author haftmann Sat, 07 Mar 2009 10:06:58 +0100 changeset 30333 e9971af27b11 parent 30324 9afd9a9d0812 (current diff) parent 30332 0d07f4823d3a (diff) child 30334 a2f236a717fa child 30348 5598523c482a child 30352 047f183c43b0
merged
```--- a/NEWS	Fri Mar 06 23:25:08 2009 +0100
+++ b/NEWS	Sat Mar 07 10:06:58 2009 +0100
@@ -220,6 +220,9 @@

*** HOL ***

+* Theory Library/Diagonalize.thy provides constructive version of
+Cantor's first diagonalization argument.
+
* New predicate "strict_mono" classifies strict functions on partial orders.
With strict functions on linear orders, reasoning about (in)equalities is
facilitated by theorems "strict_mono_eq", "strict_mono_less_eq" and "strict_mono_less".```
```--- a/src/HOL/Finite_Set.thy	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/Finite_Set.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -3060,6 +3060,30 @@
by (simp add: Max fold1_strict_below_iff [folded dual_max])
qed

+lemma Min_eqI:
+  assumes "finite A"
+  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<ge> x"
+    and "x \<in> A"
+  shows "Min A = x"
+proof (rule antisym)
+  from `x \<in> A` have "A \<noteq> {}" by auto
+  with assms show "Min A \<ge> x" by simp
+next
+  from assms show "x \<ge> Min A" by simp
+qed
+
+lemma Max_eqI:
+  assumes "finite A"
+  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
+    and "x \<in> A"
+  shows "Max A = x"
+proof (rule antisym)
+  from `x \<in> A` have "A \<noteq> {}" by auto
+  with assms show "Max A \<le> x" by simp
+next
+  from assms show "x \<le> Max A" by simp
+qed
+
lemma Min_antimono:
assumes "M \<subseteq> N" and "M \<noteq> {}" and "finite N"
shows "Min N \<le> Min M"```
```--- a/src/HOL/IsaMakefile	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/IsaMakefile	Sat Mar 07 10:06:58 2009 +0100
@@ -318,7 +318,7 @@
Library/Finite_Cartesian_Product.thy \
Library/FrechetDeriv.thy \
Library/Fundamental_Theorem_Algebra.thy \
-  Library/Inner_Product.thy \
+  Library/Inner_Product.thy Library/Lattice_Syntax.thy \
Library/Nat_Int_Bij.thy Library/Multiset.thy Library/Permutation.thy	\
Library/Primes.thy Library/Pocklington.thy Library/Quotient.thy	\
@@ -332,13 +332,13 @@
Library/List_lexord.thy Library/Commutative_Ring.thy			\
Library/comm_ring.ML Library/Coinductive_List.thy			\
Library/AssocList.thy	Library/Formal_Power_Series.thy	\
-  Library/Binomial.thy Library/Eval_Witness.thy	\
+  Library/Binomial.thy Library/Eval_Witness.thy				\
Library/Code_Index.thy Library/Code_Char.thy				\
Library/Code_Char_chr.thy Library/Code_Integer.thy			\
-  Library/Mapping.thy	Library/Numeral_Type.thy	Library/Reflection.thy		\
-  Library/Boolean_Algebra.thy Library/Countable.thy	\
-  Library/RBT.thy	Library/Univ_Poly.thy	\
-  Library/Random.thy	Library/Quickcheck.thy	\
+  Library/Mapping.thy Library/Numeral_Type.thy Library/Reflection.thy	\
+  Library/Boolean_Algebra.thy Library/Countable.thy			\
+  Library/Diagonalize.thy Library/RBT.thy Library/Univ_Poly.thy		\
+  Library/Random.thy Library/Quickcheck.thy	\
Library/Poly_Deriv.thy \
Library/Polynomial.thy \
Library/Product_plus.thy \```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Diagonalize.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -0,0 +1,149 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* A constructive version of Cantor's first diagonalization argument. *}
