--- a/NEWS Sun Aug 21 06:18:23 2022 +0000
+++ b/NEWS Sun Aug 21 06:18:23 2022 +0000
@@ -34,6 +34,9 @@
*** HOL ***
+* Moved auxiliary computation constant "divmod_nat" to theory
+"Euclidean_Division". Minor INCOMPATIBILITY.
+
* Renamed attribute "arith_split" to "linarith_split". Minor
INCOMPATIBILITY.
@@ -44,7 +47,7 @@
integers, sacrificing pattern patching in exchange for dramatically
increased performance for comparisons.
-* New theory HOL-Library.NList of fixed length lists
+* New theory HOL-Library.NList of fixed length lists.
* Rule split_of_bool_asm is not split any longer, analogously to
split_if_asm. INCOMPATIBILITY.
--- a/src/HOL/Divides.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Divides.thy Sun Aug 21 06:18:23 2022 +0000
@@ -541,439 +541,6 @@
qed
-subsection \<open>Numeral division with a pragmatic type class\<close>
-
-text \<open>
- The following type class contains everything necessary to formulate
- a division algorithm in ring structures with numerals, restricted
- to its positive segments.
-\<close>
-
-class unique_euclidean_semiring_with_nat_division = unique_euclidean_semiring_with_nat +
- fixes divmod :: \<open>num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a\<close>
- and divmod_step :: \<open>'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a\<close> \<comment> \<open>
- These are conceptually definitions but force generated code
- to be monomorphic wrt. particular instances of this class which
- yields a significant speedup.\<close>
- assumes divmod_def: \<open>divmod m n = (numeral m div numeral n, numeral m mod numeral n)\<close>
- and divmod_step_def [simp]: \<open>divmod_step l (q, r) =
- (if euclidean_size l \<le> euclidean_size r then (2 * q + 1, r - l)
- else (2 * q, r))\<close> \<comment> \<open>
- This is a formulation of one step (referring to one digit position)
- in school-method division: compare the dividend at the current
- digit position with the remainder from previous division steps
- and evaluate accordingly.\<close>
-begin
-
-lemma fst_divmod:
- \<open>fst (divmod m n) = numeral m div numeral n\<close>
- by (simp add: divmod_def)
-
-lemma snd_divmod:
- \<open>snd (divmod m n) = numeral m mod numeral n\<close>
- by (simp add: divmod_def)
-
-text \<open>
- Following a formulation of school-method division.
- If the divisor is smaller than the dividend, terminate.
- If not, shift the dividend to the right until termination
- occurs and then reiterate single division steps in the
- opposite direction.
-\<close>
-
-lemma divmod_divmod_step:
- \<open>divmod m n = (if m < n then (0, numeral m)
- else divmod_step (numeral n) (divmod m (Num.Bit0 n)))\<close>
-proof (cases \<open>m < n\<close>)
- case True
- then show ?thesis
- by (simp add: prod_eq_iff fst_divmod snd_divmod flip: of_nat_numeral of_nat_div of_nat_mod)
-next
- case False
- define r s t where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close> \<open>t = 2 * s\<close>
- then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close> \<open>numeral (num.Bit0 n) = of_nat t\<close>
- and \<open>\<not> s \<le> r mod s\<close>
- by (simp_all add: not_le)
- have t: \<open>2 * (r div t) = r div s - r div s mod 2\<close>
- \<open>r mod t = s * (r div s mod 2) + r mod s\<close>
- by (simp add: Rings.minus_mod_eq_mult_div Groups.mult.commute [of 2] Euclidean_Division.div_mult2_eq \<open>t = 2 * s\<close>)
- (use mod_mult2_eq [of r s 2] in \<open>simp add: ac_simps \<open>t = 2 * s\<close>\<close>)
- have rs: \<open>r div s mod 2 = 0 \<or> r div s mod 2 = Suc 0\<close>
- by auto
- from \<open>\<not> s \<le> r mod s\<close> have \<open>s \<le> r mod t \<Longrightarrow>
- r div s = Suc (2 * (r div t)) \<and>
- r mod s = r mod t - s\<close>
- using rs
- by (auto simp add: t)
- moreover have \<open>r mod t < s \<Longrightarrow>
- r div s = 2 * (r div t) \<and>
- r mod s = r mod t\<close>
- using rs
- by (auto simp add: t)
- ultimately show ?thesis
- by (simp add: divmod_def prod_eq_iff split_def Let_def
- not_less mod_eq_0_iff_dvd Rings.mod_eq_0_iff_dvd False not_le *)
- (simp add: flip: of_nat_numeral of_nat_mult add.commute [of 1] of_nat_div of_nat_mod of_nat_Suc of_nat_diff)
-qed
-
-text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
-
-lemma divmod_trivial [simp]:
- "divmod m Num.One = (numeral m, 0)"
- "divmod num.One (num.Bit0 n) = (0, Numeral1)"
- "divmod num.One (num.Bit1 n) = (0, Numeral1)"
- using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
-
-text \<open>Division by an even number is a right-shift\<close>
-
-lemma divmod_cancel [simp]:
- \<open>divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))\<close> (is ?P)
- \<open>divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))\<close> (is ?Q)
-proof -
- define r s where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close>
- then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close>
- \<open>numeral (num.Bit0 m) = of_nat (2 * r)\<close> \<open>numeral (num.Bit0 n) = of_nat (2 * s)\<close>
- \<open>numeral (num.Bit1 m) = of_nat (Suc (2 * r))\<close>
- by simp_all
- show ?P and ?Q
- by (simp_all add: divmod_def *)
- (simp_all flip: of_nat_numeral of_nat_div of_nat_mod of_nat_mult add.commute [of 1] of_nat_Suc
- add: Euclidean_Division.mod_mult_mult1 div_mult2_eq [of _ 2] mod_mult2_eq [of _ 2])
-qed
-
-text \<open>The really hard work\<close>
-
-lemma divmod_steps [simp]:
- "divmod (num.Bit0 m) (num.Bit1 n) =
- (if m \<le> n then (0, numeral (num.Bit0 m))
- else divmod_step (numeral (num.Bit1 n))
- (divmod (num.Bit0 m)
- (num.Bit0 (num.Bit1 n))))"
- "divmod (num.Bit1 m) (num.Bit1 n) =
- (if m < n then (0, numeral (num.Bit1 m))
- else divmod_step (numeral (num.Bit1 n))
- (divmod (num.Bit1 m)
- (num.Bit0 (num.Bit1 n))))"
- by (simp_all add: divmod_divmod_step)
-
-lemmas divmod_algorithm_code = divmod_trivial divmod_cancel divmod_steps
-
-text \<open>Special case: divisibility\<close>
-
-definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
-where
- "divides_aux qr \<longleftrightarrow> snd qr = 0"
-
-lemma divides_aux_eq [simp]:
- "divides_aux (q, r) \<longleftrightarrow> r = 0"
- by (simp add: divides_aux_def)
-
-lemma dvd_numeral_simp [simp]:
- "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
- by (simp add: divmod_def mod_eq_0_iff_dvd)
-
-text \<open>Generic computation of quotient and remainder\<close>
-
-lemma numeral_div_numeral [simp]:
- "numeral k div numeral l = fst (divmod k l)"
- by (simp add: fst_divmod)
-
-lemma numeral_mod_numeral [simp]:
- "numeral k mod numeral l = snd (divmod k l)"
- by (simp add: snd_divmod)
-
-lemma one_div_numeral [simp]:
- "1 div numeral n = fst (divmod num.One n)"
- by (simp add: fst_divmod)
-
-lemma one_mod_numeral [simp]:
- "1 mod numeral n = snd (divmod num.One n)"
- by (simp add: snd_divmod)
-
-text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
-
-lemma cong_exp_iff_simps:
- "numeral n mod numeral Num.One = 0
- \<longleftrightarrow> True"
- "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
- \<longleftrightarrow> numeral n mod numeral q = 0"
- "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
- \<longleftrightarrow> False"
- "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
- \<longleftrightarrow> True"
- "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
- \<longleftrightarrow> True"
- "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
- \<longleftrightarrow> False"
- "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
- \<longleftrightarrow> (numeral n mod numeral q) = 0"
- "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
- \<longleftrightarrow> False"
- "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
- \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
- "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
- \<longleftrightarrow> False"
- "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
- \<longleftrightarrow> (numeral m mod numeral q) = 0"
- "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
- \<longleftrightarrow> False"
- "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
- \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
- by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
-
-end
-
-instantiation nat :: unique_euclidean_semiring_with_nat_division
-begin
-
-definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
-where
- divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
-
-definition divmod_step_nat :: "nat \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
-where
- "divmod_step_nat l qr = (let (q, r) = qr
- in if r \<ge> l then (2 * q + 1, r - l)
- else (2 * q, r))"
-
-instance
- by standard (simp_all add: divmod'_nat_def divmod_step_nat_def)
-
-end
-
-declare divmod_algorithm_code [where ?'a = nat, code]
-
-lemma Suc_0_div_numeral [simp]:
- fixes k l :: num
- shows "Suc 0 div numeral k = fst (divmod Num.One k)"
- by (simp_all add: fst_divmod)
-
-lemma Suc_0_mod_numeral [simp]:
- fixes k l :: num
- shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
- by (simp_all add: snd_divmod)
-
-instantiation int :: unique_euclidean_semiring_with_nat_division
-begin
-
-definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
-where
- "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
-
-definition divmod_step_int :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
-where
- "divmod_step_int l qr = (let (q, r) = qr
- in if \<bar>l\<bar> \<le> \<bar>r\<bar> then (2 * q + 1, r - l)
- else (2 * q, r))"
-
-instance
- by standard (auto simp add: divmod_int_def divmod_step_int_def)
-
-end
-
-declare divmod_algorithm_code [where ?'a = int, code]
-
-context
-begin
-
-qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
-where
- "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
-
-qualified lemma adjust_div_eq [simp, code]:
- "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
- by (simp add: adjust_div_def)
-
-qualified definition adjust_mod :: "num \<Rightarrow> int \<Rightarrow> int"
-where
- [simp]: "adjust_mod l r = (if r = 0 then 0 else numeral l - r)"
-
-lemma minus_numeral_div_numeral [simp]:
- "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
-proof -
- have "int (fst (divmod m n)) = fst (divmod m n)"
- by (simp only: fst_divmod divide_int_def) auto
- then show ?thesis
- by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
-qed
-
-lemma minus_numeral_mod_numeral [simp]:
- "- numeral m mod numeral n = adjust_mod n (snd (divmod m n) :: int)"
-proof (cases "snd (divmod m n) = (0::int)")
- case True
- then show ?thesis
- by (simp add: mod_eq_0_iff_dvd divides_aux_def)
-next
- case False
- then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
- by (simp only: snd_divmod modulo_int_def) auto
- then show ?thesis
- by (simp add: divides_aux_def adjust_div_def)
- (simp add: divides_aux_def modulo_int_def)
-qed
-
-lemma numeral_div_minus_numeral [simp]:
- "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
-proof -
- have "int (fst (divmod m n)) = fst (divmod m n)"
- by (simp only: fst_divmod divide_int_def) auto
- then show ?thesis
- by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
-qed
-
-lemma numeral_mod_minus_numeral [simp]:
- "numeral m mod - numeral n = - adjust_mod n (snd (divmod m n) :: int)"
-proof (cases "snd (divmod m n) = (0::int)")
- case True
- then show ?thesis
- by (simp add: mod_eq_0_iff_dvd divides_aux_def)
-next
- case False
- then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
- by (simp only: snd_divmod modulo_int_def) auto
- then show ?thesis
- by (simp add: divides_aux_def adjust_div_def)
- (simp add: divides_aux_def modulo_int_def)
-qed
-
-lemma minus_one_div_numeral [simp]:
- "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
- using minus_numeral_div_numeral [of Num.One n] by simp
-
-lemma minus_one_mod_numeral [simp]:
- "- 1 mod numeral n = adjust_mod n (snd (divmod Num.One n) :: int)"
