--- a/src/HOL/IsaMakefile Tue Feb 21 08:15:42 2012 +0100
+++ b/src/HOL/IsaMakefile Tue Feb 21 09:17:53 2012 +0100
@@ -1059,9 +1059,9 @@
ex/Lagrange.thy ex/List_to_Set_Comprehension_Examples.thy \
ex/LocaleTest2.thy ex/MT.thy ex/MergeSort.thy ex/Meson_Test.thy \
ex/MonoidGroup.thy ex/Multiquote.thy ex/NatSum.thy \
- ex/Normalization_by_Evaluation.thy ex/Numeral.thy ex/PER.thy \
- ex/PresburgerEx.thy ex/Primrec.thy ex/Quickcheck_Examples.thy \
- ex/Quickcheck_Lattice_Examples.thy \
+ ex/Normalization_by_Evaluation.thy ex/Numeral_Representation.thy \
+ ex/PER.thy ex/PresburgerEx.thy ex/Primrec.thy \
+ ex/Quickcheck_Examples.thy ex/Quickcheck_Lattice_Examples.thy \
ex/Quickcheck_Narrowing_Examples.thy ex/Quicksort.thy ex/ROOT.ML \
ex/Records.thy ex/ReflectionEx.thy ex/Refute_Examples.thy \
ex/SAT_Examples.thy ex/Serbian.thy ex/Set_Theory.thy \
--- a/src/HOL/ex/Numeral.thy Tue Feb 21 08:15:42 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1116 +0,0 @@
-(* Title: HOL/ex/Numeral.thy
- Author: Florian Haftmann
-*)
-
-header {* An experimental alternative numeral representation. *}
-
-theory Numeral
-imports Main
-begin
-
-subsection {* The @{text num} type *}
-
-datatype num = One | Dig0 num | Dig1 num
-
-text {* Increment function for type @{typ num} *}
-
-primrec inc :: "num \<Rightarrow> num" where
- "inc One = Dig0 One"
-| "inc (Dig0 x) = Dig1 x"
-| "inc (Dig1 x) = Dig0 (inc x)"
-
-text {* Converting between type @{typ num} and type @{typ nat} *}
-
-primrec nat_of_num :: "num \<Rightarrow> nat" where
- "nat_of_num One = Suc 0"
-| "nat_of_num (Dig0 x) = nat_of_num x + nat_of_num x"
-| "nat_of_num (Dig1 x) = Suc (nat_of_num x + nat_of_num x)"
-
-primrec num_of_nat :: "nat \<Rightarrow> num" where
- "num_of_nat 0 = One"
-| "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
-
-lemma nat_of_num_pos: "0 < nat_of_num x"
- by (induct x) simp_all
-
-lemma nat_of_num_neq_0: "nat_of_num x \<noteq> 0"
- by (induct x) simp_all
-
-lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
- by (induct x) simp_all
-
-lemma num_of_nat_double:
- "0 < n \<Longrightarrow> num_of_nat (n + n) = Dig0 (num_of_nat n)"
- by (induct n) simp_all
-
-text {*
- Type @{typ num} is isomorphic to the strictly positive
- natural numbers.
-*}
-
-lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
- by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
-
-lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
- by (induct n) (simp_all add: nat_of_num_inc)
-
-lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
-proof
- assume "nat_of_num x = nat_of_num y"
- then have "num_of_nat (nat_of_num x) = num_of_nat (nat_of_num y)" by simp
- then show "x = y" by (simp add: nat_of_num_inverse)
-qed simp
-
-lemma num_induct [case_names One inc]:
- fixes P :: "num \<Rightarrow> bool"
- assumes One: "P One"
- and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
- shows "P x"
-proof -
- obtain n where n: "Suc n = nat_of_num x"
- by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
- have "P (num_of_nat (Suc n))"
- proof (induct n)
- case 0 show ?case using One by simp
- next
- case (Suc n)
- then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
- then show "P (num_of_nat (Suc (Suc n)))" by simp
- qed
- with n show "P x"
- by (simp add: nat_of_num_inverse)
-qed
-
-text {*
- From now on, there are two possible models for @{typ num}: as
- positive naturals (rule @{text "num_induct"}) and as digit
- representation (rules @{text "num.induct"}, @{text "num.cases"}).
-
- It is not entirely clear in which context it is better to use the
- one or the other, or whether the construction should be reversed.
-*}
-
-
-subsection {* Numeral operations *}
-
-ML {*
-structure Dig_Simps = Named_Thms
-(
- val name = @{binding numeral}
- val description = "simplification rules for numerals"
-)
-*}
-
-setup Dig_Simps.setup
-
-instantiation num :: "{plus,times,ord}"
-begin
-
-definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
- "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
-
-definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
- "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
-
-definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
- "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
-
-definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
- "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
-
-instance ..
-
-end
-
-lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
- unfolding plus_num_def
- by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
-
-lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
- unfolding times_num_def
- by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
-
-lemma Dig_plus [numeral, simp, code]:
- "One + One = Dig0 One"
- "One + Dig0 m = Dig1 m"
- "One + Dig1 m = Dig0 (m + One)"
- "Dig0 n + One = Dig1 n"
- "Dig0 n + Dig0 m = Dig0 (n + m)"
- "Dig0 n + Dig1 m = Dig1 (n + m)"
- "Dig1 n + One = Dig0 (n + One)"
- "Dig1 n + Dig0 m = Dig1 (n + m)"
- "Dig1 n + Dig1 m = Dig0 (n + m + One)"
- by (simp_all add: num_eq_iff nat_of_num_add)
-
-lemma Dig_times [numeral, simp, code]:
- "One * One = One"
- "One * Dig0 n = Dig0 n"
- "One * Dig1 n = Dig1 n"
- "Dig0 n * One = Dig0 n"
- "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
- "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
- "Dig1 n * One = Dig1 n"
- "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
- "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
- by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult
- left_distrib right_distrib)
-
-lemma less_eq_num_code [numeral, simp, code]:
- "One \<le> n \<longleftrightarrow> True"
- "Dig0 m \<le> One \<longleftrightarrow> False"
- "Dig1 m \<le> One \<longleftrightarrow> False"
- "Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
- "Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
- "Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
- "Dig1 m \<le> Dig0 n \<longleftrightarrow> m < n"
- using nat_of_num_pos [of n] nat_of_num_pos [of m]
- by (auto simp add: less_eq_num_def less_num_def)
-
-lemma less_num_code [numeral, simp, code]:
- "m < One \<longleftrightarrow> False"
- "One < One \<longleftrightarrow> False"
- "One < Dig0 n \<longleftrightarrow> True"
- "One < Dig1 n \<longleftrightarrow> True"
- "Dig0 m < Dig0 n \<longleftrightarrow> m < n"
- "Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
- "Dig1 m < Dig1 n \<longleftrightarrow> m < n"
- "Dig1 m < Dig0 n \<longleftrightarrow> m < n"
- using nat_of_num_pos [of n] nat_of_num_pos [of m]
- by (auto simp add: less_eq_num_def less_num_def)
-
-text {* Rules using @{text One} and @{text inc} as constructors *}
-
-lemma add_One: "x + One = inc x"
- by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
-
-lemma add_inc: "x + inc y = inc (x + y)"
- by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
-
-lemma mult_One: "x * One = x"
- by (simp add: num_eq_iff nat_of_num_mult)
-
-lemma mult_inc: "x * inc y = x * y + x"
- by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
-
-text {* A double-and-decrement function *}
-
-primrec DigM :: "num \<Rightarrow> num" where
- "DigM One = One"
-| "DigM (Dig0 n) = Dig1 (DigM n)"
-| "DigM (Dig1 n) = Dig1 (Dig0 n)"
-
-lemma DigM_plus_one: "DigM n + One = Dig0 n"
- by (induct n) simp_all
-
-lemma add_One_commute: "One + n = n + One"
- by (induct n) simp_all
-
-lemma one_plus_DigM: "One + DigM n = Dig0 n"
- by (simp add: add_One_commute DigM_plus_one)
-
-text {* Squaring and exponentiation *}
-
-primrec square :: "num \<Rightarrow> num" where
- "square One = One"
-| "square (Dig0 n) = Dig0 (Dig0 (square n))"
-| "square (Dig1 n) = Dig1 (Dig0 (square n + n))"
-
-primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
- "pow x One = x"
-| "pow x (Dig0 y) = square (pow x y)"
-| "pow x (Dig1 y) = x * square (pow x y)"
-
-
-subsection {* Binary numerals *}
-
-text {*
- We embed binary representations into a generic algebraic
- structure using @{text of_num}.