+
+theory Diagonalize
+imports Main
+begin
+
+subsection {* Summation from @{text "0"} to @{text "n"} *}
+
+definition sum :: "nat \<Rightarrow> nat" where
+  "sum n = n * Suc n div 2"
+
+lemma sum_0:
+  "sum 0 = 0"
+  unfolding sum_def by simp
+
+lemma sum_Suc:
+  "sum (Suc n) = Suc n + sum n"
+  unfolding sum_def by simp
+
+lemma sum2:
+  "2 * sum n = n * Suc n"
+proof -
+  have "2 dvd n * Suc n"
+  proof (cases "2 dvd n")
+    case True then show ?thesis by simp
+  next
+    case False then have "2 dvd Suc n" by arith
+    then show ?thesis by (simp del: mult_Suc_right)
+  qed
+  then have "n * Suc n div 2 * 2 = n * Suc n"
+    by (rule dvd_div_mult_self [of "2::nat"])
+  then show ?thesis by (simp add: sum_def)
+qed
+
+lemma sum_strict_mono:
+  "strict_mono sum"
+proof (rule strict_monoI)
+  fix m n :: nat
+  assume "m < n"
+  then have "m * Suc m < n * Suc n" by (intro mult_strict_mono) simp_all
+  then have "2 * sum m < 2 * sum n" by (simp add: sum2)
+  then show "sum m < sum n" by auto
+qed
+
+lemma sum_not_less_self:
+  "n \<le> sum n"
+proof -
+  have "2 * n \<le> n * Suc n" by auto
+  with sum2 have "2 * n \<le> 2 * sum n" by simp
+  then show ?thesis by simp
+qed
+
+lemma sum_rest_aux:
+  assumes "q \<le> n"
+  assumes "sum m \<le> sum n + q"
+  shows "m \<le> n"
+proof (rule ccontr)
+  assume "\<not> m \<le> n"
+  then have "n < m" by simp
+  then have "m \<ge> Suc n" by simp
+  then have "m = m - Suc n + Suc n" by simp
+  then have "m = Suc (n + (m - Suc n))" by simp
+  then obtain r where "m = Suc (n + r)" by auto
+  with `sum m \<le> sum n + q` have "sum (Suc (n + r)) \<le> sum n + q" by simp
+  then have "sum (n + r) + Suc (n + r) \<le> sum n + q" unfolding sum_Suc by simp
+  with `m = Suc (n + r)` have "sum (n + r) + m \<le> sum n + q" by simp
+  have "sum n \<le> sum (n + r)" unfolding strict_mono_less_eq [OF sum_strict_mono] by simp
+  moreover from `q \<le> n` `n < m` have "q < m" by simp
+  ultimately have "sum n + q < sum (n + r) + m" by auto
+  with `sum (n + r) + m \<le> sum n + q` show False
+    by auto
+qed
+
+lemma sum_rest:
+  assumes "q \<le> n"
+  shows "sum m \<le> sum n + q \<longleftrightarrow> m \<le> n"
+using assms apply (auto intro: sum_rest_aux)
+apply (simp add: strict_mono_less_eq [OF sum_strict_mono, symmetric, of m n])
+done
+
+
+subsection {* Diagonalization: an injective embedding of two @{typ "nat"}s to one @{typ "nat"} *}
+
+definition diagonalize :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
+  "diagonalize m n = sum (m + n) + m"
+
+lemma diagonalize_inject:
+  assumes "diagonalize a b = diagonalize c d"
+  shows "a = c" and "b = d"
+proof -
+  from assms have diageq: "sum (a + b) + a = sum (c + d) + c"
+  have "a + b = c + d \<or> a + b \<ge> Suc (c + d) \<or> c + d \<ge> Suc (a + b)" by arith
+  then have "a = c \<and> b = d"
+  proof (elim disjE)
+    assume sumeq: "a + b = c + d"
+    then have "a = c" using diageq by auto
+    moreover from sumeq this have "b = d" by auto
+    ultimately show ?thesis ..
+  next
+    assume "a + b \<ge> Suc (c + d)"
+    with strict_mono_less_eq [OF sum_strict_mono]
+      have "sum (a + b) \<ge> sum (Suc (c + d))" by simp
+    with diageq show ?thesis by (simp add: sum_Suc)
+  next
+    assume "c + d \<ge> Suc (a + b)"
+    with strict_mono_less_eq [OF sum_strict_mono]