- using minus_numeral_mod_numeral [of Num.One n] by simp
-
-lemma one_div_minus_numeral [simp]:
- "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
- using numeral_div_minus_numeral [of Num.One n] by simp
-
-lemma one_mod_minus_numeral [simp]:
- "1 mod - numeral n = - adjust_mod n (snd (divmod Num.One n) :: int)"
- using numeral_mod_minus_numeral [of Num.One n] by simp
-
-end
-
-lemma divmod_BitM_2_eq [simp]:
- \<open>divmod (Num.BitM m) (Num.Bit0 Num.One) = (numeral m - 1, (1 :: int))\<close>
- by (cases m) simp_all
-
-lemma div_positive_int:
- "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
- using that by (simp add: nonneg1_imp_zdiv_pos_iff)
-
-
-subsubsection \<open>Dedicated simproc for calculation\<close>
-
-lemma euclidean_size_nat_less_eq_iff:
- \<open>euclidean_size m \<le> euclidean_size n \<longleftrightarrow> m \<le> n\<close> for m n :: nat
- by simp
-
-lemma euclidean_size_int_less_eq_iff:
- \<open>euclidean_size k \<le> euclidean_size l \<longleftrightarrow> \<bar>k\<bar> \<le> \<bar>l\<bar>\<close> for k l :: int
- by auto
-
-text \<open>
- There is space for improvement here: the calculation itself
- could be carried out outside the logic, and a generic simproc
- (simplifier setup) for generic calculation would be helpful.
-\<close>
-
-simproc_setup numeral_divmod
- ("0 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "0 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "0 div - 1 :: int" | "0 mod - 1 :: int" |
- "0 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
- "1 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "1 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "1 div - 1 :: int" | "1 mod - 1 :: int" |
- "1 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
- "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
- "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
- "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
- "numeral a div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "numeral a div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
- "numeral a div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
- "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
- "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
- "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
- "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
- "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
- "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
-\<open> let
- val if_cong = the (Code.get_case_cong \<^theory> \<^const_name>\<open>If\<close>);
- fun successful_rewrite ctxt ct =
- let
- val thm = Simplifier.rewrite ctxt ct
- in if Thm.is_reflexive thm then NONE else SOME thm end;
- in fn phi =>
- let
- val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
- one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
- one_div_minus_numeral one_mod_minus_numeral
- numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
- numeral_div_minus_numeral numeral_mod_minus_numeral
- div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
- numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
- divmod_cancel divmod_steps divmod_step_def fst_conv snd_conv numeral_One
- case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
- minus_minus numeral_times_numeral mult_zero_right mult_1_right
- euclidean_size_nat_less_eq_iff euclidean_size_int_less_eq_iff diff_nat_numeral nat_numeral}
- @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
- fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
- (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
- in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
- end
-\<close>
-
-
-subsubsection \<open>Code generation\<close>
-
-definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
- where "divmod_nat m n = (m div n, m mod n)"
-
-lemma fst_divmod_nat [simp]:
- "fst (divmod_nat m n) = m div n"
- by (simp add: divmod_nat_def)
-
-lemma snd_divmod_nat [simp]:
- "snd (divmod_nat m n) = m mod n"
- by (simp add: divmod_nat_def)
-
-lemma divmod_nat_if [code]:
- "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
- let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
- by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
-
-lemma [code]:
- "m div n = fst (divmod_nat m n)"
- "m mod n = snd (divmod_nat m n)"
- by simp_all
-
-lemma [code]:
- fixes k :: int
- shows
- "k div 0 = 0"
- "k mod 0 = k"
- "0 div k = 0"
- "0 mod k = 0"
- "k div Int.Pos Num.One = k"
- "k mod Int.Pos Num.One = 0"
- "k div Int.Neg Num.One = - k"
- "k mod Int.Neg Num.One = 0"
- "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
- "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
- "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
- "Int.Neg m mod Int.Pos n = Divides.adjust_mod n (snd (divmod m n) :: int)"
- "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
- "Int.Pos m mod Int.Neg n = - Divides.adjust_mod n (snd (divmod m n) :: int)"
- "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
- "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
- by simp_all
-
code_identifier
code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
@@ -1090,4 +657,8 @@
lemma zmod_eq_0D [dest!]: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
using that by auto
+lemma div_positive_int:
+ "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
+ using that by (simp add: nonneg1_imp_zdiv_pos_iff)
+
end
--- a/src/HOL/Euclidean_Division.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Euclidean_Division.thy Sun Aug 21 06:18:23 2022 +0000
@@ -2290,7 +2290,402 @@
using that by (simp add: modulo_int_def sgn_if)
-subsection \<open>Code generation\<close>
+subsection \<open>Generic symbolic computations\<close>
+
+text \<open>
+ The following type class contains everything necessary to formulate
+ a division algorithm in ring structures with numerals, restricted
+ to its positive segments.