-*}
-
-class semiring_numeral = semiring + monoid_mult
-begin
-
-primrec of_num :: "num \<Rightarrow> 'a" where
- of_num_One [numeral]: "of_num One = 1"
-| "of_num (Dig0 n) = of_num n + of_num n"
-| "of_num (Dig1 n) = of_num n + of_num n + 1"
-
-lemma of_num_inc: "of_num (inc n) = of_num n + 1"
- by (induct n) (simp_all add: add_ac)
-
-lemma of_num_add: "of_num (m + n) = of_num m + of_num n"
- by (induct n rule: num_induct) (simp_all add: add_One add_inc of_num_inc add_ac)
-
-lemma of_num_mult: "of_num (m * n) = of_num m * of_num n"
- by (induct n rule: num_induct) (simp_all add: mult_One mult_inc of_num_add of_num_inc right_distrib)
-
-declare of_num.simps [simp del]
-
-end
-
-ML {*
-fun mk_num k =
- if k > 1 then
- let
- val (l, b) = Integer.div_mod k 2;
- val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
- in bit $ (mk_num l) end
- else if k = 1 then @{term One}
- else error ("mk_num " ^ string_of_int k);
-
-fun dest_num @{term One} = 1
- | dest_num (@{term Dig0} $ n) = 2 * dest_num n
- | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1
- | dest_num t = raise TERM ("dest_num", [t]);
-
-fun mk_numeral phi T k = Morphism.term phi (Const (@{const_name of_num}, @{typ num} --> T))
- $ mk_num k
-
-fun dest_numeral phi (u $ t) =
- if Term.aconv_untyped (u, Morphism.term phi (Const (@{const_name of_num}, dummyT)))
- then (range_type (fastype_of u), dest_num t)
- else raise TERM ("dest_numeral", [u, t]);
-*}
-
-syntax
- "_Numerals" :: "xnum_token \<Rightarrow> 'a" ("_")
-
-parse_translation {*
-let
- fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
- of (0, 1) => Const (@{const_name One}, dummyT)
- | (n, 0) => Const (@{const_name Dig0}, dummyT) $ num_of_int n
- | (n, 1) => Const (@{const_name Dig1}, dummyT) $ num_of_int n
- else raise Match;
- fun numeral_tr [Free (num, _)] =
- let
- val {leading_zeros, value, ...} = Lexicon.read_xnum num;
- val _ = leading_zeros = 0 andalso value > 0
- orelse error ("Bad numeral: " ^ num);
- in Const (@{const_name of_num}, @{typ num} --> dummyT) $ num_of_int value end
- | numeral_tr ts = raise TERM ("numeral_tr", ts);
-in [(@{syntax_const "_Numerals"}, numeral_tr)] end
-*}
-
-typed_print_translation (advanced) {*
-let
- fun dig b n = b + 2 * n;
- fun int_of_num' (Const (@{const_syntax Dig0}, _) $ n) =
- dig 0 (int_of_num' n)
- | int_of_num' (Const (@{const_syntax Dig1}, _) $ n) =
- dig 1 (int_of_num' n)
- | int_of_num' (Const (@{const_syntax One}, _)) = 1;
- fun num_tr' ctxt T [n] =
- let
- val k = int_of_num' n;
- val t' = Syntax.const @{syntax_const "_Numerals"} $ Syntax.free ("#" ^ string_of_int k);
- in
- case T of
- Type (@{type_name fun}, [_, T']) =>
- if not (Config.get ctxt show_types) andalso can Term.dest_Type T' then t'
- else Syntax.const @{syntax_const "_constrain"} $ t' $ Syntax_Phases.term_of_typ ctxt T'
- | T' => if T' = dummyT then t' else raise Match
- end;
-in [(@{const_syntax of_num}, num_tr')] end
-*}
-
-
-subsection {* Class-specific numeral rules *}
-
-subsubsection {* Class @{text semiring_numeral} *}
-
-context semiring_numeral
-begin
-
-abbreviation "Num1 \<equiv> of_num One"
-
-text {*
- Alas, there is still the duplication of @{term 1}, although the
- duplicated @{term 0} has disappeared. We could get rid of it by
- replacing the constructor @{term 1} in @{typ num} by two
- constructors @{text two} and @{text three}, resulting in a further
- blow-up. But it could be worth the effort.
-*}
-
-lemma of_num_plus_one [numeral]:
- "of_num n + 1 = of_num (n + One)"
- by (simp only: of_num_add of_num_One)
-
-lemma of_num_one_plus [numeral]:
- "1 + of_num n = of_num (One + n)"
- by (simp only: of_num_add of_num_One)
-
-lemma of_num_plus [numeral]:
- "of_num m + of_num n = of_num (m + n)"
- by (simp only: of_num_add)
-
-lemma of_num_times_one [numeral]:
- "of_num n * 1 = of_num n"
- by simp
-
-lemma of_num_one_times [numeral]:
- "1 * of_num n = of_num n"
- by simp
-
-lemma of_num_times [numeral]:
- "of_num m * of_num n = of_num (m * n)"
- unfolding of_num_mult ..
-
-end
-
-
-subsubsection {* Structures with a zero: class @{text semiring_1} *}
-
-context semiring_1
-begin
-
-subclass semiring_numeral ..
-
-lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
- by (induct n)
- (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)
-
-declare of_nat_1 [numeral]
-
-lemma Dig_plus_zero [numeral]:
- "0 + 1 = 1"
- "0 + of_num n = of_num n"
- "1 + 0 = 1"
- "of_num n + 0 = of_num n"
- by simp_all
-
-lemma Dig_times_zero [numeral]:
- "0 * 1 = 0"
- "0 * of_num n = 0"
- "1 * 0 = 0"
- "of_num n * 0 = 0"
- by simp_all
-
-end
-
-lemma nat_of_num_of_num: "nat_of_num = of_num"
-proof
- fix n
- have "of_num n = nat_of_num n"
- by (induct n) (simp_all add: of_num.simps)
- then show "nat_of_num n = of_num n" by simp
-qed
-
-
-subsubsection {* Equality: class @{text semiring_char_0} *}
-
-context semiring_char_0
-begin
-
-lemma of_num_eq_iff [numeral]: "of_num m = of_num n \<longleftrightarrow> m = n"
- unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
- of_nat_eq_iff num_eq_iff ..
-
-lemma of_num_eq_one_iff [numeral]: "of_num n = 1 \<longleftrightarrow> n = One"
- using of_num_eq_iff [of n One] by (simp add: of_num_One)
-
-lemma one_eq_of_num_iff [numeral]: "1 = of_num n \<longleftrightarrow> One = n"
- using of_num_eq_iff [of One n] by (simp add: of_num_One)
-
-end
-
-
-subsubsection {* Comparisons: class @{text linordered_semidom} *}
-
-text {*
- Perhaps the underlying structure could even
- be more general than @{text linordered_semidom}.
-*}
-
-context linordered_semidom
-begin
-
-lemma of_num_pos [numeral]: "0 < of_num n"
- by (induct n) (simp_all add: of_num.simps add_pos_pos)
-
-lemma of_num_not_zero [numeral]: "of_num n \<noteq> 0"
- using of_num_pos [of n] by simp
-
-lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
-proof -
- have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
- unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
- then show ?thesis by (simp add: of_nat_of_num)
-qed
-
-lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n \<le> One"
- using of_num_less_eq_iff [of n One] by (simp add: of_num_One)
-
-lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
- using of_num_less_eq_iff [of One n] by (simp add: of_num_One)
-
-lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
-proof -
- have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
- unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
- then show ?thesis by (simp add: of_nat_of_num)
-qed
-
-lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
- using of_num_less_iff [of n One] by (simp add: of_num_One)
-
-lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> One < n"
- using of_num_less_iff [of One n] by (simp add: of_num_One)
-
-lemma of_num_nonneg [numeral]: "0 \<le> of_num n"
- by (induct n) (simp_all add: of_num.simps add_nonneg_nonneg)
-
-lemma of_num_less_zero_iff [numeral]: "\<not> of_num n < 0"
- by (simp add: not_less of_num_nonneg)
-
-lemma of_num_le_zero_iff [numeral]: "\<not> of_num n \<le> 0"
- by (simp add: not_le of_num_pos)
-
-end
-
-context linordered_idom
-begin
-
-lemma minus_of_num_less_of_num_iff: "- of_num m < of_num n"
-proof -
- have "- of_num m < 0" by (simp add: of_num_pos)
- also have "0 < of_num n" by (simp add: of_num_pos)
- finally show ?thesis .
-qed
-
-lemma minus_of_num_not_equal_of_num: "- of_num m \<noteq> of_num n"
- using minus_of_num_less_of_num_iff [of m n] by simp
-
-lemma minus_of_num_less_one_iff: "- of_num n < 1"
- using minus_of_num_less_of_num_iff [of n One] by (simp add: of_num_One)
-
-lemma minus_one_less_of_num_iff: "- 1 < of_num n"
- using minus_of_num_less_of_num_iff [of One n] by (simp add: of_num_One)
-
-lemma minus_one_less_one_iff: "- 1 < 1"
- using minus_of_num_less_of_num_iff [of One One] by (simp add: of_num_One)
-
-lemma minus_of_num_le_of_num_iff: "- of_num m \<le> of_num n"
- by (simp add: less_imp_le minus_of_num_less_of_num_iff)
-
-lemma minus_of_num_le_one_iff: "- of_num n \<le> 1"
- by (simp add: less_imp_le minus_of_num_less_one_iff)
-
-lemma minus_one_le_of_num_iff: "- 1 \<le> of_num n"
- by (simp add: less_imp_le minus_one_less_of_num_iff)
-
-lemma minus_one_le_one_iff: "- 1 \<le> 1"
- by (simp add: less_imp_le minus_one_less_one_iff)
-
-lemma of_num_le_minus_of_num_iff: "\<not> of_num m \<le> - of_num n"
- by (simp add: not_le minus_of_num_less_of_num_iff)
-
-lemma one_le_minus_of_num_iff: "\<not> 1 \<le> - of_num n"
- by (simp add: not_le minus_of_num_less_one_iff)
-
-lemma of_num_le_minus_one_iff: "\<not> of_num n \<le> - 1"
- by (simp add: not_le minus_one_less_of_num_iff)
-
-lemma one_le_minus_one_iff: "\<not> 1 \<le> - 1"
- by (simp add: not_le minus_one_less_one_iff)
-
-lemma of_num_less_minus_of_num_iff: "\<not> of_num m < - of_num n"
- by (simp add: not_less minus_of_num_le_of_num_iff)
