--- a/src/HOL/ex/Numeral_Representation.thy Mon Jun 10 16:04:34 2013 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,973 +0,0 @@
-(* Title: HOL/ex/Numeral_Representation.thy
- Author: Florian Haftmann
-*)
-
-header {* First experiments for a numeral representation (now obsolete). *}
-
-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
- distrib_right distrib_left)
-
-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 distrib_left)
-
-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"}, K numeral_tr)] end
-*}
-
-typed_print_translation {*
- 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 Printer.type_emphasis ctxt T' then
- Syntax.const @{syntax_const "_constrain"} $ t' $
- Syntax_Phases.term_of_typ ctxt T'
- else 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 only: 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 only: 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 only: not_less minus_one_le_of_num_iff)
-
-lemma one_less_minus_one_iff: "\<not> 1 < - 1"
- by (simp only: 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 distrib_right distrib_left 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 = simpset_of (put_simpset HOL_ss @{context} 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]: "(1::int) \<equiv> Numeral1" by simp
-
-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
-
-end
-