+\<close>
+
+class unique_euclidean_semiring_with_nat_division = unique_euclidean_semiring_with_nat +
+ fixes divmod :: \<open>num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a\<close>
+ and divmod_step :: \<open>'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a\<close> \<comment> \<open>
+ These are conceptually definitions but force generated code
+ to be monomorphic wrt. particular instances of this class which
+ yields a significant speedup.\<close>
+ assumes divmod_def: \<open>divmod m n = (numeral m div numeral n, numeral m mod numeral n)\<close>
+ and divmod_step_def [simp]: \<open>divmod_step l (q, r) =
+ (if euclidean_size l \<le> euclidean_size r then (2 * q + 1, r - l)
+ else (2 * q, r))\<close> \<comment> \<open>
+ This is a formulation of one step (referring to one digit position)
+ in school-method division: compare the dividend at the current
+ digit position with the remainder from previous division steps
+ and evaluate accordingly.\<close>
+begin
+
+lemma fst_divmod:
+ \<open>fst (divmod m n) = numeral m div numeral n\<close>
+ by (simp add: divmod_def)
+
+lemma snd_divmod:
+ \<open>snd (divmod m n) = numeral m mod numeral n\<close>
+ by (simp add: divmod_def)
+
+text \<open>
+ Following a formulation of school-method division.
+ If the divisor is smaller than the dividend, terminate.
+ If not, shift the dividend to the right until termination
+ occurs and then reiterate single division steps in the
+ opposite direction.
+\<close>
+
+lemma divmod_divmod_step:
+ \<open>divmod m n = (if m < n then (0, numeral m)
+ else divmod_step (numeral n) (divmod m (Num.Bit0 n)))\<close>
+proof (cases \<open>m < n\<close>)
+ case True
+ then show ?thesis
+ by (simp add: prod_eq_iff fst_divmod snd_divmod flip: of_nat_numeral of_nat_div of_nat_mod)
+next
+ case False
+ define r s t where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close> \<open>t = 2 * s\<close>
+ then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close> \<open>numeral (num.Bit0 n) = of_nat t\<close>
+ and \<open>\<not> s \<le> r mod s\<close>
+ by (simp_all add: not_le)
+ have t: \<open>2 * (r div t) = r div s - r div s mod 2\<close>
+ \<open>r mod t = s * (r div s mod 2) + r mod s\<close>
+ by (simp add: Rings.minus_mod_eq_mult_div Groups.mult.commute [of 2] Euclidean_Division.div_mult2_eq \<open>t = 2 * s\<close>)
+ (use mod_mult2_eq [of r s 2] in \<open>simp add: ac_simps \<open>t = 2 * s\<close>\<close>)
+ have rs: \<open>r div s mod 2 = 0 \<or> r div s mod 2 = Suc 0\<close>
+ by auto
+ from \<open>\<not> s \<le> r mod s\<close> have \<open>s \<le> r mod t \<Longrightarrow>
+ r div s = Suc (2 * (r div t)) \<and>
+ r mod s = r mod t - s\<close>
+ using rs
+ by (auto simp add: t)
+ moreover have \<open>r mod t < s \<Longrightarrow>
+ r div s = 2 * (r div t) \<and>
+ r mod s = r mod t\<close>
+ using rs
+ by (auto simp add: t)
+ ultimately show ?thesis
+ by (simp add: divmod_def prod_eq_iff split_def Let_def
+ not_less mod_eq_0_iff_dvd Rings.mod_eq_0_iff_dvd False not_le *)
+ (simp add: flip: of_nat_numeral of_nat_mult add.commute [of 1] of_nat_div of_nat_mod of_nat_Suc of_nat_diff)
+qed
+
+text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
+
+lemma divmod_trivial [simp]:
+ "divmod m Num.One = (numeral m, 0)"
+ "divmod num.One (num.Bit0 n) = (0, Numeral1)"
+ "divmod num.One (num.Bit1 n) = (0, Numeral1)"
+ using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
+
+text \<open>Division by an even number is a right-shift\<close>
+
+lemma divmod_cancel [simp]:
+ \<open>divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))\<close> (is ?P)
+ \<open>divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))\<close> (is ?Q)
+proof -
+ define r s where \<open>r = (numeral m :: nat)\<close> \<open>s = (numeral n :: nat)\<close>
+ then have *: \<open>numeral m = of_nat r\<close> \<open>numeral n = of_nat s\<close>
+ \<open>numeral (num.Bit0 m) = of_nat (2 * r)\<close> \<open>numeral (num.Bit0 n) = of_nat (2 * s)\<close>
+ \<open>numeral (num.Bit1 m) = of_nat (Suc (2 * r))\<close>
+ by simp_all
+ have **: \<open>Suc (2 * r) div 2 = r\<close>
+ by simp
+ show ?P and ?Q
+ by (simp_all add: divmod_def *)
+ (simp_all flip: of_nat_numeral of_nat_div of_nat_mod of_nat_mult add.commute [of 1] of_nat_Suc
+ add: Euclidean_Division.mod_mult_mult1 div_mult2_eq [of _ 2] mod_mult2_eq [of _ 2] **)
+qed
+
+text \<open>The really hard work\<close>
+
+lemma divmod_steps [simp]:
+ "divmod (num.Bit0 m) (num.Bit1 n) =
+ (if m \<le> n then (0, numeral (num.Bit0 m))
+ else divmod_step (numeral (num.Bit1 n))
+ (divmod (num.Bit0 m)
+ (num.Bit0 (num.Bit1 n))))"
+ "divmod (num.Bit1 m) (num.Bit1 n) =