-
-lemma one_less_minus_of_num_iff: "\<not> 1 < - of_num n"
- by (simp add: not_less minus_of_num_le_one_iff)
-
-lemma of_num_less_minus_one_iff: "\<not> of_num n < - 1"
- by (simp add: not_less minus_one_le_of_num_iff)
-
-lemma one_less_minus_one_iff: "\<not> 1 < - 1"
- by (simp add: not_less minus_one_le_one_iff)
-
-lemmas le_signed_numeral_special [numeral] =
- minus_of_num_le_of_num_iff
- minus_of_num_le_one_iff
- minus_one_le_of_num_iff
- minus_one_le_one_iff
- of_num_le_minus_of_num_iff
- one_le_minus_of_num_iff
- of_num_le_minus_one_iff
- one_le_minus_one_iff
-
-lemmas less_signed_numeral_special [numeral] =
- minus_of_num_less_of_num_iff
- minus_of_num_not_equal_of_num
- minus_of_num_less_one_iff
- minus_one_less_of_num_iff
- minus_one_less_one_iff
- of_num_less_minus_of_num_iff
- one_less_minus_of_num_iff
- of_num_less_minus_one_iff
- one_less_minus_one_iff
-
-end
-
-subsubsection {* Structures with subtraction: class @{text semiring_1_minus} *}
-
-class semiring_minus = semiring + minus + zero +
- assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
- assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
-begin
-
-lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"
- by (simp add: add_ac minus_inverts_plus1 [of b a])
-
-lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"
- by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])
-
-end
-
-class semiring_1_minus = semiring_1 + semiring_minus
-begin
-
-lemma Dig_of_num_pos:
- assumes "k + n = m"
- shows "of_num m - of_num n = of_num k"
- using assms by (simp add: of_num_plus minus_inverts_plus1)
-
-lemma Dig_of_num_zero:
- shows "of_num n - of_num n = 0"
- by (rule minus_inverts_plus1) simp
-
-lemma Dig_of_num_neg:
- assumes "k + m = n"
- shows "of_num m - of_num n = 0 - of_num k"
- by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)
-
-lemmas Dig_plus_eval =
- of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject
-
-simproc_setup numeral_minus ("of_num m - of_num n") = {*
- let
- (*TODO proper implicit use of morphism via pattern antiquotations*)
- fun cdest_of_num ct = (List.last o snd o Drule.strip_comb) ct;
- fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
- fun attach_num ct = (dest_num (Thm.term_of ct), ct);
- fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
- val simplify = Raw_Simplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
- fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq}
- OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
- [Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
- in fn phi => fn _ => fn ct => case try cdifference ct
- of NONE => (NONE)
- | SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
- then Raw_Simplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
- else mk_meta_eq (let
- val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
- in
- (if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
- else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
- end) end)
- end
-*}
-
-lemma Dig_of_num_minus_zero [numeral]:
- "of_num n - 0 = of_num n"
- by (simp add: minus_inverts_plus1)
-
-lemma Dig_one_minus_zero [numeral]:
- "1 - 0 = 1"
- by (simp add: minus_inverts_plus1)
-
-lemma Dig_one_minus_one [numeral]:
- "1 - 1 = 0"
- by (simp add: minus_inverts_plus1)
-
-lemma Dig_of_num_minus_one [numeral]:
- "of_num (Dig0 n) - 1 = of_num (DigM n)"
- "of_num (Dig1 n) - 1 = of_num (Dig0 n)"
- by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
-
-lemma Dig_one_minus_of_num [numeral]:
- "1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
- "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
- by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
-
-end
-
-
-subsubsection {* Structures with negation: class @{text ring_1} *}
-
-context ring_1
-begin
-
-subclass semiring_1_minus proof
-qed (simp_all add: algebra_simps)
-
-lemma Dig_zero_minus_of_num [numeral]:
- "0 - of_num n = - of_num n"
- by simp
-
-lemma Dig_zero_minus_one [numeral]:
- "0 - 1 = - 1"
- by simp
-
-lemma Dig_uminus_uminus [numeral]:
- "- (- of_num n) = of_num n"
- by simp
-
-lemma Dig_plus_uminus [numeral]:
- "of_num m + - of_num n = of_num m - of_num n"
- "- of_num m + of_num n = of_num n - of_num m"
- "- of_num m + - of_num n = - (of_num m + of_num n)"
- "of_num m - - of_num n = of_num m + of_num n"
- "- of_num m - of_num n = - (of_num m + of_num n)"
- "- of_num m - - of_num n = of_num n - of_num m"
- by (simp_all add: diff_minus add_commute)
-
-lemma Dig_times_uminus [numeral]:
- "- of_num n * of_num m = - (of_num n * of_num m)"
- "of_num n * - of_num m = - (of_num n * of_num m)"
- "- of_num n * - of_num m = of_num n * of_num m"
- by simp_all
-
-lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
-by (induct n)
- (simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)
-
-declare of_int_1 [numeral]
-
-end
-
-
-subsubsection {* Structures with exponentiation *}
-
-lemma of_num_square: "of_num (square x) = of_num x * of_num x"
-by (induct x)
- (simp_all add: of_num.simps of_num_add algebra_simps)
-
-lemma of_num_pow: "of_num (pow x y) = of_num x ^ of_num y"
-by (induct y)
- (simp_all add: of_num.simps of_num_square of_num_mult power_add)
-
-lemma power_of_num [numeral]: "of_num x ^ of_num y = of_num (pow x y)"
- unfolding of_num_pow ..
-
-lemma power_zero_of_num [numeral]:
- "0 ^ of_num n = (0::'a::semiring_1)"
- using of_num_pos [where n=n and ?'a=nat]
- by (simp add: power_0_left)
-
-lemma power_minus_Dig0 [numeral]:
- fixes x :: "'a::ring_1"
- shows "(- x) ^ of_num (Dig0 n) = x ^ of_num (Dig0 n)"
- by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
-
-lemma power_minus_Dig1 [numeral]:
- fixes x :: "'a::ring_1"
- shows "(- x) ^ of_num (Dig1 n) = - (x ^ of_num (Dig1 n))"
- by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
-
-declare power_one [numeral]
-
-
-subsubsection {* Greetings to @{typ nat}. *}
-
-instance nat :: semiring_1_minus proof
-qed simp_all
-
-lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
- unfolding of_num_plus_one [symmetric] by simp
-
-lemma nat_number:
- "1 = Suc 0"
- "of_num One = Suc 0"
- "of_num (Dig0 n) = Suc (of_num (DigM n))"
- "of_num (Dig1 n) = Suc (of_num (Dig0 n))"
- by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)
-
-declare diff_0_eq_0 [numeral]
-
-
-subsection {* Proof tools setup *}
-
-subsubsection {* Numeral equations as default simplification rules *}
-
-declare (in semiring_numeral) of_num_One [simp]
-declare (in semiring_numeral) of_num_plus_one [simp]
-declare (in semiring_numeral) of_num_one_plus [simp]
-declare (in semiring_numeral) of_num_plus [simp]
-declare (in semiring_numeral) of_num_times [simp]
-
-declare (in semiring_1) of_nat_of_num [simp]
-
-declare (in semiring_char_0) of_num_eq_iff [simp]
-declare (in semiring_char_0) of_num_eq_one_iff [simp]
-declare (in semiring_char_0) one_eq_of_num_iff [simp]
-
-declare (in linordered_semidom) of_num_pos [simp]
-declare (in linordered_semidom) of_num_not_zero [simp]
-declare (in linordered_semidom) of_num_less_eq_iff [simp]
-declare (in linordered_semidom) of_num_less_eq_one_iff [simp]
-declare (in linordered_semidom) one_less_eq_of_num_iff [simp]
-declare (in linordered_semidom) of_num_less_iff [simp]
-declare (in linordered_semidom) of_num_less_one_iff [simp]
-declare (in linordered_semidom) one_less_of_num_iff [simp]
-declare (in linordered_semidom) of_num_nonneg [simp]
-declare (in linordered_semidom) of_num_less_zero_iff [simp]
-declare (in linordered_semidom) of_num_le_zero_iff [simp]
-
-declare (in linordered_idom) le_signed_numeral_special [simp]
-declare (in linordered_idom) less_signed_numeral_special [simp]
-
-declare (in semiring_1_minus) Dig_of_num_minus_one [simp]
-declare (in semiring_1_minus) Dig_one_minus_of_num [simp]
-
-declare (in ring_1) Dig_plus_uminus [simp]
-declare (in ring_1) of_int_of_num [simp]
-
-declare power_of_num [simp]
-declare power_zero_of_num [simp]
-declare power_minus_Dig0 [simp]
-declare power_minus_Dig1 [simp]
-
-declare Suc_of_num [simp]
-
-
-subsubsection {* Reorientation of equalities *}
-
-setup {*
- Reorient_Proc.add
- (fn Const(@{const_name of_num}, _) $ _ => true
- | Const(@{const_name uminus}, _) $
- (Const(@{const_name of_num}, _) $ _) => true
- | _ => false)
-*}
-
-simproc_setup reorient_num ("of_num n = x" | "- of_num m = y") = Reorient_Proc.proc
-
-
-subsubsection {* Constant folding for multiplication in semirings *}
-
-context semiring_numeral
-begin
-
-lemma mult_of_num_commute: "x * of_num n = of_num n * x"
-by (induct n)
- (simp_all only: of_num.simps left_distrib right_distrib mult_1_left mult_1_right)
-
-definition
- "commutes_with a b \<longleftrightarrow> a * b = b * a"
-
-lemma commutes_with_commute: "commutes_with a b \<Longrightarrow> a * b = b * a"
-unfolding commutes_with_def .
