# HG changeset patch
# User kleing
# Date 1187562138 -7200
# Node ID e3a2b75b1cf91634f680e63f796ef59639688717
# Parent  76f7a8c6e8423943f948c07c204116f045a5015d
boolean algebras as locales and numbers as types by Brian Huffman

diff -r 76f7a8c6e842 -r e3a2b75b1cf9 CONTRIBUTORS
--- a/CONTRIBUTORS	Sun Aug 19 21:21:37 2007 +0200
+++ b/CONTRIBUTORS	Mon Aug 20 00:22:18 2007 +0200
@@ -7,6 +7,9 @@
 Contributions to Isabelle 2007
 ------------------------------
 
+* August 2007: Brian Huffman, PSU
+  HOL/Library/Boolean_Algebra and HOL/Library/Numeral_Type
+
 * June 2007: Amine Chaieb, TUM
   Semiring normalization and Groebner Bases
 
diff -r 76f7a8c6e842 -r e3a2b75b1cf9 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Sun Aug 19 21:21:37 2007 +0200
+++ b/src/HOL/IsaMakefile	Mon Aug 20 00:22:18 2007 +0200
@@ -226,8 +226,10 @@
   Library/SCT_Interpretation.thy \
   Library/SCT_Implementation.thy Library/Size_Change_Termination.thy \
   Library/SCT_Examples.thy Library/sct.ML \
-  Library/Pure_term.thy Library/Eval.thy Library/Eval_Witness.thy Library/Pretty_Int.thy \
-  Library/Pretty_Char.thy Library/Pretty_Char_chr.thy Library/Abstract_Rat.thy
+  Library/Pure_term.thy Library/Eval.thy Library/Eval_Witness.thy \
+  Library/Pretty_Int.thy \
+  Library/Pretty_Char.thy Library/Pretty_Char_chr.thy Library/Abstract_Rat.thy\
+  Library/Numeral_Type.thy Library/Boolean_Algebra.thy
 	@cd Library; $(ISATOOL) usedir $(OUT)/HOL Library
 
 
diff -r 76f7a8c6e842 -r e3a2b75b1cf9 src/HOL/Library/Boolean_Algebra.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Boolean_Algebra.thy	Mon Aug 20 00:22:18 2007 +0200
@@ -0,0 +1,276 @@
+(* 
+  ID:     $Id$
+  Author: Brian Huffman
+
+  Boolean algebras as locales.
+*)
+
+header {* Boolean Algebras *}
+
+theory Boolean_Algebra
+imports Main
+begin
+
+locale boolean =
+  fixes conj :: "'a => 'a => 'a" (infixr "\<sqinter>" 70)
+  fixes disj :: "'a => 'a => 'a" (infixr "\<squnion>" 65)
+  fixes compl :: "'a => 'a" ("\<sim> _" [81] 80)
+  fixes zero :: "'a" ("\<zero>")
+  fixes one  :: "'a" ("\<one>")
+  assumes conj_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
+  assumes disj_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
+  assumes conj_commute: "x \<sqinter> y = y \<sqinter> x"
+  assumes disj_commute: "x \<squnion> y = y \<squnion> x"
+  assumes conj_disj_distrib: "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
+  assumes disj_conj_distrib: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
+  assumes conj_one_right: "x \<sqinter> \<one> = x"
+  assumes disj_zero_right: "x \<squnion> \<zero> = x"
+  assumes conj_cancel_right: "x \<sqinter> \<sim> x = \<zero>"
+  assumes disj_cancel_right: "x \<squnion> \<sim> x = \<one>"
+begin
+
+lemmas disj_ac =
+  disj_assoc disj_commute