+ (if m < n then (0, numeral (num.Bit1 m))
+ else divmod_step (numeral (num.Bit1 n))
+ (divmod (num.Bit1 m)
+ (num.Bit0 (num.Bit1 n))))"
+ by (simp_all add: divmod_divmod_step)
+
+lemmas divmod_algorithm_code = divmod_trivial divmod_cancel divmod_steps
+
+text \<open>Special case: divisibility\<close>
+
+definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
+where
+ "divides_aux qr \<longleftrightarrow> snd qr = 0"
+
+lemma divides_aux_eq [simp]:
+ "divides_aux (q, r) \<longleftrightarrow> r = 0"
+ by (simp add: divides_aux_def)
+
+lemma dvd_numeral_simp [simp]:
+ "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
+ by (simp add: divmod_def mod_eq_0_iff_dvd)
+
+text \<open>Generic computation of quotient and remainder\<close>
+
+lemma numeral_div_numeral [simp]:
+ "numeral k div numeral l = fst (divmod k l)"
+ by (simp add: fst_divmod)
+
+lemma numeral_mod_numeral [simp]:
+ "numeral k mod numeral l = snd (divmod k l)"
+ by (simp add: snd_divmod)
+
+lemma one_div_numeral [simp]:
+ "1 div numeral n = fst (divmod num.One n)"
+ by (simp add: fst_divmod)
+
+lemma one_mod_numeral [simp]:
+ "1 mod numeral n = snd (divmod num.One n)"
+ by (simp add: snd_divmod)
+
+end
+
+instantiation nat :: unique_euclidean_semiring_with_nat_division
+begin
+
+definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
+where
+ divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
+
+definition divmod_step_nat :: "nat \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
+where
+ "divmod_step_nat l qr = (let (q, r) = qr
+ in if r \<ge> l then (2 * q + 1, r - l)
+ else (2 * q, r))"
+
+instance
+ by standard (simp_all add: divmod'_nat_def divmod_step_nat_def)
+
+end
+
+declare divmod_algorithm_code [where ?'a = nat, code]
+
+lemma Suc_0_div_numeral [simp]:
+ \<open>Suc 0 div numeral Num.One = 1\<close>
+ \<open>Suc 0 div numeral (Num.Bit0 n) = 0\<close>
+ \<open>Suc 0 div numeral (Num.Bit1 n) = 0\<close>
+ by simp_all
+
+lemma Suc_0_mod_numeral [simp]:
+ \<open>Suc 0 mod numeral Num.One = 0\<close>
+ \<open>Suc 0 mod numeral (Num.Bit0 n) = 1\<close>
+ \<open>Suc 0 mod numeral (Num.Bit1 n) = 1\<close>
+ by simp_all
+
+instantiation int :: unique_euclidean_semiring_with_nat_division
+begin
+
+definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
+where
+ "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
+
+definition divmod_step_int :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
+where
+ "divmod_step_int l qr = (let (q, r) = qr
+ in if \<bar>l\<bar> \<le> \<bar>r\<bar> then (2 * q + 1, r - l)
+ else (2 * q, r))"
+
+instance
+ by standard (auto simp add: divmod_int_def divmod_step_int_def)
+
+end
+
+declare divmod_algorithm_code [where ?'a = int, code]
+
+context
+begin
+
+qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
+where
+ "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
+
+qualified lemma adjust_div_eq [simp, code]:
+ "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
+ by (simp add: adjust_div_def)
+
+qualified definition adjust_mod :: "num \<Rightarrow> int \<Rightarrow> int"
+where
+ [simp]: "adjust_mod l r = (if r = 0 then 0 else numeral l - r)"
+
+lemma minus_numeral_div_numeral [simp]:
+ "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
+proof -
+ have "int (fst (divmod m n)) = fst (divmod m n)"
+ by (simp only: fst_divmod divide_int_def) auto
+ then show ?thesis
+ by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
+qed
+
+lemma minus_numeral_mod_numeral [simp]:
+ "- numeral m mod numeral n = adjust_mod n (snd (divmod m n) :: int)"
+proof (cases "snd (divmod m n) = (0::int)")
+ case True
+ then show ?thesis
+ by (simp add: mod_eq_0_iff_dvd divides_aux_def)
+next
+ case False
+ then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
+ by (simp only: snd_divmod modulo_int_def) auto
+ then show ?thesis
+ by (simp add: divides_aux_def adjust_div_def)
+ (simp add: divides_aux_def modulo_int_def)
+qed
+
+lemma numeral_div_minus_numeral [simp]:
+ "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
+proof -
+ have "int (fst (divmod m n)) = fst (divmod m n)"
+ by (simp only: fst_divmod divide_int_def) auto
+ then show ?thesis
+ by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
+qed
+
+lemma numeral_mod_minus_numeral [simp]:
+ "numeral m mod - numeral n = - adjust_mod n (snd (divmod m n) :: int)"
+proof (cases "snd (divmod m n) = (0::int)")
+ case True
+ then show ?thesis
+ by (simp add: mod_eq_0_iff_dvd divides_aux_def)
+next
+ case False
+ then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
+ by (simp only: snd_divmod modulo_int_def) auto
+ then show ?thesis
+ by (simp add: divides_aux_def adjust_div_def)