-
-lemma commutes_with_left_commute: "commutes_with a b \<Longrightarrow> a * (b * c) = b * (a * c)"
-unfolding commutes_with_def by (simp only: mult_assoc [symmetric])
-
-lemma commutes_with_numeral: "commutes_with x (of_num n)" "commutes_with (of_num n) x"
-unfolding commutes_with_def by (simp_all add: mult_of_num_commute)
-
-lemmas mult_ac_numeral =
- mult_assoc
- commutes_with_commute
- commutes_with_left_commute
- commutes_with_numeral
-
-end
-
-ML {*
-structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
-struct
- val assoc_ss = HOL_ss addsimps @{thms mult_ac_numeral}
- val eq_reflection = eq_reflection
- fun is_numeral (Const(@{const_name of_num}, _) $ _) = true
- | is_numeral _ = false;
-end;
-
-structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
-*}
-
-simproc_setup semiring_assoc_fold' ("(a::'a::semiring_numeral) * b") =
- {* fn phi => fn ss => fn ct =>
- Semiring_Times_Assoc.proc ss (Thm.term_of ct) *}
-
-
-subsection {* Code generator setup for @{typ int} *}
-
-text {* Reversing standard setup *}
-
-lemma [code_unfold del]: "(0::int) \<equiv> Numeral0" by simp
-lemma [code_unfold del]: "(1::int) \<equiv> Numeral1" by simp
-declare zero_is_num_zero [code_unfold del]
-declare one_is_num_one [code_unfold del]
-
-lemma [code, code del]:
- "(1 :: int) = 1"
- "(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
- "(uminus :: int \<Rightarrow> int) = uminus"
- "(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
- "(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
- "(HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool) = HOL.equal"
- "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
- "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
- by rule+
-
-text {* Constructors *}
-
-definition Pls :: "num \<Rightarrow> int" where
- [simp, code_post]: "Pls n = of_num n"
-
-definition Mns :: "num \<Rightarrow> int" where
- [simp, code_post]: "Mns n = - of_num n"
-
-code_datatype "0::int" Pls Mns
-
-lemmas [code_unfold] = Pls_def [symmetric] Mns_def [symmetric]
-
-text {* Auxiliary operations *}
-
-definition dup :: "int \<Rightarrow> int" where
- [simp]: "dup k = k + k"
-
-lemma Dig_dup [code]:
- "dup 0 = 0"
- "dup (Pls n) = Pls (Dig0 n)"
- "dup (Mns n) = Mns (Dig0 n)"
- by (simp_all add: of_num.simps)
-
-definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
- [simp]: "sub m n = (of_num m - of_num n)"
-
-lemma Dig_sub [code]:
- "sub One One = 0"
- "sub (Dig0 m) One = of_num (DigM m)"
- "sub (Dig1 m) One = of_num (Dig0 m)"
- "sub One (Dig0 n) = - of_num (DigM n)"
- "sub One (Dig1 n) = - of_num (Dig0 n)"
- "sub (Dig0 m) (Dig0 n) = dup (sub m n)"
- "sub (Dig1 m) (Dig1 n) = dup (sub m n)"
- "sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
- "sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
- by (simp_all add: algebra_simps num_eq_iff nat_of_num_add)
-
-text {* Implementations *}
-
-lemma one_int_code [code]:
- "1 = Pls One"
- by simp
-
-lemma plus_int_code [code]:
- "k + 0 = (k::int)"
- "0 + l = (l::int)"
- "Pls m + Pls n = Pls (m + n)"
- "Pls m + Mns n = sub m n"
- "Mns m + Pls n = sub n m"
- "Mns m + Mns n = Mns (m + n)"
- by simp_all
-
-lemma uminus_int_code [code]:
- "uminus 0 = (0::int)"
- "uminus (Pls m) = Mns m"
- "uminus (Mns m) = Pls m"
- by simp_all
-
-lemma minus_int_code [code]:
- "k - 0 = (k::int)"
- "0 - l = uminus (l::int)"
- "Pls m - Pls n = sub m n"
- "Pls m - Mns n = Pls (m + n)"
- "Mns m - Pls n = Mns (m + n)"
- "Mns m - Mns n = sub n m"
- by simp_all
-
-lemma times_int_code [code]:
- "k * 0 = (0::int)"
- "0 * l = (0::int)"
- "Pls m * Pls n = Pls (m * n)"
- "Pls m * Mns n = Mns (m * n)"
- "Mns m * Pls n = Mns (m * n)"
- "Mns m * Mns n = Pls (m * n)"
- by simp_all
-
-lemma eq_int_code [code]:
- "HOL.equal 0 (0::int) \<longleftrightarrow> True"
- "HOL.equal 0 (Pls l) \<longleftrightarrow> False"
- "HOL.equal 0 (Mns l) \<longleftrightarrow> False"
- "HOL.equal (Pls k) 0 \<longleftrightarrow> False"
- "HOL.equal (Pls k) (Pls l) \<longleftrightarrow> HOL.equal k l"
- "HOL.equal (Pls k) (Mns l) \<longleftrightarrow> False"
- "HOL.equal (Mns k) 0 \<longleftrightarrow> False"
- "HOL.equal (Mns k) (Pls l) \<longleftrightarrow> False"
- "HOL.equal (Mns k) (Mns l) \<longleftrightarrow> HOL.equal k l"
- by (auto simp add: equal dest: sym)
-
-lemma [code nbe]:
- "HOL.equal (k::int) k \<longleftrightarrow> True"
- by (fact equal_refl)
-
-lemma less_eq_int_code [code]:
- "0 \<le> (0::int) \<longleftrightarrow> True"
- "0 \<le> Pls l \<longleftrightarrow> True"
- "0 \<le> Mns l \<longleftrightarrow> False"
- "Pls k \<le> 0 \<longleftrightarrow> False"
- "Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
- "Pls k \<le> Mns l \<longleftrightarrow> False"
- "Mns k \<le> 0 \<longleftrightarrow> True"
- "Mns k \<le> Pls l \<longleftrightarrow> True"
- "Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
- by simp_all
-
-lemma less_int_code [code]:
- "0 < (0::int) \<longleftrightarrow> False"
- "0 < Pls l \<longleftrightarrow> True"
- "0 < Mns l \<longleftrightarrow> False"
- "Pls k < 0 \<longleftrightarrow> False"
- "Pls k < Pls l \<longleftrightarrow> k < l"
- "Pls k < Mns l \<longleftrightarrow> False"
- "Mns k < 0 \<longleftrightarrow> True"
- "Mns k < Pls l \<longleftrightarrow> True"
- "Mns k < Mns l \<longleftrightarrow> l < k"
- by simp_all
-
-hide_const (open) sub dup
-
-text {* Pretty literals *}
-
-ML {*
-local open Code_Thingol in
-
-fun add_code print target =
- let
- fun dest_num one' dig0' dig1' thm =
- let
- fun dest_dig (IConst (c, _)) = if c = dig0' then 0
- else if c = dig1' then 1
- else Code_Printer.eqn_error thm "Illegal numeral expression: illegal dig"
- | dest_dig _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal digit";
- fun dest_num (IConst (c, _)) = if c = one' then 1
- else Code_Printer.eqn_error thm "Illegal numeral expression: illegal leading digit"
- | dest_num (t1 `$ t2) = 2 * dest_num t2 + dest_dig t1
- | dest_num _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal term";
- in dest_num end;
- fun pretty sgn literals [one', dig0', dig1'] _ thm _ _ [(t, _)] =
- (Code_Printer.str o print literals o sgn o dest_num one' dig0' dig1' thm) t
- fun add_syntax (c, sgn) = Code_Target.add_const_syntax target c
- (SOME (Code_Printer.complex_const_syntax
- (1, ([@{const_name One}, @{const_name Dig0}, @{const_name Dig1}],
- pretty sgn))));
- in
- add_syntax (@{const_name Pls}, I)
- #> add_syntax (@{const_name Mns}, (fn k => ~ k))
- end;
-
-end
-*}
-
-hide_const (open) One Dig0 Dig1
-
-
-subsection {* Toy examples *}
-
-definition "foo \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat)"
-definition "bar \<longleftrightarrow> #4 * #2 + #7 \<ge> (#8 :: int) - #3"
-
-code_thms foo bar
-export_code foo bar checking SML OCaml? Haskell? Scala?
-
-text {* This is an ad-hoc @{text Code_Integer} setup. *}
-
-setup {*
- fold (add_code Code_Printer.literal_numeral)
- [Code_ML.target_SML, Code_ML.target_OCaml, Code_Haskell.target, Code_Scala.target]
-*}
-
-code_type int
- (SML "IntInf.int")
- (OCaml "Big'_int.big'_int")
- (Haskell "Integer")
- (Scala "BigInt")
- (Eval "int")
-
-code_const "0::int"
- (SML "0/ :/ IntInf.int")
- (OCaml "Big'_int.zero")
- (Haskell "0")
- (Scala "BigInt(0)")
- (Eval "0/ :/ int")
-
-code_const Int.pred
- (SML "IntInf.- ((_), 1)")
- (OCaml "Big'_int.pred'_big'_int")
- (Haskell "!(_/ -/ 1)")
- (Scala "!(_ -/ 1)")
- (Eval "!(_/ -/ 1)")
-
-code_const Int.succ
- (SML "IntInf.+ ((_), 1)")
- (OCaml "Big'_int.succ'_big'_int")
- (Haskell "!(_/ +/ 1)")
- (Scala "!(_ +/ 1)")
- (Eval "!(_/ +/ 1)")
-
-code_const "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"
- (SML "IntInf.+ ((_), (_))")
- (OCaml "Big'_int.add'_big'_int")
- (Haskell infixl 6 "+")
- (Scala infixl 7 "+")
- (Eval infixl 8 "+")
-
-code_const "uminus \<Colon> int \<Rightarrow> int"
- (SML "IntInf.~")
- (OCaml "Big'_int.minus'_big'_int")
- (Haskell "negate")
- (Scala "!(- _)")
- (Eval "~/ _")
-
-code_const "op - \<Colon> int \<Rightarrow> int \<Rightarrow> int"
- (SML "IntInf.- ((_), (_))")
- (OCaml "Big'_int.sub'_big'_int")
- (Haskell infixl 6 "-")
- (Scala infixl 7 "-")
- (Eval infixl 8 "-")
-
-code_const "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"
- (SML "IntInf.* ((_), (_))")
- (OCaml "Big'_int.mult'_big'_int")
- (Haskell infixl 7 "*")
- (Scala infixl 8 "*")
- (Eval infixl 9 "*")
-
-code_const pdivmod
- (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
- (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
- (Haskell "divMod/ (abs _)/ (abs _)")
- (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
- (Eval "Integer.div'_mod/ (abs _)/ (abs _)")
-
-code_const "HOL.equal \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
- (SML "!((_ : IntInf.int) = _)")
- (OCaml "Big'_int.eq'_big'_int")
- (Haskell infix 4 "==")
- (Scala infixl 5 "==")
- (Eval infixl 6 "=")
-
-code_const "op \<le> \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
- (SML "IntInf.<= ((_), (_))")
- (OCaml "Big'_int.le'_big'_int")
- (Haskell infix 4 "<=")
- (Scala infixl 4 "<=")
- (Eval infixl 6 "<=")
-
-code_const "op < \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
- (SML "IntInf.< ((_), (_))")
- (OCaml "Big'_int.lt'_big'_int")
- (Haskell infix 4 "<")
- (Scala infixl 4 "<")
- (Eval infixl 6 "<")
-
-code_const Code_Numeral.int_of
- (SML "IntInf.fromInt")
- (OCaml "_")
- (Haskell "toInteger")
- (Scala "!_.as'_BigInt")
- (Eval "_")
-
-export_code foo bar checking SML OCaml? Haskell? Scala?