+  mk_left_commute [of "disj", OF disj_assoc disj_commute]
+
+lemmas conj_ac =
+  conj_assoc conj_commute
+  mk_left_commute [of "conj", OF conj_assoc conj_commute]
+
+lemma dual: "boolean disj conj compl one zero"
+apply (rule boolean.intro)
+apply (rule disj_assoc)
+apply (rule conj_assoc)
+apply (rule disj_commute)
+apply (rule conj_commute)
+apply (rule disj_conj_distrib)
+apply (rule conj_disj_distrib)
+apply (rule disj_zero_right)
+apply (rule conj_one_right)
+apply (rule disj_cancel_right)
+apply (rule conj_cancel_right)
+done
+
+text {* Complement *}
+
+lemma complement_unique:
+  assumes 1: "a \<sqinter> x = \<zero>"
+  assumes 2: "a \<squnion> x = \<one>"
+  assumes 3: "a \<sqinter> y = \<zero>"
+  assumes 4: "a \<squnion> y = \<one>"
+  shows "x = y"
+proof -
+  have "(a \<sqinter> x) \<squnion> (x \<sqinter> y) = (a \<sqinter> y) \<squnion> (x \<sqinter> y)" using 1 3 by simp
+  hence "(x \<sqinter> a) \<squnion> (x \<sqinter> y) = (y \<sqinter> a) \<squnion> (y \<sqinter> x)" using conj_commute by simp
+  hence "x \<sqinter> (a \<squnion> y) = y \<sqinter> (a \<squnion> x)" using conj_disj_distrib by simp
+  hence "x \<sqinter> \<one> = y \<sqinter> \<one>" using 2 4 by simp
+  thus "x = y" using conj_one_right by simp
+qed
+
+lemma compl_unique: "[| x \<sqinter> y = \<zero>; x \<squnion> y = \<one> |] ==> \<sim> x = y"
+by (rule complement_unique [OF conj_cancel_right disj_cancel_right])
+
+lemma double_compl [simp]: "\<sim> (\<sim> x) = x"
+proof (rule compl_unique)
+  from conj_cancel_right show "\<sim> x \<sqinter> x = \<zero>" by (simp add: conj_commute)
+  from disj_cancel_right show "\<sim> x \<squnion> x = \<one>" by (simp add: disj_commute)
+qed
+
+lemma compl_eq_compl_iff [simp]: "(\<sim> x = \<sim> y) = (x = y)"
+by (rule inj_eq [OF inj_on_inverseI], rule double_compl)
+
+text {* Conjunction *}
+
+lemma conj_absorb: "x \<sqinter> x = x"
+proof -
+  have "x \<sqinter> x = (x \<sqinter> x) \<squnion> \<zero>" using disj_zero_right by simp
+  also have "... = (x \<sqinter> x) \<squnion> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp
+  also have "... = x \<sqinter> (x \<squnion> \<sim> x)" using conj_disj_distrib by simp
+  also have "... = x \<sqinter> \<one>" using disj_cancel_right by simp
+  also have "... = x" using conj_one_right by simp
+  finally show ?thesis .
+qed
+
+lemma conj_zero_right [simp]: "x \<sqinter> \<zero> = \<zero>"
+proof -
+  have "x \<sqinter> \<zero> = x \<sqinter> (x \<sqinter> \<sim> x)" using conj_cancel_right by simp
+  also have "... = (x \<sqinter> x) \<sqinter> \<sim> x" using conj_assoc by simp
+  also have "... = x \<sqinter> \<sim> x" using conj_absorb by simp
+  also have "... = \<zero>" using conj_cancel_right by simp
+  finally show ?thesis .