+      have "sum (c + d) \<ge> sum (Suc (a + b))" by simp
+    with diageq show ?thesis by (simp add: sum_Suc)
+  qed
+  then show "a = c" and "b = d" by auto
+qed
+
+
+subsection {* The reverse diagonalization: reconstruction a pair of @{typ nat}s from one @{typ nat} *}
+
+text {* The inverse of the @{const sum} function *}
+
+definition tupelize :: "nat \<Rightarrow> nat \<times> nat" where
+  "tupelize q = (let d = Max {d. sum d \<le> q}; m = q - sum d
+     in (m, d - m))"
+
+lemma tupelize_diagonalize:
+  "tupelize (diagonalize m n) = (m, n)"
+proof -
+  from sum_rest
+    have "\<And>r. sum r \<le> sum (m + n) + m \<longleftrightarrow> r \<le> m + n" by simp
+  then have "Max {d. sum d \<le> (sum (m + n) + m)} = m + n"
+    by (auto intro: Max_eqI)
+  then show ?thesis
+    by (simp add: tupelize_def diagonalize_def)
+qed
+
+lemma snd_tupelize:
+  "snd (tupelize n) \<le> n"
+proof -
+  have "sum 0 \<le> n" by (simp add: sum_0)
+  then have "Max {m \<Colon> nat. sum m \<le> n} \<le> Max {m \<Colon> nat. m \<le> n}"
+    by (intro Max_mono [of "{m. sum m \<le> n}" "{m. m \<le> n}"])
+      (auto intro: Max_mono order_trans sum_not_less_self)
+  also have "Max {m \<Colon> nat. m \<le> n} \<le> n"
+    by (subst Max_le_iff) auto
+  finally have "Max {m. sum m \<le> n} \<le> n" .
+  then show ?thesis by (simp add: tupelize_def Let_def)
+qed
+
+end```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Lattice_Syntax.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -0,0 +1,17 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* Pretty syntax for lattice operations *}
+
+(*<*)
+theory Lattice_Syntax
+imports Set
+begin
+
+notation
+  inf  (infixl "\<sqinter>" 70) and
+  sup  (infixl "\<squnion>" 65) and
+  Inf  ("\<Sqinter>_"  900) and
+  Sup  ("\<Squnion>_"  900)
+
+end
+(*>*)
\ No newline at end of file```
```--- a/src/HOL/Library/Library.thy	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/Library/Library.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -17,6 +17,7 @@
ContNotDenum
Countable
Determinants
+  Diagonalize
Efficient_Nat
Enum
Eval_Witness
@@ -28,6 +29,7 @@
Fundamental_Theorem_Algebra
Infinite_Set
Inner_Product
+  Lattice_Syntax
ListVector
Mapping
Multiset```
```--- a/src/HOL/Option.thy	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/Option.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -5,7 +5,7 @@

theory Option
-imports Datatype
+imports Datatype Finite_Set
begin

datatype 'a option = None | Some 'a
@@ -30,6 +30,9 @@
lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV"
by (rule set_ext, case_tac x) auto

+instance option :: (finite) finite proof
+
lemma inj_Some [simp]: "inj_on Some A"
by (rule inj_onI) simp
```
```--- a/src/HOL/Plain.thy	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/Plain.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -9,9 +9,6 @@
include @{text Hilbert_Choice}.
*}

-instance option :: (finite) finite
-  by default (simp add: insert_None_conv_UNIV [symmetric])
-

end```
```--- a/src/HOL/Predicate.thy	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/Predicate.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -1,15 +1,40 @@
(*  Title:      HOL/Predicate.thy
-    ID:         \$Id\$
-    Author:     Stefan Berghofer, TU Muenchen
+    Author:     Stefan Berghofer and Lukas Bulwahn and Florian Haftmann, TU Muenchen
*)