+ (simp add: divides_aux_def modulo_int_def)
+qed
+
+lemma minus_one_div_numeral [simp]:
+ "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
+ using minus_numeral_div_numeral [of Num.One n] by simp
+
+lemma minus_one_mod_numeral [simp]:
+ "- 1 mod numeral n = adjust_mod n (snd (divmod Num.One n) :: int)"
+ using minus_numeral_mod_numeral [of Num.One n] by simp
+
+lemma one_div_minus_numeral [simp]:
+ "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
+ using numeral_div_minus_numeral [of Num.One n] by simp
+
+lemma one_mod_minus_numeral [simp]:
+ "1 mod - numeral n = - adjust_mod n (snd (divmod Num.One n) :: int)"
+ using numeral_mod_minus_numeral [of Num.One n] by simp
+
+lemma [code]:
+ fixes k :: int
+ shows
+ "k div 0 = 0"
+ "k mod 0 = k"
+ "0 div k = 0"
+ "0 mod k = 0"
+ "k div Int.Pos Num.One = k"
+ "k mod Int.Pos Num.One = 0"
+ "k div Int.Neg Num.One = - k"
+ "k mod Int.Neg Num.One = 0"
+ "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
+ "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
+ "Int.Neg m div Int.Pos n = - (adjust_div (divmod m n) :: int)"
+ "Int.Neg m mod Int.Pos n = adjust_mod n (snd (divmod m n) :: int)"
+ "Int.Pos m div Int.Neg n = - (adjust_div (divmod m n) :: int)"
+ "Int.Pos m mod Int.Neg n = - adjust_mod n (snd (divmod m n) :: int)"
+ "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
+ "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
+ by simp_all
+
+end
+
+lemma divmod_BitM_2_eq [simp]:
+ \<open>divmod (Num.BitM m) (Num.Bit0 Num.One) = (numeral m - 1, (1 :: int))\<close>
+ by (cases m) simp_all
+
+
+subsubsection \<open>Computation by simplification\<close>
+
+lemma euclidean_size_nat_less_eq_iff:
+ \<open>euclidean_size m \<le> euclidean_size n \<longleftrightarrow> m \<le> n\<close> for m n :: nat
+ by simp
+
+lemma euclidean_size_int_less_eq_iff:
+ \<open>euclidean_size k \<le> euclidean_size l \<longleftrightarrow> \<bar>k\<bar> \<le> \<bar>l\<bar>\<close> for k l :: int
+ by auto
+
+simproc_setup numeral_divmod
+ ("0 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "0 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "0 div - 1 :: int" | "0 mod - 1 :: int" |
+ "0 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
+ "1 div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "1 div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "1 div - 1 :: int" | "1 mod - 1 :: int" |
+ "1 div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
+ "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
+ "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
+ "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
+ "numeral a div 0 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "numeral a div 1 :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
+ "numeral a div numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_with_nat_division" |
+ "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
+ "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
+ "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
+ "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
+ "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
+ "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") = \<open>
+ let
+ val if_cong = the (Code.get_case_cong \<^theory> \<^const_name>\<open>If\<close>);
+ fun successful_rewrite ctxt ct =
+ let
+ val thm = Simplifier.rewrite ctxt ct
+ in if Thm.is_reflexive thm then NONE else SOME thm end;
+ in fn phi =>
+ let
+ val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
+ one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
+ one_div_minus_numeral one_mod_minus_numeral
+ numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
+ numeral_div_minus_numeral numeral_mod_minus_numeral
+ div_minus_minus mod_minus_minus Euclidean_Division.adjust_div_eq of_bool_eq one_neq_zero
+ numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
+ divmod_cancel divmod_steps divmod_step_def fst_conv snd_conv numeral_One
+ case_prod_beta rel_simps Euclidean_Division.adjust_mod_def div_minus1_right mod_minus1_right
+ minus_minus numeral_times_numeral mult_zero_right mult_1_right
+ euclidean_size_nat_less_eq_iff euclidean_size_int_less_eq_iff diff_nat_numeral nat_numeral}
+ @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
+ fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
+ (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
+ in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
+ end
+\<close> \<comment> \<open>
+ There is space for improvement here: the calculation itself
+ could be carried out outside the logic, and a generic simproc
+ (simplifier setup) for generic calculation would be helpful.