+qed
+
+lemma compl_one [simp]: "\<sim> \<one> = \<zero>"
+by (rule compl_unique [OF conj_zero_right disj_zero_right])
+
+lemma conj_zero_left [simp]: "\<zero> \<sqinter> x = \<zero>"
+by (subst conj_commute) (rule conj_zero_right)
+
+lemma conj_one_left [simp]: "\<one> \<sqinter> x = x"
+by (subst conj_commute) (rule conj_one_right)
+
+lemma conj_cancel_left [simp]: "\<sim> x \<sqinter> x = \<zero>"
+by (subst conj_commute) (rule conj_cancel_right)
+
+lemma conj_left_absorb [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
+by (simp add: conj_assoc [symmetric] conj_absorb)
+
+lemma conj_disj_distrib2:
+  "(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" 
+by (simp add: conj_commute conj_disj_distrib)
+
+lemmas conj_disj_distribs =
+   conj_disj_distrib conj_disj_distrib2
+
+text {* Disjunction *}
+
+lemma disj_absorb [simp]: "x \<squnion> x = x"
+by (rule boolean.conj_absorb [OF dual])
+
+lemma disj_one_right [simp]: "x \<squnion> \<one> = \<one>"
+by (rule boolean.conj_zero_right [OF dual])
+
+lemma compl_zero [simp]: "\<sim> \<zero> = \<one>"
+by (rule boolean.compl_one [OF dual])
+
+lemma disj_zero_left [simp]: "\<zero> \<squnion> x = x"
+by (rule boolean.conj_one_left [OF dual])
+
+lemma disj_one_left [simp]: "\<one> \<squnion> x = \<one>"
+by (rule boolean.conj_zero_left [OF dual])
+
+lemma disj_cancel_left [simp]: "\<sim> x \<squnion> x = \<one>"
+by (rule boolean.conj_cancel_left [OF dual])
+
+lemma disj_left_absorb [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
+by (rule boolean.conj_left_absorb [OF dual])
+
+lemma disj_conj_distrib2:
+  "(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
+by (rule boolean.conj_disj_distrib2 [OF dual])
+
+lemmas disj_conj_distribs =
+   disj_conj_distrib disj_conj_distrib2
+
+text {* De Morgan's Laws *}
+
+lemma de_Morgan_conj [simp]: "\<sim> (x \<sqinter> y) = \<sim> x \<squnion> \<sim> y"
+proof (rule compl_unique)
+  have "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = ((x \<sqinter> y) \<sqinter> \<sim> x) \<squnion> ((x \<sqinter> y) \<sqinter> \<sim> y)"
+    by (rule conj_disj_distrib)
+  also have "... = (y \<sqinter> (x \<sqinter> \<sim> x)) \<squnion> (x \<sqinter> (y \<sqinter> \<sim> y))"
+    by (simp add: conj_ac)
+  finally show "(x \<sqinter> y) \<sqinter> (\<sim> x \<squnion> \<sim> y) = \<zero>"
+    by (simp add: conj_cancel_right conj_zero_right disj_zero_right)
+next
+  have "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = (x \<squnion> (\<sim> x \<squnion> \<sim> y)) \<sqinter> (y \<squnion> (\<sim> x \<squnion> \<sim> y))"
+    by (rule disj_conj_distrib2)
+  also have "... = (\<sim> y \<squnion> (x \<squnion> \<sim> x)) \<sqinter> (\<sim> x \<squnion> (y \<squnion> \<sim> y))"
+    by (simp add: disj_ac)
+  finally show "(x \<sqinter> y) \<squnion> (\<sim> x \<squnion> \<sim> y) = \<one>"
+    by (simp add: disj_cancel_right disj_one_right conj_one_right)
+qed
+
+lemma de_Morgan_disj [simp]: "\<sim> (x \<squnion> y) = \<sim> x \<sqinter> \<sim> y"
+by (rule boolean.de_Morgan_conj [OF dual])
+
+end
+
+text {* Symmetric Difference *}
+
+locale boolean_xor = boolean +
+  fixes xor :: "'a => 'a => 'a"  (infixr "\<oplus>" 65)
+  assumes xor_def: "x \<oplus> y = (x \<sqinter> \<sim> y) \<squnion> (\<sim> x \<sqinter> y)"
+begin
+
+lemma xor_def2:
+  "x \<oplus> y = (x \<squnion> y) \<sqinter> (\<sim> x \<squnion> \<sim> y)"