+header {* Predicates as relations and enumerations *}

theory Predicate
imports Inductive Relation
begin

-subsection {* Equality and Subsets *}
+notation
+  inf (infixl "\<sqinter>" 70) and
+  sup (infixl "\<squnion>" 65) and
+  Inf ("\<Sqinter>_"  900) and
+  Sup ("\<Squnion>_"  900) and
+  top ("\<top>") and
+  bot ("\<bottom>")
+
+
+subsection {* Predicates as (complete) lattices *}
+
+subsubsection {* @{const sup} on @{typ bool} *}
+
+lemma sup_boolI1:
+  "P \<Longrightarrow> P \<squnion> Q"
+
+lemma sup_boolI2:
+  "Q \<Longrightarrow> P \<squnion> Q"
+
+lemma sup_boolE:
+  "P \<squnion> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
+  by (auto simp add: sup_bool_eq)
+
+
+subsubsection {* Equality and Subsets *}

lemma pred_equals_eq: "((\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S)) = (R = S)"
@@ -24,7 +49,7 @@
by fast

-subsection {* Top and bottom elements *}
+subsubsection {* Top and bottom elements *}

lemma top1I [intro!]: "top x"
@@ -39,7 +64,7 @@

-subsection {* The empty set *}
+subsubsection {* The empty set *}

lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
@@ -48,7 +73,7 @@

-subsection {* Binary union *}
+subsubsection {* Binary union *}

lemma sup1_iff [simp]: "sup A B x \<longleftrightarrow> A x | B x"
@@ -92,7 +117,7 @@
by simp iprover

-subsection {* Binary intersection *}
+subsubsection {* Binary intersection *}

lemma inf1_iff [simp]: "inf A B x \<longleftrightarrow> A x \<and> B x"
@@ -131,7 +156,7 @@
by simp

-subsection {* Unions of families *}
+subsubsection {* Unions of families *}

lemma SUP1_iff [simp]: "(SUP x:A. B x) b = (EX x:A. B x b)"
by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast
@@ -158,7 +183,7 @@

-subsection {* Intersections of families *}
+subsubsection {* Intersections of families *}

lemma INF1_iff [simp]: "(INF x:A. B x) b = (ALL x:A. B x b)"
by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast
@@ -191,7 +216,9 @@

-subsection {* Composition of two relations *}
+subsection {* Predicates as relations *}
+
+subsubsection {* Composition  *}

inductive
pred_comp  :: "['b => 'c => bool, 'a => 'b => bool] => 'a => 'c => bool"
@@ -207,7 +234,7 @@
by (auto simp add: expand_fun_eq elim: pred_compE)

-subsection {* Converse *}
+subsubsection {* Converse *}

inductive
conversep :: "('a => 'b => bool) => 'b => 'a => bool"
@@ -253,7 +280,7 @@

-subsection {* Domain *}
+subsubsection {* Domain *}

inductive
DomainP :: "('a => 'b => bool) => 'a => bool"
@@ -267,7 +294,7 @@
by (blast intro!: Orderings.order_antisym predicate1I)

-subsection {* Range *}
+subsubsection {* Range *}

inductive
RangeP :: "('a => 'b => bool) => 'b => bool"
@@ -281,7 +308,7 @@
by (blast intro!: Orderings.order_antisym predicate1I)

-subsection {* Inverse image *}
+subsubsection {* Inverse image *}

definition
inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where
@@ -294,7 +321,7 @@

-subsection {* The Powerset operator *}
+subsubsection {* Powerset *}

definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" where
"Powp A == \<lambda>B. \<forall>x \<in> B. A x"
@@ -305,7 +332,7 @@
lemmas Powp_mono [mono] = Pow_mono [to_pred pred_subset_eq]

-subsection {* Properties of relations - predicate versions *}
+subsubsection {* Properties of relations *}

abbreviation antisymP :: "('a => 'a => bool) => bool" where
"antisymP r == antisym {(x, y). r x y}"
@@ -316,4 +343,264 @@
abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
"single_valuedP r == single_valued {(x, y). r x y}"