+\<close>
+
+
+subsubsection \<open>Code generation\<close>
+
+context
+begin
+
+qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
+ where "divmod_nat m n = (m div n, m mod n)"
+
+qualified lemma divmod_nat_if [code]:
+ "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
+ let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
+ by (simp add: divmod_nat_def prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
+
+qualified lemma [code]:
+ "m div n = fst (divmod_nat m n)"
+ "m mod n = snd (divmod_nat m n)"
+ by (simp_all add: divmod_nat_def)
+
+end
code_identifier
code_module Euclidean_Division \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
--- a/src/HOL/Library/Code_Abstract_Char.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Library/Code_Abstract_Char.thy Sun Aug 21 06:18:23 2022 +0000
@@ -18,7 +18,7 @@
lemma char_of_integer_code [code]:
\<open>integer_of_char (char_of_integer k) = (if 0 \<le> k \<and> k < 256 then k else k mod 256)\<close>
- by (simp add: integer_of_char_def char_of_integer_def take_bit_eq_mod unique_euclidean_semiring_numeral_class.mod_less)
+ by (simp add: integer_of_char_def char_of_integer_def integer_eq_iff integer_less_eq_iff integer_less_iff)
lemma of_char_code [code]:
\<open>of_char c = of_nat (nat_of_integer (integer_of_char c))\<close>
@@ -104,7 +104,7 @@
then have \<open>(0 :: integer) \<le> of_char c\<close>
by (simp only: of_nat_0 of_nat_of_char)
ultimately show ?thesis
- by (simp add: Let_def integer_of_char_def take_bit_eq_mod unique_euclidean_semiring_numeral_class.mod_less)
+ by (simp add: Let_def integer_of_char_def take_bit_eq_mod integer_eq_iff integer_less_eq_iff integer_less_iff)
next
case False
then have \<open>(128 :: integer) \<le> of_char c\<close>
@@ -117,7 +117,7 @@
then have \<open>of_char c = k + 128\<close>
by simp
ultimately show ?thesis
- by (simp add: Let_def integer_of_char_def take_bit_eq_mod unique_euclidean_semiring_numeral_class.mod_less)
+ by (simp add: Let_def integer_of_char_def take_bit_eq_mod integer_eq_iff integer_less_eq_iff integer_less_iff)
qed
lemma equal_char_code [code]:
--- a/src/HOL/Library/Code_Binary_Nat.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Library/Code_Binary_Nat.thy Sun Aug 21 06:18:23 2022 +0000
@@ -127,13 +127,13 @@
"nat_of_num k < nat_of_num l \<longleftrightarrow> k < l"
by (simp_all add: nat_of_num_numeral)
-declare [[code drop: Divides.divmod_nat]]
+declare [[code drop: Euclidean_Division.divmod_nat]]
lemma divmod_nat_code [code]:
- "Divides.divmod_nat (nat_of_num k) (nat_of_num l) = divmod k l"
- "Divides.divmod_nat m 0 = (0, m)"
- "Divides.divmod_nat 0 n = (0, 0)"
- by (simp_all add: prod_eq_iff nat_of_num_numeral)
+ "Euclidean_Division.divmod_nat (nat_of_num k) (nat_of_num l) = divmod k l"
+ "Euclidean_Division.divmod_nat m 0 = (0, m)"
+ "Euclidean_Division.divmod_nat 0 n = (0, 0)"
+ by (simp_all add: Euclidean_Division.divmod_nat_def nat_of_num_numeral)
end
--- a/src/HOL/Library/Code_Target_Nat.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Library/Code_Target_Nat.thy Sun Aug 21 06:18:23 2022 +0000
@@ -98,13 +98,13 @@
begin
lemma divmod_nat_code [code]: \<^marker>\<open>contributor \<open>René Thiemann\<close>\<close> \<^marker>\<open>contributor \<open>Akihisa Yamada\<close>\<close>
- "Divides.divmod_nat m n = (
+ "Euclidean_Division.divmod_nat m n = (
let k = integer_of_nat m; l = integer_of_nat n
in map_prod nat_of_integer nat_of_integer
(if k = 0 then (0, 0)
else if l = 0 then (0, k) else
Code_Numeral.divmod_abs k l))"
- by (simp add: prod_eq_iff Let_def; transfer)
+ by (simp add: prod_eq_iff Let_def Euclidean_Division.divmod_nat_def; transfer)
(simp add: nat_div_distrib nat_mod_distrib)
end
@@ -136,15 +136,12 @@
lemma (in semiring_1) of_nat_code_if:
"of_nat n = (if n = 0 then 0
else let
- (m, q) = Divides.divmod_nat n 2;
+ (m, q) = Euclidean_Division.divmod_nat n 2;
m' = 2 * of_nat m
in if q = 0 then m' else m' + 1)"
-proof -
- from div_mult_mod_eq have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
- show ?thesis
- by (simp add: Let_def divmod_nat_def of_nat_add [symmetric])
- (simp add: * mult.commute of_nat_mult add.commute)
-qed
+ by (cases n)
+ (simp_all add: Let_def Euclidean_Division.divmod_nat_def ac_simps
+ flip: of_nat_numeral of_nat_mult minus_mod_eq_mult_div)
declare of_nat_code_if [code]
--- a/src/HOL/Library/RBT_Impl.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Library/RBT_Impl.thy Sun Aug 21 06:18:23 2022 +0000
@@ -1154,24 +1154,24 @@
else if n = 1 then
case kvs of (k, v) # kvs' \<Rightarrow>
(Branch R Empty k v Empty, kvs')
- else let (n', r) = Divides.divmod_nat n 2 in
+ else let (n', r) = Euclidean_Division.divmod_nat n 2 in
if r = 0 then
case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
apfst (Branch B t1 k v) (rbtreeify_f n' kvs'))"
-by (subst rbtreeify_f.simps) (simp only: Let_def divmod_nat_def prod.case)
