11    then show ?thesis using h(1) by blast
9.12 @@ -2995,7 +2995,7 @@
9.13    {fix f f':: "'a => 'a"
9.14      assume lf: "linear f" "linear f'" and f: "f o f' = id"
9.15      from f have sf: "surj f"
9.16 -      apply (auto simp add: o_def fun_eq_iff id_def surj_def)
9.17 +      apply (auto simp add: o_def id_def surj_def)
9.18        by metis
9.19      from linear_surjective_isomorphism[OF lf(1) sf] lf f
9.20      have "f' o f = id" unfolding fun_eq_iff o_def id_def
```
```    10.1 --- a/src/HOL/Nominal/Examples/Support.thy	Fri Nov 26 09:15:49 2010 +0100
10.2 +++ b/src/HOL/Nominal/Examples/Support.thy	Fri Nov 26 10:04:04 2010 +0100
10.3 @@ -47,7 +47,7 @@
10.4    also have "\<dots> = (\<lambda>n. atom n) ` ({n. \<exists>i. n = 2*i \<or> n = 2*i+1})" by auto
10.5    also have "\<dots> = (\<lambda>n. atom n) ` (UNIV::nat set)" using even_or_odd by auto
10.6    also have "\<dots> = (UNIV::atom set)" using atom.exhaust
10.7 -    by (rule_tac  surj_range) (auto simp add: surj_def)
10.8 +    by (auto simp add: surj_def)
10.9    finally show "EVEN \<union> ODD = UNIV" by simp
10.10  qed
10.11
```
```    11.1 --- a/src/HOL/Probability/Sigma_Algebra.thy	Fri Nov 26 09:15:49 2010 +0100
11.2 +++ b/src/HOL/Probability/Sigma_Algebra.thy	Fri Nov 26 10:04:04 2010 +0100
11.3 @@ -132,7 +132,7 @@
11.4        by (auto intro!: exI[of _ "from_nat i"])
11.5    qed
11.6    have **: "range ?A' = range A"
11.7 -    using surj_range[OF surj_from_nat]
11.8 +    using surj_from_nat
11.9      by (auto simp: image_compose intro!: imageI)
11.10    show ?thesis unfolding * ** ..
11.11  qed
```
```    12.1 --- a/src/HOL/Product_Type.thy	Fri Nov 26 09:15:49 2010 +0100
12.2 +++ b/src/HOL/Product_Type.thy	Fri Nov 26 10:04:04 2010 +0100
12.3 @@ -1135,6 +1135,7 @@
12.4  qed
12.5
12.6  lemma map_pair_surj:
12.7 +  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
12.8    assumes "surj f" and "surj g"
12.9    shows "surj (map_pair f g)"
12.10  unfolding surj_def
```
```    13.1 --- a/src/HOL/Set.thy	Fri Nov 26 09:15:49 2010 +0100
13.2 +++ b/src/HOL/Set.thy	Fri Nov 26 10:04:04 2010 +0100
13.3 @@ -622,6 +622,8 @@
13.4  lemma Pow_top: "A \<in> Pow A"
13.5    by simp
13.6
13.7 +lemma Pow_not_empty: "Pow A \<noteq> {}"
13.8 +  using Pow_top by blast
13.9
13.10  subsubsection {* Set complement *}
13.11
13.12 @@ -972,6 +974,21 @@
13.13  lemmas [symmetric, rulify] = atomize_ball
13.14    and [symmetric, defn] = atomize_ball
13.15
13.16 +lemma image_Pow_mono:
13.17 +  assumes "f ` A \<le> B"
13.18 +  shows "(image f) ` (Pow A) \<le> Pow B"
13.19 +using assms by blast
13.20 +
13.21 +lemma image_Pow_surj:
13.22 +  assumes "f ` A = B"
13.23 +  shows "(image f) ` (Pow A) = Pow B"
13.24 +using assms unfolding Pow_def proof(auto)
13.25 +  fix Y assume *: "Y \<le> f ` A"
13.26 +  obtain X where X_def: "X = {x \<in> A. f x \<in> Y}" by blast
13.27 +  have "f ` X = Y \<and> X \<le> A" unfolding X_def using * by auto
13.28 +  thus "Y \<in> (image f) ` {X. X \<le> A}" by blast
13.29 +qed
13.30 +
13.31  subsubsection {* Derived rules involving subsets. *}
13.32
13.33  text {* @{text insert}. *}
```
```    14.1 --- a/src/HOL/SetInterval.thy	Fri Nov 26 09:15:49 2010 +0100
14.2 +++ b/src/HOL/SetInterval.thy	Fri Nov 26 10:04:04 2010 +0100
14.3 @@ -159,6 +159,10 @@
14.4   apply simp_all
14.5  done
14.6
14.7 +lemma lessThan_strict_subset_iff:
14.8 +  fixes m n :: "'a::linorder"
14.9 +  shows "{..<m} < {..<n} \<longleftrightarrow> m < n"
14.10 +  by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)
14.11
14.12  subsection {*Two-sided intervals*}
14.13
14.14 @@ -262,6 +266,18 @@
14.16  done
14.17
14.18 +lemma atLeastLessThan_inj:
14.19 +  fixes a b c d :: "'a::linorder"
14.20 +  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
14.21 +  shows "a = c" "b = d"
14.22 +using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le linorder_antisym_conv2 subset_refl)+
14.23 +
14.24 +lemma atLeastLessThan_eq_iff:
14.25 +  fixes a b c d :: "'a::linorder"
14.26 +  assumes "a < b" "c < d"
14.27 +  shows "{a ..< b} = {c ..< d} \<longleftrightarrow> a = c \<and> b = d"
14.28 +  using atLeastLessThan_inj assms by auto
14.29 +
14.30  subsubsection {* Intersection *}
14.31
14.32  context linorder
14.33 @@ -705,6 +721,39 @@
14.34    "finite M \<Longrightarrow> \<exists>h. bij_betw h {1 .. card M} M"
14.35  by (rule finite_same_card_bij) auto
14.36
14.37 +lemma bij_betw_iff_card:
14.38 +  assumes FIN: "finite A" and FIN': "finite B"
14.39 +  shows BIJ: "(\<exists>f. bij_betw f A B) \<longleftrightarrow> (card A = card B)"
14.40 +using assms
14.42 +  assume *: "card A = card B"
14.43 +  obtain f where "bij_betw f A {0 ..< card A}"
14.44 +  using FIN ex_bij_betw_finite_nat by blast
14.45 +  moreover obtain g where "bij_betw g {0 ..< card B} B"
14.46 +  using FIN' ex_bij_betw_nat_finite by blast
14.47 +  ultimately have "bij_betw (g o f) A B"
14.48 +  using * by (auto simp add: bij_betw_trans)
14.49 +  thus "(\<exists>f. bij_betw f A B)" by blast
14.50 +qed
14.51 +
14.52 +lemma inj_on_iff_card_le:
14.53 +  assumes FIN: "finite A" and FIN': "finite B"
14.54 +  shows "(\<exists>f. inj_on f A \<and> f ` A \<le> B) = (card A \<le> card B)"
14.55 +proof (safe intro!: card_inj_on_le)
14.56 +  assume *: "card A \<le> card B"
14.57 +  obtain f where 1: "inj_on f A" and 2: "f ` A = {0 ..< card A}"
14.58 +  using FIN ex_bij_betw_finite_nat unfolding bij_betw_def by force
14.59 +  moreover obtain g where "inj_on g {0 ..< card B}" and 3: "g ` {0 ..< card B} = B"
14.60 +  using FIN' ex_bij_betw_nat_finite unfolding bij_betw_def by force
14.61 +  ultimately have "inj_on g (f ` A)" using subset_inj_on[of g _ "f ` A"] * by force
14.62 +  hence "inj_on (g o f) A" using 1 comp_inj_on by blast
14.63 +  moreover
14.64 +  {have "{0 ..< card A} \<le> {0 ..< card B}" using * by force
14.65 +   with 2 have "f ` A  \<le> {0 ..< card B}" by blast
14.66 +   hence "(g o f) ` A \<le> B" unfolding comp_def using 3 by force
14.67 +  }
14.68 +  ultimately show "(\<exists>f. inj_on f A \<and> f ` A \<le> B)" by blast
14.69 +qed (insert assms, auto)
14.70
14.71  subsection {* Intervals of integers *}
14.72
```
```    15.1 --- a/src/HOL/UNITY/Comp/Alloc.thy	Fri Nov 26 09:15:49 2010 +0100
15.2 +++ b/src/HOL/UNITY/Comp/Alloc.thy	Fri Nov 26 10:04:04 2010 +0100
15.3 @@ -358,7 +358,7 @@
15.4    done
15.5
15.6  lemma surj_sysOfAlloc [iff]: "surj sysOfAlloc"
15.7 -  apply (simp add: surj_iff fun_eq_iff o_apply)
15.8 +  apply (simp add: surj_iff_all)
15.9    apply record_auto
15.10    done
15.11
15.12 @@ -386,7 +386,7 @@
15.13    done
15.14
15.15  lemma surj_sysOfClient [iff]: "surj sysOfClient"
15.16 -  apply (simp add: surj_iff fun_eq_iff o_apply)
15.17 +  apply (simp add: surj_iff_all)