+
+subsection {* Predicates as enumerations *}
+
+subsubsection {* The type of predicate enumerations (a monad) *}
+
+datatype 'a pred = Pred "'a \<Rightarrow> bool"
+
+primrec eval :: "'a pred \<Rightarrow> 'a \<Rightarrow> bool" where
+  eval_pred: "eval (Pred f) = f"
+
+lemma Pred_eval [simp]:
+  "Pred (eval x) = x"
+  by (cases x) simp
+
+lemma eval_inject: "eval x = eval y \<longleftrightarrow> x = y"
+  by (cases x) auto
+
+definition single :: "'a \<Rightarrow> 'a pred" where
+  "single x = Pred ((op =) x)"
+
+definition bind :: "'a pred \<Rightarrow> ('a \<Rightarrow> 'b pred) \<Rightarrow> 'b pred" (infixl "\<guillemotright>=" 70) where
+  "P \<guillemotright>= f = Pred (\<lambda>x. (\<exists>y. eval P y \<and> eval (f y) x))"
+
+instantiation pred :: (type) complete_lattice
+begin
+
+definition
+  "P \<le> Q \<longleftrightarrow> eval P \<le> eval Q"
+
+definition
+  "P < Q \<longleftrightarrow> eval P < eval Q"
+
+definition
+  "\<bottom> = Pred \<bottom>"
+
+definition
+  "\<top> = Pred \<top>"
+
+definition
+  "P \<sqinter> Q = Pred (eval P \<sqinter> eval Q)"
+
+definition
+  "P \<squnion> Q = Pred (eval P \<squnion> eval Q)"
+
+definition
+  "\<Sqinter>A = Pred (INFI A eval)"
+
+definition
+  "\<Squnion>A = Pred (SUPR A eval)"
+
+instance by default
+  (auto simp add: less_eq_pred_def less_pred_def
+    inf_pred_def sup_pred_def bot_pred_def top_pred_def
+    Inf_pred_def Sup_pred_def,
+    auto simp add: le_fun_def less_fun_def le_bool_def less_bool_def
+    eval_inject mem_def)
+
end
+
+lemma bind_bind:
+  "(P \<guillemotright>= Q) \<guillemotright>= R = P \<guillemotright>= (\<lambda>x. Q x \<guillemotright>= R)"
+  by (auto simp add: bind_def expand_fun_eq)
+
+lemma bind_single:
+  "P \<guillemotright>= single = P"
+  by (simp add: bind_def single_def)
+
+lemma single_bind:
+  "single x \<guillemotright>= P = P x"
+  by (simp add: bind_def single_def)
+
+lemma bottom_bind:
+  "\<bottom> \<guillemotright>= P = \<bottom>"
+  by (auto simp add: bot_pred_def bind_def expand_fun_eq)
+
+lemma sup_bind:
+  "(P \<squnion> Q) \<guillemotright>= R = P \<guillemotright>= R \<squnion> Q \<guillemotright>= R"
+  by (auto simp add: bind_def sup_pred_def expand_fun_eq)
+
+lemma Sup_bind: "(\<Squnion>A \<guillemotright>= f) = \<Squnion>((\<lambda>x. x \<guillemotright>= f) ` A)"
+  by (auto simp add: bind_def Sup_pred_def expand_fun_eq)
+
+lemma pred_iffI:
+  assumes "\<And>x. eval A x \<Longrightarrow> eval B x"
+  and "\<And>x. eval B x \<Longrightarrow> eval A x"
+  shows "A = B"
+proof -
+  from assms have "\<And>x. eval A x \<longleftrightarrow> eval B x" by blast
+  then show ?thesis by (cases A, cases B) (simp add: expand_fun_eq)
+qed
+
+lemma singleI: "eval (single x) x"
+  unfolding single_def by simp
+
+lemma singleI_unit: "eval (single ()) x"
+  by simp (rule singleI)
+
+lemma singleE: "eval (single x) y \<Longrightarrow> (y = x \<Longrightarrow> P) \<Longrightarrow> P"
+  unfolding single_def by simp
+
+lemma singleE': "eval (single x) y \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
+  by (erule singleE) simp
+
+lemma bindI: "eval P x \<Longrightarrow> eval (Q x) y \<Longrightarrow> eval (P \<guillemotright>= Q) y"
+  unfolding bind_def by auto
+
+lemma bindE: "eval (R \<guillemotright>= Q) y \<Longrightarrow> (\<And>x. eval R x \<Longrightarrow> eval (Q x) y \<Longrightarrow> P) \<Longrightarrow> P"