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Numeral_Representation.thy Tue Feb 21 09:17:53 2012 +0100
@@ -0,0 +1,1116 @@
+(* Title: HOL/ex/Numeral_Representation.thy
+ Author: Florian Haftmann
+*)
+
+header {* An experimental alternative numeral representation. *}
+
+theory Numeral_Representation
+imports Main
+begin
+
+subsection {* The @{text num} type *}
+
+datatype num = One | Dig0 num | Dig1 num
+
+text {* Increment function for type @{typ num} *}
+
+primrec inc :: "num \<Rightarrow> num" where
+ "inc One = Dig0 One"
+| "inc (Dig0 x) = Dig1 x"
+| "inc (Dig1 x) = Dig0 (inc x)"
+
+text {* Converting between type @{typ num} and type @{typ nat} *}
+
+primrec nat_of_num :: "num \<Rightarrow> nat" where
+ "nat_of_num One = Suc 0"
+| "nat_of_num (Dig0 x) = nat_of_num x + nat_of_num x"
+| "nat_of_num (Dig1 x) = Suc (nat_of_num x + nat_of_num x)"
+
+primrec num_of_nat :: "nat \<Rightarrow> num" where
+ "num_of_nat 0 = One"
+| "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
+
+lemma nat_of_num_pos: "0 < nat_of_num x"
+ by (induct x) simp_all
+
+lemma nat_of_num_neq_0: "nat_of_num x \<noteq> 0"
+ by (induct x) simp_all
+
+lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
+ by (induct x) simp_all
+
+lemma num_of_nat_double:
+ "0 < n \<Longrightarrow> num_of_nat (n + n) = Dig0 (num_of_nat n)"
+ by (induct n) simp_all
+
+text {*
+ Type @{typ num} is isomorphic to the strictly positive
+ natural numbers.
+*}
+
+lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
+ by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
+
+lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
+ by (induct n) (simp_all add: nat_of_num_inc)
+
+lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
+proof
+ assume "nat_of_num x = nat_of_num y"
+ then have "num_of_nat (nat_of_num x) = num_of_nat (nat_of_num y)" by simp
+ then show "x = y" by (simp add: nat_of_num_inverse)
+qed simp
+
+lemma num_induct [case_names One inc]:
+ fixes P :: "num \<Rightarrow> bool"
+ assumes One: "P One"
+ and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
+ shows "P x"
+proof -
+ obtain n where n: "Suc n = nat_of_num x"
+ by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
+ have "P (num_of_nat (Suc n))"
+ proof (induct n)
+ case 0 show ?case using One by simp
+ next
+ case (Suc n)
+ then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
+ then show "P (num_of_nat (Suc (Suc n)))" by simp
+ qed
+ with n show "P x"
+ by (simp add: nat_of_num_inverse)
+qed
+
+text {*
+ From now on, there are two possible models for @{typ num}: as
+ positive naturals (rule @{text "num_induct"}) and as digit
+ representation (rules @{text "num.induct"}, @{text "num.cases"}).
+
+ It is not entirely clear in which context it is better to use the
+ one or the other, or whether the construction should be reversed.
+*}
+
+
+subsection {* Numeral operations *}
+
+ML {*
+structure Dig_Simps = Named_Thms
+(
+ val name = @{binding numeral}
+ val description = "simplification rules for numerals"
+)
+*}
+
+setup Dig_Simps.setup
+
+instantiation num :: "{plus,times,ord}"
+begin
+
+definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
+ "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
+
+definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
+ "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
+
+definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
+ "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
+
+definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
+ "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
+
+instance ..
+
+end
+
+lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
+ unfolding plus_num_def
+ by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
+
+lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
+ unfolding times_num_def
+ by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
+
+lemma Dig_plus [numeral, simp, code]:
+ "One + One = Dig0 One"
+ "One + Dig0 m = Dig1 m"
+ "One + Dig1 m = Dig0 (m + One)"
+ "Dig0 n + One = Dig1 n"
+ "Dig0 n + Dig0 m = Dig0 (n + m)"
+ "Dig0 n + Dig1 m = Dig1 (n + m)"
+ "Dig1 n + One = Dig0 (n + One)"
+ "Dig1 n + Dig0 m = Dig1 (n + m)"
+ "Dig1 n + Dig1 m = Dig0 (n + m + One)"
+ by (simp_all add: num_eq_iff nat_of_num_add)
+
+lemma Dig_times [numeral, simp, code]:
+ "One * One = One"
+ "One * Dig0 n = Dig0 n"
+ "One * Dig1 n = Dig1 n"
+ "Dig0 n * One = Dig0 n"
+ "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
+ "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
+ "Dig1 n * One = Dig1 n"
+ "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
+ "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
+ by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult
+ left_distrib right_distrib)
+
+lemma less_eq_num_code [numeral, simp, code]:
+ "One \<le> n \<longleftrightarrow> True"
+ "Dig0 m \<le> One \<longleftrightarrow> False"
+ "Dig1 m \<le> One \<longleftrightarrow> False"
+ "Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
+ "Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
+ "Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
+ "Dig1 m \<le> Dig0 n \<longleftrightarrow> m < n"
+ using nat_of_num_pos [of n] nat_of_num_pos [of m]
+ by (auto simp add: less_eq_num_def less_num_def)
+
+lemma less_num_code [numeral, simp, code]:
+ "m < One \<longleftrightarrow> False"
+ "One < One \<longleftrightarrow> False"
+ "One < Dig0 n \<longleftrightarrow> True"
+ "One < Dig1 n \<longleftrightarrow> True"
+ "Dig0 m < Dig0 n \<longleftrightarrow> m < n"
+ "Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
+ "Dig1 m < Dig1 n \<longleftrightarrow> m < n"
+ "Dig1 m < Dig0 n \<longleftrightarrow> m < n"
+ using nat_of_num_pos [of n] nat_of_num_pos [of m]
+ by (auto simp add: less_eq_num_def less_num_def)
+
+text {* Rules using @{text One} and @{text inc} as constructors *}
+
+lemma add_One: "x + One = inc x"
+ by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
+
+lemma add_inc: "x + inc y = inc (x + y)"
+ by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
+
+lemma mult_One: "x * One = x"
+ by (simp add: num_eq_iff nat_of_num_mult)
+
+lemma mult_inc: "x * inc y = x * y + x"
+ by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
+
+text {* A double-and-decrement function *}
+
+primrec DigM :: "num \<Rightarrow> num" where
+ "DigM One = One"
+| "DigM (Dig0 n) = Dig1 (DigM n)"
+| "DigM (Dig1 n) = Dig1 (Dig0 n)"
+
+lemma DigM_plus_one: "DigM n + One = Dig0 n"
+ by (induct n) simp_all
+
+lemma add_One_commute: "One + n = n + One"
+ by (induct n) simp_all
+
+lemma one_plus_DigM: "One + DigM n = Dig0 n"
+ by (simp add: add_One_commute DigM_plus_one)
+
+text {* Squaring and exponentiation *}
+
+primrec square :: "num \<Rightarrow> num" where
+ "square One = One"
+| "square (Dig0 n) = Dig0 (Dig0 (square n))"
+| "square (Dig1 n) = Dig1 (Dig0 (square n + n))"
+
+primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
+ "pow x One = x"
+| "pow x (Dig0 y) = square (pow x y)"
+| "pow x (Dig1 y) = x * square (pow x y)"
+
+
+subsection {* Binary numerals *}
+
+text {*
+ We embed binary representations into a generic algebraic
+ structure using @{text of_num}.
+*}
+
+class semiring_numeral = semiring + monoid_mult
+begin
+
+primrec of_num :: "num \<Rightarrow> 'a" where
+ of_num_One [numeral]: "of_num One = 1"
+| "of_num (Dig0 n) = of_num n + of_num n"
+| "of_num (Dig1 n) = of_num n + of_num n + 1"
+
+lemma of_num_inc: "of_num (inc n) = of_num n + 1"
+ by (induct n) (simp_all add: add_ac)
+
+lemma of_num_add: "of_num (m + n) = of_num m + of_num n"
+ by (induct n rule: num_induct) (simp_all add: add_One add_inc of_num_inc add_ac)
+
+lemma of_num_mult: "of_num (m * n) = of_num m * of_num n"
+ by (induct n rule: num_induct) (simp_all add: mult_One mult_inc of_num_add of_num_inc right_distrib)
+
+declare of_num.simps [simp del]
+
+end
+
+ML {*
+fun mk_num k =
+ if k > 1 then
+ let
+ val (l, b) = Integer.div_mod k 2;
+ val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
+ in bit $ (mk_num l) end
+ else if k = 1 then @{term One}
+ else error ("mk_num " ^ string_of_int k);
+
+fun dest_num @{term One} = 1
+ | dest_num (@{term Dig0} $ n) = 2 * dest_num n
+ | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1
+ | dest_num t = raise TERM ("dest_num", [t]);
+
+fun mk_numeral phi T k = Morphism.term phi (Const (@{const_name of_num}, @{typ num} --> T))
+ $ mk_num k
+
+fun dest_numeral phi (u $ t) =
+ if Term.aconv_untyped (u, Morphism.term phi (Const (@{const_name of_num}, dummyT)))
+ then (range_type (fastype_of u), dest_num t)
+ else raise TERM ("dest_numeral", [u, t]);
+*}
+
+syntax
+ "_Numerals" :: "xnum_token \<Rightarrow> 'a" ("_")
+
+parse_translation {*
+let
+ fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
+ of (0, 1) => Const (@{const_name One}, dummyT)
+ | (n, 0) => Const (@{const_name Dig0}, dummyT) $ num_of_int n
+ | (n, 1) => Const (@{const_name Dig1}, dummyT) $ num_of_int n
+ else raise Match;
+ fun numeral_tr [Free (num, _)] =
+ let
+ val {leading_zeros, value, ...} = Lexicon.read_xnum num;
+ val _ = leading_zeros = 0 andalso value > 0
+ orelse error ("Bad numeral: " ^ num);
+ in Const (@{const_name of_num}, @{typ num} --> dummyT) $ num_of_int value end
+ | numeral_tr ts = raise TERM ("numeral_tr", ts);
+in [(@{syntax_const "_Numerals"}, numeral_tr)] end
+*}
+
+typed_print_translation (advanced) {*
+let
+ fun dig b n = b + 2 * n;
+ fun int_of_num' (Const (@{const_syntax Dig0}, _) $ n) =
+ dig 0 (int_of_num' n)
+ | int_of_num' (Const (@{const_syntax Dig1}, _) $ n) =
+ dig 1 (int_of_num' n)
+ | int_of_num' (Const (@{const_syntax One}, _)) = 1;
+ fun num_tr' ctxt T [n] =
+ let
+ val k = int_of_num' n;
+ val t' = Syntax.const @{syntax_const "_Numerals"} $ Syntax.free ("#" ^ string_of_int k);
+ in
+ case T of
+ Type (@{type_name fun}, [_, T']) =>
+ if not (Config.get ctxt show_types) andalso can Term.dest_Type T' then t'
+ else Syntax.const @{syntax_const "_constrain"} $ t' $ Syntax_Phases.term_of_typ ctxt T'
+ | T' => if T' = dummyT then t' else raise Match
+ end;
+in [(@{const_syntax of_num}, num_tr')] end
+*}
+
+
+subsection {* Class-specific numeral rules *}
+
+subsubsection {* Class @{text semiring_numeral} *}
+
+context semiring_numeral
+begin
+
+abbreviation "Num1 \<equiv> of_num One"
+
+text {*
+ Alas, there is still the duplication of @{term 1}, although the
+ duplicated @{term 0} has disappeared. We could get rid of it by
+ replacing the constructor @{term 1} in @{typ num} by two
+ constructors @{text two} and @{text three}, resulting in a further
+ blow-up. But it could be worth the effort.
