--- 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/Library.thy Library/List_Prefix.thy Library/State_Monad.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"
+ by (simp add: diagonalize_def)
+ 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] 900) and
+ Sup ("\<Squnion>_" [900] 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 @@
header {* Datatype option *}
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
+qed (simp add: insert_None_conv_UNIV [symmetric])
+
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])
-
ML {* path_add "~~/src/HOL/Library" *}
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 *}
+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] 900) and
+ Sup ("\<Squnion>_" [900] 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"
+ by (simp add: sup_bool_eq)
+
+lemma sup_boolI2:
+ "Q \<Longrightarrow> P \<squnion> Q"
+ by (simp add: sup_bool_eq)
+
+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)"
by (simp add: mem_def)
@@ -24,7 +49,7 @@
by fast
-subsection {* Top and bottom elements *}
+subsubsection {* Top and bottom elements *}
lemma top1I [intro!]: "top x"
by (simp add: top_fun_eq top_bool_eq)
@@ -39,7 +64,7 @@
by (simp add: bot_fun_eq bot_bool_eq)
-subsection {* The empty set *}
+subsubsection {* The empty set *}
lemma bot_empty_eq: "bot = (\<lambda>x. x \<in> {})"
by (auto simp add: expand_fun_eq)
@@ -48,7 +73,7 @@
by (auto simp add: expand_fun_eq)
-subsection {* Binary union *}
+subsubsection {* Binary union *}
lemma sup1_iff [simp]: "sup A B x \<longleftrightarrow> A x | B x"
by (simp add: sup_fun_eq sup_bool_eq)
@@ -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"
by (simp add: inf_fun_eq inf_bool_eq)
@@ -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 @@
by (simp add: expand_fun_eq)
-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 @@
by (simp add: expand_fun_eq)
-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 @@
by (auto simp add: expand_fun_eq)
-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 @@
by (simp add: inv_imagep_def)
-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
+ by (simp add: bottom_bind)
+next
+ case Insert show ?case
+ by (simp add: single_bind sup_bind)
+next
+ case Join then show ?case
+ by (simp add: sup_bind)
+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] 900) and
+ Sup ("\<Squnion>_" [900] 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]