15.18    apply record_auto
15.19    done
15.20
15.21 @@ -410,7 +410,7 @@
15.22    done
15.23
15.24  lemma surj_client_map [iff]: "surj client_map"
15.25 -  apply (simp add: surj_iff fun_eq_iff o_apply)
15.26 +  apply (simp add: surj_iff_all)
15.27    apply record_auto
15.28    done
15.29
15.30 @@ -682,7 +682,7 @@
15.31  lemma fst_lift_map_eq_fst [simp]: "fst (lift_map i x) i = fst x"
15.32  apply (insert fst_o_lift_map [of i])
15.33  apply (drule fun_cong [where x=x])
15.36  done
15.37
15.38  lemma fst_o_lift_map' [simp]:
15.39 @@ -702,7 +702,7 @@
15.40    RS guarantees_PLam_I
15.41    RS (bij_sysOfClient RS rename_rename_guarantees_eq RS iffD2)
15.42    |> simplify (simpset() addsimps [lift_image_eq_rename, o_def, split_def,
15.43 -                                   surj_rename RS surj_range])
15.44 +                                   surj_rename])
15.45
15.46  However, the "preserves" property remains to be discharged, and the unfolding
15.47  of "o" and "sub" complicates subsequent reasoning.
15.48 @@ -723,7 +723,7 @@
15.49      asm_simp_tac
15.51                        @{thm rename_guarantees_eq_rename_inv},
15.52 -                      @{thm bij_imp_bij_inv}, @{thm surj_rename} RS @{thm surj_range},
15.53 +                      @{thm bij_imp_bij_inv}, @{thm surj_rename},
15.54                        @{thm inv_inv_eq}]) 1,
15.55      asm_simp_tac
15.56          (@{simpset} addsimps [@{thm o_def}, @{thm non_dummy_def}, @{thm guarantees_Int_right}]) 1]
15.57 @@ -798,9 +798,9 @@
15.58  lemmas rename_Alloc_Increasing =
15.59    Alloc_Increasing
15.60      [THEN rename_guarantees_sysOfAlloc_I,
15.61 -     simplified surj_rename [THEN surj_range] o_def sub_apply
15.62 +     simplified surj_rename o_def sub_apply
15.63                  rename_image_Increasing bij_sysOfAlloc
15.64 -                allocGiv_o_inv_sysOfAlloc_eq'];
15.65 +                allocGiv_o_inv_sysOfAlloc_eq']
15.66
15.67  lemma System_Increasing_allocGiv:
15.68       "i < Nclients ==> System : Increasing (sub i o allocGiv)"
15.69 @@ -879,7 +879,7 @@
15.70              \<le> NbT + (\<Sum>i \<in> lessThan Nclients. (tokens o sub i o allocRel) s)}"
15.72    apply (insert Alloc_Safety [THEN rename_guarantees_sysOfAlloc_I])
15.73 -  apply (simp add: o_def);
15.74 +  apply (simp add: o_def)
15.75    apply (erule component_guaranteesD)
15.76    apply (auto simp add: System_Increasing_allocRel [simplified sub_apply o_def])
15.77    done
```
```    16.1 --- a/src/HOL/UNITY/Extend.thy	Fri Nov 26 09:15:49 2010 +0100
16.2 +++ b/src/HOL/UNITY/Extend.thy	Fri Nov 26 10:04:04 2010 +0100
16.3 @@ -127,7 +127,7 @@
16.4       assumes surj_h: "surj h"
16.5           and prem:   "!! x y. g (h(x,y)) = x"
16.6       shows "fst (inv h z) = g z"
16.7 -by (metis UNIV_I f_inv_into_f pair_collapse prem surj_h surj_range)
16.8 +by (metis UNIV_I f_inv_into_f pair_collapse prem surj_h)
16.9
16.10
16.11  subsection{*Trivial properties of f, g, h*}
```
```    17.1 --- a/src/HOL/UNITY/Rename.thy	Fri Nov 26 09:15:49 2010 +0100
17.2 +++ b/src/HOL/UNITY/Rename.thy	Fri Nov 26 10:04:04 2010 +0100
17.3 @@ -60,7 +60,8 @@
17.4  lemma bij_extend_act: "bij h ==> bij (extend_act (%(x,u::'c). h x))"
17.5  apply (rule bijI)
17.6  apply (rule Extend.inj_extend_act)
17.7 -apply (auto simp add: bij_extend_act_eq_project_act)
17.8 +apply simp