+*}
+
+lemma of_num_plus_one [numeral]:
+ "of_num n + 1 = of_num (n + One)"
+ by (simp only: of_num_add of_num_One)
+
+lemma of_num_one_plus [numeral]:
+ "1 + of_num n = of_num (One + n)"
+ by (simp only: of_num_add of_num_One)
+
+lemma of_num_plus [numeral]:
+ "of_num m + of_num n = of_num (m + n)"
+ by (simp only: of_num_add)
+
+lemma of_num_times_one [numeral]:
+ "of_num n * 1 = of_num n"
+ by simp
+
+lemma of_num_one_times [numeral]:
+ "1 * of_num n = of_num n"
+ by simp
+
+lemma of_num_times [numeral]:
+ "of_num m * of_num n = of_num (m * n)"
+ unfolding of_num_mult ..
+
+end
+
+
+subsubsection {* Structures with a zero: class @{text semiring_1} *}
+
+context semiring_1
+begin
+
+subclass semiring_numeral ..
+
+lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
+ by (induct n)
+ (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)
+
+declare of_nat_1 [numeral]
+
+lemma Dig_plus_zero [numeral]:
+ "0 + 1 = 1"
+ "0 + of_num n = of_num n"
+ "1 + 0 = 1"
+ "of_num n + 0 = of_num n"
+ by simp_all
+
+lemma Dig_times_zero [numeral]:
+ "0 * 1 = 0"
+ "0 * of_num n = 0"
+ "1 * 0 = 0"
+ "of_num n * 0 = 0"
+ by simp_all
+
+end
+
+lemma nat_of_num_of_num: "nat_of_num = of_num"
+proof
+ fix n
+ have "of_num n = nat_of_num n"
+ by (induct n) (simp_all add: of_num.simps)
+ then show "nat_of_num n = of_num n" by simp
+qed
+
+
+subsubsection {* Equality: class @{text semiring_char_0} *}
+
+context semiring_char_0
+begin
+
+lemma of_num_eq_iff [numeral]: "of_num m = of_num n \<longleftrightarrow> m = n"
+ unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
+ of_nat_eq_iff num_eq_iff ..
+
+lemma of_num_eq_one_iff [numeral]: "of_num n = 1 \<longleftrightarrow> n = One"
+ using of_num_eq_iff [of n One] by (simp add: of_num_One)
+
+lemma one_eq_of_num_iff [numeral]: "1 = of_num n \<longleftrightarrow> One = n"
+ using of_num_eq_iff [of One n] by (simp add: of_num_One)
+
+end
+
+
+subsubsection {* Comparisons: class @{text linordered_semidom} *}
+
+text {*
+ Perhaps the underlying structure could even
+ be more general than @{text linordered_semidom}.
+*}
+
+context linordered_semidom
+begin
+
+lemma of_num_pos [numeral]: "0 < of_num n"
+ by (induct n) (simp_all add: of_num.simps add_pos_pos)
+
+lemma of_num_not_zero [numeral]: "of_num n \<noteq> 0"
+ using of_num_pos [of n] by simp
+
+lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
+proof -
+ have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
+ unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
+ then show ?thesis by (simp add: of_nat_of_num)
+qed
+
+lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n \<le> One"
+ using of_num_less_eq_iff [of n One] by (simp add: of_num_One)
+
+lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
+ using of_num_less_eq_iff [of One n] by (simp add: of_num_One)
+
+lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
+proof -
+ have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
+ unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
+ then show ?thesis by (simp add: of_nat_of_num)
+qed
+
+lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
+ using of_num_less_iff [of n One] by (simp add: of_num_One)
+
+lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> One < n"
+ using of_num_less_iff [of One n] by (simp add: of_num_One)
+
+lemma of_num_nonneg [numeral]: "0 \<le> of_num n"
+ by (induct n) (simp_all add: of_num.simps add_nonneg_nonneg)
+
+lemma of_num_less_zero_iff [numeral]: "\<not> of_num n < 0"
+ by (simp add: not_less of_num_nonneg)
+
+lemma of_num_le_zero_iff [numeral]: "\<not> of_num n \<le> 0"
+ by (simp add: not_le of_num_pos)
+
+end
+
+context linordered_idom
+begin
+
+lemma minus_of_num_less_of_num_iff: "- of_num m < of_num n"
+proof -
+ have "- of_num m < 0" by (simp add: of_num_pos)
+ also have "0 < of_num n" by (simp add: of_num_pos)
+ finally show ?thesis .
+qed
+
+lemma minus_of_num_not_equal_of_num: "- of_num m \<noteq> of_num n"
+ using minus_of_num_less_of_num_iff [of m n] by simp
+
+lemma minus_of_num_less_one_iff: "- of_num n < 1"
+ using minus_of_num_less_of_num_iff [of n One] by (simp add: of_num_One)
+
+lemma minus_one_less_of_num_iff: "- 1 < of_num n"
+ using minus_of_num_less_of_num_iff [of One n] by (simp add: of_num_One)
+
+lemma minus_one_less_one_iff: "- 1 < 1"
+ using minus_of_num_less_of_num_iff [of One One] by (simp add: of_num_One)
+
+lemma minus_of_num_le_of_num_iff: "- of_num m \<le> of_num n"
+ by (simp add: less_imp_le minus_of_num_less_of_num_iff)
+
+lemma minus_of_num_le_one_iff: "- of_num n \<le> 1"
+ by (simp add: less_imp_le minus_of_num_less_one_iff)
+
+lemma minus_one_le_of_num_iff: "- 1 \<le> of_num n"
+ by (simp add: less_imp_le minus_one_less_of_num_iff)
+
+lemma minus_one_le_one_iff: "- 1 \<le> 1"
+ by (simp add: less_imp_le minus_one_less_one_iff)
+
+lemma of_num_le_minus_of_num_iff: "\<not> of_num m \<le> - of_num n"
+ by (simp add: not_le minus_of_num_less_of_num_iff)
+
+lemma one_le_minus_of_num_iff: "\<not> 1 \<le> - of_num n"
+ by (simp add: not_le minus_of_num_less_one_iff)
+
+lemma of_num_le_minus_one_iff: "\<not> of_num n \<le> - 1"
+ by (simp add: not_le minus_one_less_of_num_iff)
+
+lemma one_le_minus_one_iff: "\<not> 1 \<le> - 1"
+ by (simp add: not_le minus_one_less_one_iff)
+
+lemma of_num_less_minus_of_num_iff: "\<not> of_num m < - of_num n"
+ by (simp add: not_less minus_of_num_le_of_num_iff)
+
+lemma one_less_minus_of_num_iff: "\<not> 1 < - of_num n"
+ by (simp add: not_less minus_of_num_le_one_iff)
+
+lemma of_num_less_minus_one_iff: "\<not> of_num n < - 1"
+ by (simp add: not_less minus_one_le_of_num_iff)
+
+lemma one_less_minus_one_iff: "\<not> 1 < - 1"
+ by (simp add: not_less minus_one_le_one_iff)
+
+lemmas le_signed_numeral_special [numeral] =
+ minus_of_num_le_of_num_iff
+ minus_of_num_le_one_iff
+ minus_one_le_of_num_iff
+ minus_one_le_one_iff
+ of_num_le_minus_of_num_iff
+ one_le_minus_of_num_iff
+ of_num_le_minus_one_iff
+ one_le_minus_one_iff
+
+lemmas less_signed_numeral_special [numeral] =
+ minus_of_num_less_of_num_iff
+ minus_of_num_not_equal_of_num
+ minus_of_num_less_one_iff
+ minus_one_less_of_num_iff
+ minus_one_less_one_iff
+ of_num_less_minus_of_num_iff
+ one_less_minus_of_num_iff
+ of_num_less_minus_one_iff
+ one_less_minus_one_iff
+
+end
+
+subsubsection {* Structures with subtraction: class @{text semiring_1_minus} *}
+
+class semiring_minus = semiring + minus + zero +
+ assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
+ assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
+begin
+
+lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"
+ by (simp add: add_ac minus_inverts_plus1 [of b a])
+
+lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"
+ by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])
+
+end
+
+class semiring_1_minus = semiring_1 + semiring_minus
+begin
+
+lemma Dig_of_num_pos:
+ assumes "k + n = m"
+ shows "of_num m - of_num n = of_num k"
+ using assms by (simp add: of_num_plus minus_inverts_plus1)
+
+lemma Dig_of_num_zero:
+ shows "of_num n - of_num n = 0"
+ by (rule minus_inverts_plus1) simp
+