+by (simp add: xor_def conj_disj_distribs
+              disj_ac conj_ac conj_cancel_right disj_zero_left)
+
+lemma xor_commute: "x \<oplus> y = y \<oplus> x"
+by (simp add: xor_def conj_commute disj_commute)
+
+lemma xor_assoc: "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
+proof -
+  let ?t = "(x \<sqinter> y \<sqinter> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> \<sim> z) \<squnion>
+            (\<sim> x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (\<sim> x \<sqinter> \<sim> y \<sqinter> z)"
+  have "?t \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> y \<sqinter> \<sim> y) =
+        ?t \<squnion> (x \<sqinter> y \<sqinter> \<sim> y) \<squnion> (x \<sqinter> z \<sqinter> \<sim> z)"
+    by (simp add: conj_cancel_right conj_zero_right)
+  thus "(x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
+    apply (simp add: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+    apply (simp add: conj_disj_distribs conj_ac disj_ac)
+    done
+qed
+
+lemmas xor_ac =
+  xor_assoc xor_commute
+  mk_left_commute [of "xor", OF xor_assoc xor_commute]
+
+lemma xor_zero_right [simp]: "x \<oplus> \<zero> = x"
+by (simp add: xor_def compl_zero conj_one_right conj_zero_right disj_zero_right)
+
+lemma xor_zero_left [simp]: "\<zero> \<oplus> x = x"
+by (subst xor_commute) (rule xor_zero_right)
+
+lemma xor_one_right [simp]: "x \<oplus> \<one> = \<sim> x"
+by (simp add: xor_def compl_one conj_zero_right conj_one_right disj_zero_left)
+
+lemma xor_one_left [simp]: "\<one> \<oplus> x = \<sim> x"
+by (subst xor_commute) (rule xor_one_right)
+
+lemma xor_self [simp]: "x \<oplus> x = \<zero>"
+by (simp add: xor_def conj_cancel_right conj_cancel_left disj_zero_right)
+
+lemma xor_left_self [simp]: "x \<oplus> (x \<oplus> y) = y"
+by (simp add: xor_assoc [symmetric] xor_self xor_zero_left)
+
+lemma xor_compl_left: "\<sim> x \<oplus> y = \<sim> (x \<oplus> y)"
+apply (simp add: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+apply (simp add: conj_disj_distribs)
+apply (simp add: conj_cancel_right conj_cancel_left)
+apply (simp add: disj_zero_left disj_zero_right)
+apply (simp add: disj_ac conj_ac)
+done
+
+lemma xor_compl_right: "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)"
+apply (simp add: xor_def de_Morgan_disj de_Morgan_conj double_compl)
+apply (simp add: conj_disj_distribs)
+apply (simp add: conj_cancel_right conj_cancel_left)
+apply (simp add: disj_zero_left disj_zero_right)
+apply (simp add: disj_ac conj_ac)
+done
+
+lemma xor_cancel_right [simp]: "x \<oplus> \<sim> x = \<one>"
+by (simp add: xor_compl_right xor_self compl_zero)
+
+lemma xor_cancel_left [simp]: "\<sim> x \<oplus> x = \<one>"
+by (subst xor_commute) (rule xor_cancel_right)
+
+lemma conj_xor_distrib: "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+proof -
+  have "(x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z) =
+        (y \<sqinter> x \<sqinter> \<sim> x) \<squnion> (z \<sqinter> x \<sqinter> \<sim> x) \<squnion> (x \<sqinter> y \<sqinter> \<sim> z) \<squnion> (x \<sqinter> \<sim> y \<sqinter> z)"
+    by (simp add: conj_cancel_right conj_zero_right disj_zero_left)
+  thus "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+    by (simp (no_asm_use) add:
+        xor_def de_Morgan_disj de_Morgan_conj double_compl
+        conj_disj_distribs conj_ac disj_ac)
