--- a/src/HOL/Euclidean_Division.thy Fri Oct 28 06:34:25 2022 +0000
+++ b/src/HOL/Euclidean_Division.thy Fri Oct 28 06:34:26 2022 +0000
@@ -844,7 +844,7 @@
end
-subsection \<open>Euclidean division on \<^typ>\<open>nat\<close>\<close>
+subsection \<open>Division on \<^typ>\<open>nat\<close>\<close>
instantiation nat :: normalization_semidom
begin
@@ -1652,7 +1652,7 @@
-subsection \<open>Elementary euclidean division on \<^typ>\<open>int\<close>\<close>
+subsection \<open>Division on \<^typ>\<open>int\<close>\<close>
subsubsection \<open>Basic instantiation\<close>
@@ -1956,286 +1956,6 @@
qed
-subsection \<open>Special case: euclidean rings containing the natural numbers\<close>
-
-class unique_euclidean_semiring_with_nat = semidom + semiring_char_0 + unique_euclidean_semiring +
- assumes of_nat_div: "of_nat (m div n) = of_nat m div of_nat n"
- and division_segment_of_nat [simp]: "division_segment (of_nat n) = 1"
- and division_segment_euclidean_size [simp]: "division_segment a * of_nat (euclidean_size a) = a"
-begin
-
-lemma division_segment_eq_iff:
- "a = b" if "division_segment a = division_segment b"
- and "euclidean_size a = euclidean_size b"
- using that division_segment_euclidean_size [of a] by simp
-
-lemma euclidean_size_of_nat [simp]:
- "euclidean_size (of_nat n) = n"
-proof -
- have "division_segment (of_nat n) * of_nat (euclidean_size (of_nat n)) = of_nat n"
- by (fact division_segment_euclidean_size)
- then show ?thesis by simp
-qed
-
-lemma of_nat_euclidean_size:
- "of_nat (euclidean_size a) = a div division_segment a"
-proof -
- have "of_nat (euclidean_size a) = division_segment a * of_nat (euclidean_size a) div division_segment a"
- by (subst nonzero_mult_div_cancel_left) simp_all
- also have "\<dots> = a div division_segment a"
- by simp
- finally show ?thesis .
-qed
-
-lemma division_segment_1 [simp]:
- "division_segment 1 = 1"
- using division_segment_of_nat [of 1] by simp
-
-lemma division_segment_numeral [simp]:
- "division_segment (numeral k) = 1"
- using division_segment_of_nat [of "numeral k"] by simp
-
-lemma euclidean_size_1 [simp]:
- "euclidean_size 1 = 1"
- using euclidean_size_of_nat [of 1] by simp
-
-lemma euclidean_size_numeral [simp]:
- "euclidean_size (numeral k) = numeral k"
- using euclidean_size_of_nat [of "numeral k"] by simp
-
-lemma of_nat_dvd_iff:
- "of_nat m dvd of_nat n \<longleftrightarrow> m dvd n" (is "?P \<longleftrightarrow> ?Q")
-proof (cases "m = 0")
- case True
- then show ?thesis
- by simp
-next
- case False
- show ?thesis
- proof
- assume ?Q
- then show ?P
- by auto
- next
- assume ?P
- with False have "of_nat n = of_nat n div of_nat m * of_nat m"
- by simp
- then have "of_nat n = of_nat (n div m * m)"
- by (simp add: of_nat_div)
- then have "n = n div m * m"
- by (simp only: of_nat_eq_iff)
- then have "n = m * (n div m)"
- by (simp add: ac_simps)
- then show ?Q ..
- qed
-qed
-
-lemma of_nat_mod:
- "of_nat (m mod n) = of_nat m mod of_nat n"
-proof -
- have "of_nat m div of_nat n * of_nat n + of_nat m mod of_nat n = of_nat m"
- by (simp add: div_mult_mod_eq)
- also have "of_nat m = of_nat (m div n * n + m mod n)"
- by simp
- finally show ?thesis
- by (simp only: of_nat_div of_nat_mult of_nat_add) simp
-qed
-
-lemma one_div_two_eq_zero [simp]:
- "1 div 2 = 0"
-proof -
- from of_nat_div [symmetric] have "of_nat 1 div of_nat 2 = of_nat 0"
- by (simp only:) simp
- then show ?thesis
- by simp
-qed
-
-lemma one_mod_two_eq_one [simp]:
- "1 mod 2 = 1"
-proof -
- from of_nat_mod [symmetric] have "of_nat 1 mod of_nat 2 = of_nat 1"
- by (simp only:) simp
- then show ?thesis
- by simp
-qed
-
-lemma one_mod_2_pow_eq [simp]:
- "1 mod (2 ^ n) = of_bool (n > 0)"
-proof -
- have "1 mod (2 ^ n) = of_nat (1 mod (2 ^ n))"
- using of_nat_mod [of 1 "2 ^ n"] by simp
- also have "\<dots> = of_bool (n > 0)"
- by simp
- finally show ?thesis .
-qed
-
-lemma one_div_2_pow_eq [simp]:
- "1 div (2 ^ n) = of_bool (n = 0)"
- using div_mult_mod_eq [of 1 "2 ^ n"] by auto
-
-lemma div_mult2_eq':
- \<open>a div (of_nat m * of_nat n) = a div of_nat m div of_nat n\<close>
-proof (cases \<open>m = 0 \<or> n = 0\<close>)
- case True
- then show ?thesis
- by auto
-next
- case False
- then have \<open>m > 0\<close> \<open>n > 0\<close>
- by simp_all
- show ?thesis
- proof (cases \<open>of_nat m * of_nat n dvd a\<close>)
- case True
- then obtain b where \<open>a = (of_nat m * of_nat n) * b\<close> ..