+lemma Dig_of_num_neg:
+ assumes "k + m = n"
+ shows "of_num m - of_num n = 0 - of_num k"
+ by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)
+
+lemmas Dig_plus_eval =
+ of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject
+
+simproc_setup numeral_minus ("of_num m - of_num n") = {*
+ let
+ (*TODO proper implicit use of morphism via pattern antiquotations*)
+ fun cdest_of_num ct = (List.last o snd o Drule.strip_comb) ct;
+ fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
+ fun attach_num ct = (dest_num (Thm.term_of ct), ct);
+ fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
+ val simplify = Raw_Simplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
+ fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq}
+ OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
+ [Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
+ in fn phi => fn _ => fn ct => case try cdifference ct
+ of NONE => (NONE)
+ | SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
+ then Raw_Simplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
+ else mk_meta_eq (let
+ val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
+ in
+ (if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
+ else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
+ end) end)
+ end
+*}
+
+lemma Dig_of_num_minus_zero [numeral]:
+ "of_num n - 0 = of_num n"
+ by (simp add: minus_inverts_plus1)
+
+lemma Dig_one_minus_zero [numeral]:
+ "1 - 0 = 1"
+ by (simp add: minus_inverts_plus1)
+
+lemma Dig_one_minus_one [numeral]:
+ "1 - 1 = 0"
+ by (simp add: minus_inverts_plus1)
+
+lemma Dig_of_num_minus_one [numeral]:
+ "of_num (Dig0 n) - 1 = of_num (DigM n)"
+ "of_num (Dig1 n) - 1 = of_num (Dig0 n)"
+ by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
+
+lemma Dig_one_minus_of_num [numeral]:
+ "1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
+ "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
+ by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)
+
+end
+
+
+subsubsection {* Structures with negation: class @{text ring_1} *}
+
+context ring_1
+begin
+
+subclass semiring_1_minus proof
+qed (simp_all add: algebra_simps)
+
+lemma Dig_zero_minus_of_num [numeral]:
+ "0 - of_num n = - of_num n"
+ by simp
+
+lemma Dig_zero_minus_one [numeral]:
+ "0 - 1 = - 1"
+ by simp
+
+lemma Dig_uminus_uminus [numeral]:
+ "- (- of_num n) = of_num n"
+ by simp
+
+lemma Dig_plus_uminus [numeral]:
+ "of_num m + - of_num n = of_num m - of_num n"
+ "- of_num m + of_num n = of_num n - of_num m"
+ "- of_num m + - of_num n = - (of_num m + of_num n)"
+ "of_num m - - of_num n = of_num m + of_num n"
+ "- of_num m - of_num n = - (of_num m + of_num n)"
+ "- of_num m - - of_num n = of_num n - of_num m"
+ by (simp_all add: diff_minus add_commute)
+
+lemma Dig_times_uminus [numeral]:
+ "- of_num n * of_num m = - (of_num n * of_num m)"
+ "of_num n * - of_num m = - (of_num n * of_num m)"
+ "- of_num n * - of_num m = of_num n * of_num m"
+ by simp_all
+
+lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
+by (induct n)
+ (simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)
+
+declare of_int_1 [numeral]
+
+end
+
+
+subsubsection {* Structures with exponentiation *}
+
+lemma of_num_square: "of_num (square x) = of_num x * of_num x"
+by (induct x)
+ (simp_all add: of_num.simps of_num_add algebra_simps)
+
+lemma of_num_pow: "of_num (pow x y) = of_num x ^ of_num y"
+by (induct y)
+ (simp_all add: of_num.simps of_num_square of_num_mult power_add)
+
+lemma power_of_num [numeral]: "of_num x ^ of_num y = of_num (pow x y)"
+ unfolding of_num_pow ..
+
+lemma power_zero_of_num [numeral]:
+ "0 ^ of_num n = (0::'a::semiring_1)"
+ using of_num_pos [where n=n and ?'a=nat]
+ by (simp add: power_0_left)
+
+lemma power_minus_Dig0 [numeral]:
+ fixes x :: "'a::ring_1"
+ shows "(- x) ^ of_num (Dig0 n) = x ^ of_num (Dig0 n)"
+ by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
+
+lemma power_minus_Dig1 [numeral]:
+ fixes x :: "'a::ring_1"
+ shows "(- x) ^ of_num (Dig1 n) = - (x ^ of_num (Dig1 n))"
+ by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)
+
+declare power_one [numeral]
+
+
+subsubsection {* Greetings to @{typ nat}. *}
+
+instance nat :: semiring_1_minus proof
+qed simp_all
+
+lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
+ unfolding of_num_plus_one [symmetric] by simp
+
+lemma nat_number:
+ "1 = Suc 0"
+ "of_num One = Suc 0"
+ "of_num (Dig0 n) = Suc (of_num (DigM n))"
+ "of_num (Dig1 n) = Suc (of_num (Dig0 n))"
+ by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)
+
+declare diff_0_eq_0 [numeral]
+
+
+subsection {* Proof tools setup *}
+
+subsubsection {* Numeral equations as default simplification rules *}
+
+declare (in semiring_numeral) of_num_One [simp]
+declare (in semiring_numeral) of_num_plus_one [simp]
+declare (in semiring_numeral) of_num_one_plus [simp]
+declare (in semiring_numeral) of_num_plus [simp]
+declare (in semiring_numeral) of_num_times [simp]
+
+declare (in semiring_1) of_nat_of_num [simp]
+
+declare (in semiring_char_0) of_num_eq_iff [simp]
+declare (in semiring_char_0) of_num_eq_one_iff [simp]
+declare (in semiring_char_0) one_eq_of_num_iff [simp]
+
+declare (in linordered_semidom) of_num_pos [simp]
+declare (in linordered_semidom) of_num_not_zero [simp]
+declare (in linordered_semidom) of_num_less_eq_iff [simp]
+declare (in linordered_semidom) of_num_less_eq_one_iff [simp]
+declare (in linordered_semidom) one_less_eq_of_num_iff [simp]
+declare (in linordered_semidom) of_num_less_iff [simp]
+declare (in linordered_semidom) of_num_less_one_iff [simp]
+declare (in linordered_semidom) one_less_of_num_iff [simp]
+declare (in linordered_semidom) of_num_nonneg [simp]
+declare (in linordered_semidom) of_num_less_zero_iff [simp]
+declare (in linordered_semidom) of_num_le_zero_iff [simp]
+
+declare (in linordered_idom) le_signed_numeral_special [simp]
+declare (in linordered_idom) less_signed_numeral_special [simp]
+
+declare (in semiring_1_minus) Dig_of_num_minus_one [simp]
+declare (in semiring_1_minus) Dig_one_minus_of_num [simp]
+
+declare (in ring_1) Dig_plus_uminus [simp]
+declare (in ring_1) of_int_of_num [simp]
+
+declare power_of_num [simp]
+declare power_zero_of_num [simp]
+declare power_minus_Dig0 [simp]
+declare power_minus_Dig1 [simp]
+
+declare Suc_of_num [simp]
+
+
+subsubsection {* Reorientation of equalities *}
+
+setup {*
+ Reorient_Proc.add
+ (fn Const(@{const_name of_num}, _) $ _ => true
+ | Const(@{const_name uminus}, _) $
+ (Const(@{const_name of_num}, _) $ _) => true
+ | _ => false)
+*}
+
+simproc_setup reorient_num ("of_num n = x" | "- of_num m = y") = Reorient_Proc.proc
+
+
+subsubsection {* Constant folding for multiplication in semirings *}
+
+context semiring_numeral
+begin
+
+lemma mult_of_num_commute: "x * of_num n = of_num n * x"
+by (induct n)
+ (simp_all only: of_num.simps left_distrib right_distrib mult_1_left mult_1_right)
+
+definition
+ "commutes_with a b \<longleftrightarrow> a * b = b * a"
+
+lemma commutes_with_commute: "commutes_with a b \<Longrightarrow> a * b = b * a"
+unfolding commutes_with_def .