+  unfolding bind_def by auto
+
+lemma botE: "eval \<bottom> x \<Longrightarrow> P"
+  unfolding bot_pred_def by auto
+
+lemma supI1: "eval A x \<Longrightarrow> eval (A \<squnion> B) x"
+  unfolding sup_pred_def by simp
+
+lemma supI2: "eval B x \<Longrightarrow> eval (A \<squnion> B) x"
+  unfolding sup_pred_def by simp
+
+lemma supE: "eval (A \<squnion> B) x \<Longrightarrow> (eval A x \<Longrightarrow> P) \<Longrightarrow> (eval B x \<Longrightarrow> P) \<Longrightarrow> P"
+  unfolding sup_pred_def by auto
+
+
+subsubsection {* Derived operations *}
+
+definition if_pred :: "bool \<Rightarrow> unit pred" where
+  if_pred_eq: "if_pred b = (if b then single () else \<bottom>)"
+
+definition eq_pred :: "'a \<Rightarrow> 'a \<Rightarrow> unit pred" where
+  eq_pred_eq: "eq_pred a b = if_pred (a = b)"
+
+definition not_pred :: "unit pred \<Rightarrow> unit pred" where
+  not_pred_eq: "not_pred P = (if eval P () then \<bottom> else single ())"
+
+lemma if_predI: "P \<Longrightarrow> eval (if_pred P) ()"
+  unfolding if_pred_eq by (auto intro: singleI)
+
+lemma if_predE: "eval (if_pred b) x \<Longrightarrow> (b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
+  unfolding if_pred_eq by (cases b) (auto elim: botE)
+
+lemma eq_predI: "eval (eq_pred a a) ()"
+  unfolding eq_pred_eq if_pred_eq by (auto intro: singleI)
+
+lemma eq_predE: "eval (eq_pred a b) x \<Longrightarrow> (a = b \<Longrightarrow> x = () \<Longrightarrow> P) \<Longrightarrow> P"
+  unfolding eq_pred_eq by (erule if_predE)
+
+lemma not_predI: "\<not> P \<Longrightarrow> eval (not_pred (Pred (\<lambda>u. P))) ()"
+  unfolding not_pred_eq eval_pred by (auto intro: singleI)
+
+lemma not_predI': "\<not> eval P () \<Longrightarrow> eval (not_pred P) ()"
+  unfolding not_pred_eq by (auto intro: singleI)
+
+lemma not_predE: "eval (not_pred (Pred (\<lambda>u. P))) x \<Longrightarrow> (\<not> P \<Longrightarrow> thesis) \<Longrightarrow> thesis"
+  unfolding not_pred_eq
+  by (auto split: split_if_asm elim: botE)
+
+lemma not_predE': "eval (not_pred P) x \<Longrightarrow> (\<not> eval P x \<Longrightarrow> thesis) \<Longrightarrow> thesis"
+  unfolding not_pred_eq
+  by (auto split: split_if_asm elim: botE)
+
+
+subsubsection {* Implementation *}
+
+datatype 'a seq = Empty | Insert "'a" "'a pred" | Join "'a pred" "'a seq"
+
+primrec pred_of_seq :: "'a seq \<Rightarrow> 'a pred" where
+    "pred_of_seq Empty = \<bottom>"
+  | "pred_of_seq (Insert x P) = single x \<squnion> P"
+  | "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
+
+definition Seq :: "(unit \<Rightarrow> 'a seq) \<Rightarrow> 'a pred" where
+  "Seq f = pred_of_seq (f ())"
+
+code_datatype Seq
+
+primrec member :: "'a seq \<Rightarrow> 'a \<Rightarrow> bool"  where
+  "member Empty x \<longleftrightarrow> False"
+  | "member (Insert y P) x \<longleftrightarrow> x = y \<or> eval P x"
+  | "member (Join P xq) x \<longleftrightarrow> eval P x \<or> member xq x"
+
+lemma eval_member:
+  "member xq = eval (pred_of_seq xq)"
+proof (induct xq)
+  case Empty show ?case
+  by (auto simp add: expand_fun_eq elim: botE)
+next
+  case Insert show ?case
+  by (auto simp add: expand_fun_eq elim: supE singleE intro: supI1 supI2 singleI)
+next
+  case Join then show ?case