+by (subst rbtreeify_f.simps) (simp only: Let_def Euclidean_Division.divmod_nat_def prod.case)
lemma rbtreeify_g_code [code]:
"rbtreeify_g n kvs =
(if n = 0 \<or> n = 1 then (Empty, kvs)
- else let (n', r) = Divides.divmod_nat n 2 in
+ else let (n', r) = Euclidean_Division.divmod_nat n 2 in
if r = 0 then
case rbtreeify_g n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
apfst (Branch B t1 k v) (rbtreeify_g n' kvs'))"
-by(subst rbtreeify_g.simps)(simp only: Let_def divmod_nat_def prod.case)
+by(subst rbtreeify_g.simps)(simp only: Let_def Euclidean_Division.divmod_nat_def prod.case)
lemma Suc_double_half: "Suc (2 * n) div 2 = n"
by simp
--- a/src/HOL/Matrix_LP/ComputeNumeral.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Matrix_LP/ComputeNumeral.thy Sun Aug 21 06:18:23 2022 +0000
@@ -51,10 +51,10 @@
one_div_minus_numeral one_mod_minus_numeral
numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
numeral_div_minus_numeral numeral_mod_minus_numeral
- div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
+ div_minus_minus mod_minus_minus Euclidean_Division.adjust_div_eq of_bool_eq one_neq_zero
numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
- divmod_steps divmod_cancel divmod_step_eq fst_conv snd_conv numeral_One
- case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
+ divmod_steps divmod_cancel divmod_step_def fst_conv snd_conv numeral_One
+ case_prod_beta rel_simps Euclidean_Division.adjust_mod_def div_minus1_right mod_minus1_right
minus_minus numeral_times_numeral mult_zero_right mult_1_right
--- a/src/HOL/Parity.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/Parity.thy Sun Aug 21 06:18:23 2022 +0000
@@ -669,6 +669,44 @@
end
+
+subsection \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
+
+context unique_euclidean_semiring_with_nat_division
+begin
+
+lemma cong_exp_iff_simps:
+ "numeral n mod numeral Num.One = 0
+ \<longleftrightarrow> True"
+ "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
+ \<longleftrightarrow> numeral n mod numeral q = 0"
+ "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
+ \<longleftrightarrow> False"
+ "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
+ \<longleftrightarrow> True"
+ "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> True"
+ "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> False"
+ "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> (numeral n mod numeral q) = 0"
+ "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> False"
+ "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
+ "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> False"
+ "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> (numeral m mod numeral q) = 0"
+ "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> False"
+ "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
+ \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
+ by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
+
+end
+
+
code_identifier
code_module Parity \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
--- a/src/HOL/ROOT Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/ROOT Sun Aug 21 06:18:23 2022 +0000
@@ -74,6 +74,7 @@
Datatype_Records
(*data refinements and dependent applications*)
AList_Mapping
+ Code_Abstract_Char
Code_Binary_Nat
Code_Prolog
Code_Real_Approx_By_Float
--- a/src/HOL/ex/Parallel_Example.thy Sun Aug 21 06:18:23 2022 +0000
+++ b/src/HOL/ex/Parallel_Example.thy Sun Aug 21 06:18:23 2022 +0000
@@ -41,11 +41,11 @@
proof -
fix ps qs q
assume "dropWhile Not ps = q # qs"
- then have "length (q # qs) = length (dropWhile Not ps)" by simp
- then have "length qs < length (dropWhile Not ps)" by simp
- moreover have "length (dropWhile Not ps) \<le> length ps"
+ then have "length qs < length (dropWhile Not ps)"
+ by simp
+ also have "length (dropWhile Not ps) \<le> length ps"
by (simp add: length_dropWhile_le)
- ultimately show "length qs < length ps" by auto
+ finally show "length qs < length ps" .
qed
primrec natify :: "nat \<Rightarrow> bool list \<Rightarrow> nat list" where
@@ -61,7 +61,7 @@
function factorise_from :: "nat \<Rightarrow> nat \<Rightarrow> nat list" where
"factorise_from k n = (if 1 < k \<and> k \<le> n
then
- let (q, r) = Divides.divmod_nat n k
+ let (q, r) = Euclidean_Division.divmod_nat n k
in if r = 0 then k # factorise_from k q
else factorise_from (Suc k) n
else [])"
@@ -69,9 +69,11 @@
termination factorise_from \<comment> \<open>tuning of this proof is left as an exercise to the reader\<close>
apply (relation "measure (\<lambda>(k, n). 2 * n - k)")
- apply (auto simp add: prod_eq_iff algebra_simps elim!: dvdE)
- apply (case_tac "k \<le> ka * 2")
- apply (auto intro: diff_less_mono)
+ apply (auto simp add: Euclidean_Division.divmod_nat_def algebra_simps elim!: dvdE)
+ subgoal for m n
+ apply (cases "m \<le> n * 2")
+ apply (auto intro: diff_less_mono)
+ done
done
definition factorise :: "nat \<Rightarrow> nat list" where