author haftmann Tue Jun 02 16:23:43 2009 +0200 (2009-06-02) changeset 31380 f25536c0bb80 parent 31379 213299656575 child 31381 b3a785a69538
 src/HOL/Finite_Set.thy file | annotate | diff | revisions src/HOL/Hilbert_Choice.thy file | annotate | diff | revisions src/HOL/Library/Fin_Fun.thy file | annotate | diff | revisions src/HOL/Map.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Finite_Set.thy	Tue Jun 02 15:53:34 2009 +0200
1.2 +++ b/src/HOL/Finite_Set.thy	Tue Jun 02 16:23:43 2009 +0200
1.3 @@ -1926,34 +1926,40 @@
1.4  But now that we have @{text setsum} things are easy:
1.5  *}
1.6
1.7 -definition card :: "'a set \<Rightarrow> nat"
1.8 -where "card A = setsum (\<lambda>x. 1) A"
1.9 +definition card :: "'a set \<Rightarrow> nat" where
1.10 +  "card A = setsum (\<lambda>x. 1) A"
1.11 +
1.12 +lemmas card_eq_setsum = card_def
1.13
1.14  lemma card_empty [simp]: "card {} = 0"
1.16 -
1.17 -lemma card_infinite [simp]: "~ finite A ==> card A = 0"
1.19 -
1.20 -lemma card_eq_setsum: "card A = setsum (%x. 1) A"
1.22 +  by (simp add: card_def)
1.23
1.24  lemma card_insert_disjoint [simp]:
1.25    "finite A ==> x \<notin> A ==> card (insert x A) = Suc(card A)"
1.27 +  by (simp add: card_def)
1.28
1.29  lemma card_insert_if:
1.30    "finite A ==> card (insert x A) = (if x:A then card A else Suc(card(A)))"
1.32 +  by (simp add: insert_absorb)
1.33 +
1.34 +lemma card_infinite [simp]: "~ finite A ==> card A = 0"
1.35 +  by (simp add: card_def)
1.36 +
1.37 +lemma card_ge_0_finite:
1.38 +  "card A > 0 \<Longrightarrow> finite A"
1.39 +  by (rule ccontr) simp
1.40
1.41  lemma card_0_eq [simp,noatp]: "finite A ==> (card A = 0) = (A = {})"
1.42 -apply auto
1.43 -apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
1.44 -done
1.45 +  apply auto
1.46 +  apply (drule_tac a = x in mk_disjoint_insert, clarify, auto)
1.47 +  done
1.48 +
1.49 +lemma finite_UNIV_card_ge_0:
1.50 +  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
1.51 +  by (rule ccontr) simp
1.52
1.53  lemma card_eq_0_iff: "(card A = 0) = (A = {} | ~ finite A)"
1.54 -by auto
1.55 -
1.56 +  by auto
1.57
1.58  lemma card_Suc_Diff1: "finite A ==> x: A ==> Suc (card (A - {x})) = card A"
1.59  apply(rule_tac t = A in insert_Diff [THEN subst], assumption)
1.60 @@ -2049,6 +2055,24 @@
1.61         finite_subset [of _ "\<Union> (insert x F)"])
1.62  done
1.63
1.64 +lemma card_eq_UNIV_imp_eq_UNIV:
1.65 +  assumes fin: "finite (UNIV :: 'a set)"
1.66 +  and card: "card A = card (UNIV :: 'a set)"
1.67 +  shows "A = (UNIV :: 'a set)"
1.68 +proof
1.69 +  show "A \<subseteq> UNIV" by simp
1.70 +  show "UNIV \<subseteq> A"
1.71 +  proof
1.72 +    fix x
1.73 +    show "x \<in> A"
1.74 +    proof (rule ccontr)
1.75 +      assume "x \<notin> A"
1.76 +      then have "A \<subset> UNIV" by auto
1.77 +      with fin have "card A < card (UNIV :: 'a set)" by (fact psubset_card_mono)
1.78 +      with card show False by simp
1.79 +    qed
1.80 +  qed
1.81 +qed
1.82
1.83  text{*The form of a finite set of given cardinality*}
1.84
1.85 @@ -2078,6 +2102,17 @@
1.86   apply(auto intro:ccontr)
1.87  done
1.88
1.89 +lemma finite_fun_UNIVD2:
1.90 +  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
1.91 +  shows "finite (UNIV :: 'b set)"
1.92 +proof -
1.93 +  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
1.94 +    by(rule finite_imageI)
1.95 +  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
1.96 +    by(rule UNIV_eq_I) auto
1.97 +  ultimately show "finite (UNIV :: 'b set)" by simp
1.98 +qed
1.99 +
1.100  lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
1.101  apply (cases "finite A")
1.102  apply (erule finite_induct)
```
```     2.1 --- a/src/HOL/Hilbert_Choice.thy	Tue Jun 02 15:53:34 2009 +0200
2.2 +++ b/src/HOL/Hilbert_Choice.thy	Tue Jun 02 16:23:43 2009 +0200
2.3 @@ -219,6 +219,25 @@
2.4  apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
2.5  done
2.6
2.7 +lemma finite_fun_UNIVD1:
2.8 +  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
2.9 +  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
2.10 +  shows "finite (UNIV :: 'a set)"
2.11 +proof -
2.12 +  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
2.13 +  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
2.14 +    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
2.15 +  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
2.16 +  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
2.17 +  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
2.18 +  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
2.19 +  proof (rule UNIV_eq_I)
2.20 +    fix x :: 'a
2.21 +    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_def)