+qed
+
+lemma conj_xor_distrib2:
+  "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)"
+proof -
+  have "x \<sqinter> (y \<oplus> z) = (x \<sqinter> y) \<oplus> (x \<sqinter> z)"
+    by (rule conj_xor_distrib)
+  thus "(y \<oplus> z) \<sqinter> x = (y \<sqinter> x) \<oplus> (z \<sqinter> x)"
+    by (simp add: conj_commute)
+qed
+
+lemmas conj_xor_distribs =
+   conj_xor_distrib conj_xor_distrib2
+
+end
+
+end
diff -r 76f7a8c6e842 -r e3a2b75b1cf9 src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Sun Aug 19 21:21:37 2007 +0200
+++ b/src/HOL/Library/Library.thy	Mon Aug 20 00:22:18 2007 +0200
@@ -6,6 +6,7 @@
   AssocList
   BigO
   Binomial
+  Boolean_Algebra
   Char_ord
   Coinductive_List
   Commutative_Ring
@@ -24,6 +25,7 @@
   NatPair
   Nat_Infinity
   Nested_Environment
+  Numeral_Type
   OptionalSugar
   Parity
   Permutation
diff -r 76f7a8c6e842 -r e3a2b75b1cf9 src/HOL/Library/Numeral_Type.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Numeral_Type.thy	Mon Aug 20 00:22:18 2007 +0200
@@ -0,0 +1,238 @@
+(*
+  ID:     $Id$
+  Author: Brian Huffman
+
+  Numeral Syntax for Types
+*)
+
+header "Numeral Syntax for Types"
+
+theory Numeral_Type
+  imports Infinite_Set
+begin
+
+subsection {* Preliminary lemmas *}
+(* These should be moved elsewhere *)
+
+lemma inj_Inl [simp]: "inj_on Inl A"
+  by (rule inj_onI, simp)
+
+lemma inj_Inr [simp]: "inj_on Inr A"
+  by (rule inj_onI, simp)
+
+lemma inj_Some [simp]: "inj_on Some A"
+  by (rule inj_onI, simp)
+
+lemma card_Plus:
+  "[| finite A; finite B |] ==> card (A <+> B) = card A + card B"
+  unfolding Plus_def
+  apply (subgoal_tac "Inl ` A \<inter> Inr ` B = {}")
+  apply (simp add: card_Un_disjoint card_image)
+  apply fast
+  done
+
+lemma (in type_definition) univ:
+  "UNIV = Abs ` A"
+proof
+  show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
+  show "UNIV \<subseteq> Abs ` A"
+  proof
+    fix x :: 'b
+    have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
+    moreover have "Rep x \<in> A" by (rule Rep)
+    ultimately show "x \<in> Abs ` A" by (rule image_eqI)
+  qed
+qed
+
+lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
+  by (simp add: univ card_image inj_on_def Abs_inject)
+
+
+subsection {* Cardinalities of types *}
+
+syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")
+
+translations "CARD(t)" => "card (UNIV::t set)"
+
+lemma card_unit: "CARD(unit) = 1"
+  unfolding univ_unit by simp
+
+lemma card_bool: "CARD(bool) = 2"
+  unfolding univ_bool by simp
+
+lemma card_prod: "CARD('a::finite \<times> 'b::finite) = CARD('a) * CARD('b)"
+  unfolding univ_prod by (simp only: card_cartesian_product)
+
+lemma card_sum: "CARD('a::finite + 'b::finite) = CARD('a) + CARD('b)"
+  unfolding univ_sum by (simp only: finite card_Plus)
+
+lemma card_option: "CARD('a::finite option) = Suc CARD('a)"
+  unfolding univ_option
+  apply (subgoal_tac "(None::'a option) \<notin> range Some")
+  apply (simp add: finite card_image)
+  apply fast
+  done
+
+lemma card_set: "CARD('a::finite set) = 2 ^ CARD('a)"
+  unfolding univ_set
+  by (simp only: card_Pow finite numeral_2_eq_2)
+
+subsection {* Numeral Types *}
+
+typedef (open) pls = "UNIV :: nat set" ..
+typedef (open) num1 = "UNIV :: unit set" ..
+typedef (open) 'a bit0 = "UNIV :: (bool * 'a) set" ..
+typedef (open) 'a bit1 = "UNIV :: (bool * 'a) option set" ..