+
+lemma commutes_with_left_commute: "commutes_with a b \<Longrightarrow> a * (b * c) = b * (a * c)"
+unfolding commutes_with_def by (simp only: mult_assoc [symmetric])
+
+lemma commutes_with_numeral: "commutes_with x (of_num n)" "commutes_with (of_num n) x"
+unfolding commutes_with_def by (simp_all add: mult_of_num_commute)
+
+lemmas mult_ac_numeral =
+ mult_assoc
+ commutes_with_commute
+ commutes_with_left_commute
+ commutes_with_numeral
+
+end
+
+ML {*
+structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
+struct
+ val assoc_ss = HOL_ss addsimps @{thms mult_ac_numeral}
+ val eq_reflection = eq_reflection
+ fun is_numeral (Const(@{const_name of_num}, _) $ _) = true
+ | is_numeral _ = false;
+end;
+
+structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
+*}
+
+simproc_setup semiring_assoc_fold' ("(a::'a::semiring_numeral) * b") =
+ {* fn phi => fn ss => fn ct =>
+ Semiring_Times_Assoc.proc ss (Thm.term_of ct) *}
+
+
+subsection {* Code generator setup for @{typ int} *}
+
+text {* Reversing standard setup *}
+
+lemma [code_unfold del]: "(0::int) \<equiv> Numeral0" by simp
+lemma [code_unfold del]: "(1::int) \<equiv> Numeral1" by simp
+declare zero_is_num_zero [code_unfold del]
+declare one_is_num_one [code_unfold del]
+
+lemma [code, code del]:
+ "(1 :: int) = 1"
+ "(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
+ "(uminus :: int \<Rightarrow> int) = uminus"
+ "(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
+ "(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
+ "(HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool) = HOL.equal"
+ "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
+ "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
+ by rule+
+
+text {* Constructors *}
+
+definition Pls :: "num \<Rightarrow> int" where
+ [simp, code_post]: "Pls n = of_num n"
+
+definition Mns :: "num \<Rightarrow> int" where
+ [simp, code_post]: "Mns n = - of_num n"
+
+code_datatype "0::int" Pls Mns
+
+lemmas [code_unfold] = Pls_def [symmetric] Mns_def [symmetric]
+
+text {* Auxiliary operations *}
+
+definition dup :: "int \<Rightarrow> int" where
+ [simp]: "dup k = k + k"
+
+lemma Dig_dup [code]:
+ "dup 0 = 0"
+ "dup (Pls n) = Pls (Dig0 n)"
+ "dup (Mns n) = Mns (Dig0 n)"
+ by (simp_all add: of_num.simps)
+
+definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
+ [simp]: "sub m n = (of_num m - of_num n)"
+
+lemma Dig_sub [code]:
+ "sub One One = 0"
+ "sub (Dig0 m) One = of_num (DigM m)"
+ "sub (Dig1 m) One = of_num (Dig0 m)"
+ "sub One (Dig0 n) = - of_num (DigM n)"
+ "sub One (Dig1 n) = - of_num (Dig0 n)"
+ "sub (Dig0 m) (Dig0 n) = dup (sub m n)"
+ "sub (Dig1 m) (Dig1 n) = dup (sub m n)"
+ "sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
+ "sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
+ by (simp_all add: algebra_simps num_eq_iff nat_of_num_add)
+
+text {* Implementations *}
+
+lemma one_int_code [code]:
+ "1 = Pls One"
+ by simp
+
+lemma plus_int_code [code]:
+ "k + 0 = (k::int)"
+ "0 + l = (l::int)"
+ "Pls m + Pls n = Pls (m + n)"
+ "Pls m + Mns n = sub m n"
+ "Mns m + Pls n = sub n m"
+ "Mns m + Mns n = Mns (m + n)"
+ by simp_all
+
+lemma uminus_int_code [code]:
+ "uminus 0 = (0::int)"
+ "uminus (Pls m) = Mns m"
+ "uminus (Mns m) = Pls m"
+ by simp_all
+
+lemma minus_int_code [code]:
+ "k - 0 = (k::int)"
+ "0 - l = uminus (l::int)"
+ "Pls m - Pls n = sub m n"
+ "Pls m - Mns n = Pls (m + n)"
+ "Mns m - Pls n = Mns (m + n)"
+ "Mns m - Mns n = sub n m"
+ by simp_all
+
+lemma times_int_code [code]:
+ "k * 0 = (0::int)"
+ "0 * l = (0::int)"
+ "Pls m * Pls n = Pls (m * n)"
+ "Pls m * Mns n = Mns (m * n)"
+ "Mns m * Pls n = Mns (m * n)"
+ "Mns m * Mns n = Pls (m * n)"
+ by simp_all
+
+lemma eq_int_code [code]:
+ "HOL.equal 0 (0::int) \<longleftrightarrow> True"
+ "HOL.equal 0 (Pls l) \<longleftrightarrow> False"
+ "HOL.equal 0 (Mns l) \<longleftrightarrow> False"
+ "HOL.equal (Pls k) 0 \<longleftrightarrow> False"
+ "HOL.equal (Pls k) (Pls l) \<longleftrightarrow> HOL.equal k l"
+ "HOL.equal (Pls k) (Mns l) \<longleftrightarrow> False"
+ "HOL.equal (Mns k) 0 \<longleftrightarrow> False"
+ "HOL.equal (Mns k) (Pls l) \<longleftrightarrow> False"
+ "HOL.equal (Mns k) (Mns l) \<longleftrightarrow> HOL.equal k l"
+ by (auto simp add: equal dest: sym)
+
+lemma [code nbe]:
+ "HOL.equal (k::int) k \<longleftrightarrow> True"
+ by (fact equal_refl)
+
+lemma less_eq_int_code [code]:
+ "0 \<le> (0::int) \<longleftrightarrow> True"
+ "0 \<le> Pls l \<longleftrightarrow> True"
+ "0 \<le> Mns l \<longleftrightarrow> False"
+ "Pls k \<le> 0 \<longleftrightarrow> False"
+ "Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
+ "Pls k \<le> Mns l \<longleftrightarrow> False"
+ "Mns k \<le> 0 \<longleftrightarrow> True"
+ "Mns k \<le> Pls l \<longleftrightarrow> True"
+ "Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
+ by simp_all
+
+lemma less_int_code [code]:
+ "0 < (0::int) \<longleftrightarrow> False"
+ "0 < Pls l \<longleftrightarrow> True"
+ "0 < Mns l \<longleftrightarrow> False"
+ "Pls k < 0 \<longleftrightarrow> False"
+ "Pls k < Pls l \<longleftrightarrow> k < l"
+ "Pls k < Mns l \<longleftrightarrow> False"
+ "Mns k < 0 \<longleftrightarrow> True"
+ "Mns k < Pls l \<longleftrightarrow> True"
+ "Mns k < Mns l \<longleftrightarrow> l < k"
+ by simp_all
+
+hide_const (open) sub dup
+
+text {* Pretty literals *}
+
+ML {*
+local open Code_Thingol in
+
+fun add_code print target =
+ let
+ fun dest_num one' dig0' dig1' thm =
+ let
+ fun dest_dig (IConst (c, _)) = if c = dig0' then 0
+ else if c = dig1' then 1
+ else Code_Printer.eqn_error thm "Illegal numeral expression: illegal dig"
+ | dest_dig _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal digit";
+ fun dest_num (IConst (c, _)) = if c = one' then 1
+ else Code_Printer.eqn_error thm "Illegal numeral expression: illegal leading digit"
+ | dest_num (t1 `$ t2) = 2 * dest_num t2 + dest_dig t1
+ | dest_num _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal term";
+ in dest_num end;
+ fun pretty sgn literals [one', dig0', dig1'] _ thm _ _ [(t, _)] =
+ (Code_Printer.str o print literals o sgn o dest_num one' dig0' dig1' thm) t
+ fun add_syntax (c, sgn) = Code_Target.add_const_syntax target c
+ (SOME (Code_Printer.complex_const_syntax
+ (1, ([@{const_name One}, @{const_name Dig0}, @{const_name Dig1}],
+ pretty sgn))));
+ in
+ add_syntax (@{const_name Pls}, I)
+ #> add_syntax (@{const_name Mns}, (fn k => ~ k))
+ end;
+
+end
+*}
+
+hide_const (open) One Dig0 Dig1
+
+
+subsection {* Toy examples *}
+
+definition "foo \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat)"
+definition "bar \<longleftrightarrow> #4 * #2 + #7 \<ge> (#8 :: int) - #3"
+
+code_thms foo bar
+export_code foo bar checking SML OCaml? Haskell? Scala?
+
+text {* This is an ad-hoc @{text Code_Integer} setup. *}
+
+setup {*
+ fold (add_code Code_Printer.literal_numeral)
+ [Code_ML.target_SML, Code_ML.target_OCaml, Code_Haskell.target, Code_Scala.target]
+*}
+
+code_type int
+ (SML "IntInf.int")
+ (OCaml "Big'_int.big'_int")
+ (Haskell "Integer")
+ (Scala "BigInt")
+ (Eval "int")
+
+code_const "0::int"
+ (SML "0/ :/ IntInf.int")
+ (OCaml "Big'_int.zero")
+ (Haskell "0")
+ (Scala "BigInt(0)")
+ (Eval "0/ :/ int")
+
+code_const Int.pred
+ (SML "IntInf.- ((_), 1)")
+ (OCaml "Big'_int.pred'_big'_int")
+ (Haskell "!(_/ -/ 1)")
+ (Scala "!(_ -/ 1)")
+ (Eval "!(_/ -/ 1)")
+
+code_const Int.succ
+ (SML "IntInf.+ ((_), 1)")
+ (OCaml "Big'_int.succ'_big'_int")
+ (Haskell "!(_/ +/ 1)")
+ (Scala "!(_ +/ 1)")
+ (Eval "!(_/ +/ 1)")
+
+code_const "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"
+ (SML "IntInf.+ ((_), (_))")
+ (OCaml "Big'_int.add'_big'_int")
+ (Haskell infixl 6 "+")
+ (Scala infixl 7 "+")
+ (Eval infixl 8 "+")
+
+code_const "uminus \<Colon> int \<Rightarrow> int"
+ (SML "IntInf.~")
+ (OCaml "Big'_int.minus'_big'_int")
+ (Haskell "negate")
+ (Scala "!(- _)")
+ (Eval "~/ _")
+
+code_const "op - \<Colon> int \<Rightarrow> int \<Rightarrow> int"
+ (SML "IntInf.- ((_), (_))")
+ (OCaml "Big'_int.sub'_big'_int")
+ (Haskell infixl 6 "-")
+ (Scala infixl 7 "-")
+ (Eval infixl 8 "-")
+
+code_const "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"
+ (SML "IntInf.* ((_), (_))")
+ (OCaml "Big'_int.mult'_big'_int")
+ (Haskell infixl 7 "*")
+ (Scala infixl 8 "*")
+ (Eval infixl 9 "*")
+
+code_const pdivmod
+ (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
+ (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
+ (Haskell "divMod/ (abs _)/ (abs _)")
+ (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
+ (Eval "Integer.div'_mod/ (abs _)/ (abs _)")
+
+code_const "HOL.equal \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
+ (SML "!((_ : IntInf.int) = _)")
+ (OCaml "Big'_int.eq'_big'_int")
+ (Haskell infix 4 "==")
+ (Scala infixl 5 "==")
+ (Eval infixl 6 "=")
+
+code_const "op \<le> \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
+ (SML "IntInf.<= ((_), (_))")
+ (OCaml "Big'_int.le'_big'_int")
+ (Haskell infix 4 "<=")
+ (Scala infixl 4 "<=")
+ (Eval infixl 6 "<=")
+
+code_const "op < \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
+ (SML "IntInf.< ((_), (_))")
+ (OCaml "Big'_int.lt'_big'_int")
+ (Haskell infix 4 "<")
+ (Scala infixl 4 "<")
+ (Eval infixl 6 "<")
+
+code_const Code_Numeral.int_of
+ (SML "IntInf.fromInt")
+ (OCaml "_")
+ (Haskell "toInteger")
+ (Scala "!_.as'_BigInt")
+ (Eval "_")
+
+export_code foo bar checking SML OCaml? Haskell? Scala?
+
+end
--- a/src/HOL/ex/ROOT.ML Tue Feb 21 08:15:42 2012 +0100
+++ b/src/HOL/ex/ROOT.ML Tue Feb 21 09:17:53 2012 +0100
@@ -17,7 +17,7 @@
use_thys [
"Iff_Oracle",
"Coercion_Examples",
- "Numeral",
+ "Numeral_Representation",
"Higher_Order_Logic",
"Abstract_NAT",
"Guess",