2.22 +    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
2.23 +  qed
2.24 +  ultimately show "finite (UNIV :: 'a set)" by simp
2.25 +qed
2.26
2.27  subsection {*Inverse of a PI-function (restricted domain)*}
2.28
```
```     3.1 --- a/src/HOL/Library/Fin_Fun.thy	Tue Jun 02 15:53:34 2009 +0200
3.2 +++ b/src/HOL/Library/Fin_Fun.thy	Tue Jun 02 16:23:43 2009 +0200
3.3 @@ -17,68 +17,12 @@
3.4    For details, see Formalising FinFuns - Generating Code for Functions as Data by A. Lochbihler in TPHOLs 2009.
3.5  *}
3.6
3.7 -subsection {* Auxiliary definitions and lemmas *}
3.8 -
3.9 -(*FIXME move these to Finite_Set.thy*)
3.10 -lemma card_ge_0_finite:
3.11 -  "card A > 0 \<Longrightarrow> finite A"
3.12 -by(rule ccontr, drule card_infinite, simp)
3.13 -
3.14 -lemma finite_UNIV_card_ge_0:
3.15 -  "finite (UNIV :: 'a set) \<Longrightarrow> card (UNIV :: 'a set) > 0"
3.16 -by(rule ccontr) simp
3.17 -
3.18 -lemma card_eq_UNIV_imp_eq_UNIV:
3.19 -  assumes fin: "finite (UNIV :: 'a set)"
3.20 -  and card: "card A = card (UNIV :: 'a set)"
3.21 -  shows "A = (UNIV :: 'a set)"
3.22 -apply -
3.23 -  proof
3.24 -  show "A \<subseteq> UNIV" by simp
3.25 -  show "UNIV \<subseteq> A"
3.26 -  proof
3.27 -    fix x
3.28 -    show "x \<in> A"
3.29 -    proof(rule ccontr)
3.30 -      assume "x \<notin> A"
3.31 -      hence "A \<subset> UNIV" by auto
3.32 -      from psubset_card_mono[OF fin this] card show False by simp
3.33 -    qed
3.34 -  qed
3.35 -qed
3.36 -
3.37 -lemma finite_fun_UNIVD2: assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
3.38 -  shows "finite (UNIV :: 'b set)"
3.39 -proof -
3.40 -  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary))"
3.41 -    by(rule finite_imageI)
3.42 -  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. f arbitrary)"
3.43 -    by(rule UNIV_eq_I) auto
3.44 -  ultimately show "finite (UNIV :: 'b set)" by simp
3.45 -qed
3.46 -
3.47 -lemma finite_fun_UNIVD1: assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
3.48 -  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
3.49 -  shows "finite (UNIV :: 'a set)"
3.50 -proof -
3.51 -  from fin have finb: "finite (UNIV :: 'b set)" by(rule finite_fun_UNIVD2)
3.52 -  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
3.53 -    by(cases "card (UNIV :: 'b set)")(auto simp add: card_eq_0_iff)
3.54 -  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - 2" by(auto)
3.55 -  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by(auto simp add: card_Suc_eq)
3.56 -  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by(rule finite_imageI)
3.57 -  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
3.58 -  proof(rule UNIV_eq_I)
3.59 -    fix x :: 'a
3.60 -    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by(simp add: inv_def)
3.61 -    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
3.62 -  qed
3.63 -  ultimately show "finite (UNIV :: 'a set)" by simp
3.64 -qed
3.65 -
3.66  (*FIXME move to Map.thy*)
3.67  lemma restrict_map_insert: "f |` (insert a A) = (f |` A)(a := f a)"
3.68 -by(auto simp add: restrict_map_def intro: ext)
3.69 +  by (auto simp add: restrict_map_def intro: ext)
3.70 +
3.71 +
3.72 +subsection {* The @{text "map_default"} operation *}
3.73
3.74  definition map_default :: "'b \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
3.75  where "map_default b f a \<equiv> case f a of None \<Rightarrow> b | Some b' \<Rightarrow> b'"
```
```     4.1 --- a/src/HOL/Map.thy	Tue Jun 02 15:53:34 2009 +0200
4.2 +++ b/src/HOL/Map.thy	Tue Jun 02 16:23:43 2009 +0200
4.3 @@ -332,6 +332,9 @@
4.4  lemma restrict_map_to_empty [simp]: "m|`{} = empty"