+  by (auto simp add: expand_fun_eq elim: supE intro: supI1 supI2)
+qed
+
+lemma eval_code [code]: "eval (Seq f) = member (f ())"
+  unfolding Seq_def by (rule sym, rule eval_member)
+
+lemma single_code [code]:
+  "single x = Seq (\<lambda>u. Insert x \<bottom>)"
+  unfolding Seq_def by simp
+
+primrec "apply" :: "('a \<Rightarrow> 'b Predicate.pred) \<Rightarrow> 'a seq \<Rightarrow> 'b seq" where
+    "apply f Empty = Empty"
+  | "apply f (Insert x P) = Join (f x) (Join (P \<guillemotright>= f) Empty)"
+  | "apply f (Join P xq) = Join (P \<guillemotright>= f) (apply f xq)"
+
+lemma apply_bind:
+  "pred_of_seq (apply f xq) = pred_of_seq xq \<guillemotright>= f"
+proof (induct xq)
+  case Empty show ?case
+next
+  case Insert show ?case
+    by (simp add: single_bind sup_bind)
+next
+  case Join then show ?case
+qed
+
+lemma bind_code [code]:
+  "Seq g \<guillemotright>= f = Seq (\<lambda>u. apply f (g ()))"
+  unfolding Seq_def by (rule sym, rule apply_bind)
+
+lemma bot_set_code [code]:
+  "\<bottom> = Seq (\<lambda>u. Empty)"
+  unfolding Seq_def by simp
+
+lemma sup_code [code]:
+  "Seq f \<squnion> Seq g = Seq (\<lambda>u. case f ()
+    of Empty \<Rightarrow> g ()
+     | Insert x P \<Rightarrow> Insert x (P \<squnion> Seq g)
+     | Join P xq \<Rightarrow> Join (Seq g) (Join P xq))" (*FIXME order!?*)
+proof (cases "f ()")
+  case Empty
+  thus ?thesis
+    unfolding Seq_def by (simp add: sup_commute [of "\<bottom>"] sup_bot)
+next
+  case Insert
+  thus ?thesis
+    unfolding Seq_def by (simp add: sup_assoc)
+next
+  case Join
+  thus ?thesis
+    unfolding Seq_def by (simp add: sup_commute [of "pred_of_seq (g ())"] sup_assoc)
+qed
+
+
+declare eq_pred_def [code, code del]
+
+no_notation
+  inf (infixl "\<sqinter>" 70) and
+  sup (infixl "\<squnion>" 65) and
+  Inf ("\<Sqinter>_"  900) and
+  Sup ("\<Squnion>_"  900) and
+  top ("\<top>") and
+  bot ("\<bottom>") and
+  bind (infixl "\<guillemotright>=" 70)
+
+hide (open) type pred seq
+hide (open) const Pred eval single bind if_pred eq_pred not_pred
+  Empty Insert Join Seq member "apply"
+
+end```
```--- a/src/HOL/document/root.tex	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/document/root.tex	Sat Mar 07 10:06:58 2009 +0100
@@ -1,9 +1,8 @@
-
-% \$Id\$

\documentclass[11pt,a4paper]{article}
\usepackage{graphicx,isabelle,isabellesym,latexsym}
\usepackage{amssymb}
+\usepackage[english]{babel}
\usepackage[only,bigsqcap]{stmaryrd}
\usepackage[latin1]{inputenc}
\usepackage{pdfsetup}```
```--- a/src/HOL/ex/ExecutableContent.thy	Fri Mar 06 23:25:08 2009 +0100
+++ b/src/HOL/ex/ExecutableContent.thy	Sat Mar 07 10:06:58 2009 +0100
@@ -24,6 +24,19 @@
"~~/src/HOL/ex/Records"
begin

+lemma [code, code del]:
+  "(size :: 'a::size Predicate.pred => nat) = size" ..
+lemma [code, code del]:
+  "pred_size = pred_size" ..
+lemma [code, code del]:
+  "pred_case = pred_case" ..
+lemma [code, code del]:
+  "pred_rec = pred_rec" ..
+lemma [code, code del]:
+  "(Code_Eval.term_of \<Colon> 'a::{type, term_of} Predicate.pred \<Rightarrow> Code_Eval.term) = Code_Eval.term_of" ..
+lemma [code, code del]:
+  "(Code_Eval.term_of \<Colon> 'a::{type, term_of} Predicate.seq \<Rightarrow> Code_Eval.term) = Code_Eval.term_of" ..
+
text {* However, some aren't executable *}

declare pair_leq_def[code del]```