+
+instance num1 :: finite
+proof
+  show "finite (UNIV::num1 set)"
+    unfolding type_definition.univ [OF type_definition_num1]
+    using finite by (rule finite_imageI)
+qed
+
+instance bit0 :: (finite) finite
+proof
+  show "finite (UNIV::'a bit0 set)"
+    unfolding type_definition.univ [OF type_definition_bit0]
+    using finite by (rule finite_imageI)
+qed
+
+instance bit1 :: (finite) finite
+proof
+  show "finite (UNIV::'a bit1 set)"
+    unfolding type_definition.univ [OF type_definition_bit1]
+    using finite by (rule finite_imageI)
+qed
+
+lemma card_num1: "CARD(num1) = 1"
+  unfolding type_definition.card [OF type_definition_num1]
+  by (simp only: card_unit)
+
+lemma card_bit0: "CARD('a::finite bit0) = 2 * CARD('a)"
+  unfolding type_definition.card [OF type_definition_bit0]
+  by (simp only: card_prod card_bool)
+
+lemma card_bit1: "CARD('a::finite bit1) = Suc (2 * CARD('a))"
+  unfolding type_definition.card [OF type_definition_bit1]
+  by (simp only: card_prod card_option card_bool)
+
+lemma card_pls: "CARD (pls) = 0"
+  by (simp add: type_definition.card [OF type_definition_pls])
+
+lemmas card_univ_simps [simp] =
+  card_unit
+  card_bool
+  card_prod
+  card_sum
+  card_option
+  card_set
+  card_num1
+  card_bit0
+  card_bit1
+  card_pls
+
+subsection {* Syntax *}
+
+
+syntax
+  "_NumeralType" :: "num_const => type"  ("_")
+  "_NumeralType0" :: type ("0")
+  "_NumeralType1" :: type ("1")
+
+translations
+  "_NumeralType1" == (type) "num1"
+  "_NumeralType0" == (type) "pls"
+
+parse_translation {*
+let
+
+val num1_const = Syntax.const "Numeral_Type.num1";
+val pls_const = Syntax.const "Numeral_Type.pls";
+val B0_const = Syntax.const "Numeral_Type.bit0";
+val B1_const = Syntax.const "Numeral_Type.bit1";
+
+fun mk_bintype n =
+  let
+    fun mk_bit n = if n = 0 then B0_const else B1_const;
+    fun bin_of n =
+      if n = 1 then num1_const
+      else if n = 0 then pls_const
+      else if n = ~1 then raise TERM ("negative type numeral", [])
+      else
+        let val (q, r) = IntInf.divMod (n, 2);
+        in mk_bit r $ bin_of q end;
+  in bin_of n end;
+
+fun numeral_tr (*"_NumeralType"*) [Const (str, _)] =
+      mk_bintype (valOf (IntInf.fromString str))
+  | numeral_tr (*"_NumeralType"*) ts = raise TERM ("numeral_tr", ts);
+
+in [("_NumeralType", numeral_tr)] end;
+*}
+
+print_translation {*
+let
+fun int_of [] = 0
+  | int_of (b :: bs) = IntInf.fromInt b + (2 * int_of bs);
+
+fun bin_of (Const ("pls", _)) = []
+  | bin_of (Const ("num1", _)) = [1]
+  | bin_of (Const ("bit0", _) $ bs) = 0 :: bin_of bs
+  | bin_of (Const ("bit1", _) $ bs) = 1 :: bin_of bs
+  | bin_of t = raise TERM("bin_of", [t]);
+
+fun bit_tr' b [t] =
+  let
+    val rev_digs = b :: bin_of t handle TERM _ => raise Match
+    val i = int_of rev_digs;
+    val num = IntInf.toString (IntInf.abs i);
+  in
+    Syntax.const "_NumeralType" $ Syntax.free num
+  end
+  | bit_tr' b _ = raise Match;
+
+in [("bit0", bit_tr' 0), ("bit1", bit_tr' 1)] end;
+*}
+
+
+subsection {* Classes with at values least 1 and 2  *}
+
+text {* Class finite already captures "at least 1" *}
+
+lemma zero_less_card_finite:
+  "0 < CARD('a::finite)"
+proof (cases "CARD('a::finite) = 0")
+  case False thus ?thesis by (simp del: card_0_eq)
+next
+  case True
+  thus ?thesis by (simp add: finite)
+qed
+
+lemma one_le_card_finite:
+  "Suc 0 <= CARD('a::finite)"
+  by (simp add: less_Suc_eq_le [symmetric] zero_less_card_finite)
+
+
+text {* Class for cardinality "at least 2" *}
+
+class card2 = finite + 
+  assumes two_le_card: "2 <= CARD('a)"
+
+lemma one_less_card: "Suc 0 < CARD('a::card2)"
+  using two_le_card [where 'a='a] by simp
+
+instance bit0 :: (finite) card2
+  by intro_classes (simp add: one_le_card_finite)
+
+instance bit1 :: (finite) card2
+  by intro_classes (simp add: one_le_card_finite)
+
+subsection {* Examples *}
+
+term "TYPE(10)"
+
+lemma "CARD(0) = 0" by simp
+lemma "CARD(17) = 17" by simp
+  
+end