- then have \<open>a = of_nat m * (of_nat n * b)\<close>
- by (simp add: ac_simps)
- then show ?thesis
- by simp
- next
- case False
- define q where \<open>q = a div (of_nat m * of_nat n)\<close>
- define r where \<open>r = a mod (of_nat m * of_nat n)\<close>
- from \<open>m > 0\<close> \<open>n > 0\<close> \<open>\<not> of_nat m * of_nat n dvd a\<close> r_def have "division_segment r = 1"
- using division_segment_of_nat [of "m * n"] by (simp add: division_segment_mod)
- with division_segment_euclidean_size [of r]
- have "of_nat (euclidean_size r) = r"
- by simp
- have "a mod (of_nat m * of_nat n) div (of_nat m * of_nat n) = 0"
- by simp
- with \<open>m > 0\<close> \<open>n > 0\<close> r_def have "r div (of_nat m * of_nat n) = 0"
- by simp
- with \<open>of_nat (euclidean_size r) = r\<close>
- have "of_nat (euclidean_size r) div (of_nat m * of_nat n) = 0"
- by simp
- then have "of_nat (euclidean_size r div (m * n)) = 0"
- by (simp add: of_nat_div)
- then have "of_nat (euclidean_size r div m div n) = 0"
- by (simp add: div_mult2_eq)
- with \<open>of_nat (euclidean_size r) = r\<close> have "r div of_nat m div of_nat n = 0"
- by (simp add: of_nat_div)
- with \<open>m > 0\<close> \<open>n > 0\<close> q_def
- have "q = (r div of_nat m + q * of_nat n * of_nat m div of_nat m) div of_nat n"
- by simp
- moreover have \<open>a = q * (of_nat m * of_nat n) + r\<close>
- by (simp add: q_def r_def div_mult_mod_eq)
- ultimately show \<open>a div (of_nat m * of_nat n) = a div of_nat m div of_nat n\<close>
- using q_def [symmetric] div_plus_div_distrib_dvd_right [of \<open>of_nat m\<close> \<open>q * (of_nat m * of_nat n)\<close> r]
- by (simp add: ac_simps)
- qed
-qed
-
-lemma mod_mult2_eq':
- "a mod (of_nat m * of_nat n) = of_nat m * (a div of_nat m mod of_nat n) + a mod of_nat m"
-proof -
- have "a div (of_nat m * of_nat n) * (of_nat m * of_nat n) + a mod (of_nat m * of_nat n) = a div of_nat m div of_nat n * of_nat n * of_nat m + (a div of_nat m mod of_nat n * of_nat m + a mod of_nat m)"
- by (simp add: combine_common_factor div_mult_mod_eq)
- moreover have "a div of_nat m div of_nat n * of_nat n * of_nat m = of_nat n * of_nat m * (a div of_nat m div of_nat n)"
- by (simp add: ac_simps)
- ultimately show ?thesis
- by (simp add: div_mult2_eq' mult_commute)
-qed
-
-lemma div_mult2_numeral_eq:
- "a div numeral k div numeral l = a div numeral (k * l)" (is "?A = ?B")
-proof -
- have "?A = a div of_nat (numeral k) div of_nat (numeral l)"
- by simp
- also have "\<dots> = a div (of_nat (numeral k) * of_nat (numeral l))"
- by (fact div_mult2_eq' [symmetric])
- also have "\<dots> = ?B"
- by simp
- finally show ?thesis .
-qed
-
-lemma numeral_Bit0_div_2:
- "numeral (num.Bit0 n) div 2 = numeral n"
-proof -
- have "numeral (num.Bit0 n) = numeral n + numeral n"
- by (simp only: numeral.simps)
- also have "\<dots> = numeral n * 2"
- by (simp add: mult_2_right)
- finally have "numeral (num.Bit0 n) div 2 = numeral n * 2 div 2"
- by simp
- also have "\<dots> = numeral n"
- by (rule nonzero_mult_div_cancel_right) simp
- finally show ?thesis .
-qed
-
-lemma numeral_Bit1_div_2:
- "numeral (num.Bit1 n) div 2 = numeral n"
-proof -
- have "numeral (num.Bit1 n) = numeral n + numeral n + 1"
- by (simp only: numeral.simps)
- also have "\<dots> = numeral n * 2 + 1"
- by (simp add: mult_2_right)
- finally have "numeral (num.Bit1 n) div 2 = (numeral n * 2 + 1) div 2"
- by simp
- also have "\<dots> = numeral n * 2 div 2 + 1 div 2"
- using dvd_triv_right by (rule div_plus_div_distrib_dvd_left)
- also have "\<dots> = numeral n * 2 div 2"
- by simp
- also have "\<dots> = numeral n"
- by (rule nonzero_mult_div_cancel_right) simp
- finally show ?thesis .
-qed
-
-lemma exp_mod_exp:
- \<open>2 ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close>
-proof -
- have \<open>(2::nat) ^ m mod 2 ^ n = of_bool (m < n) * 2 ^ m\<close> (is \<open>?lhs = ?rhs\<close>)
- by (auto simp add: not_less monoid_mult_class.power_add dest!: le_Suc_ex)
- then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
- by simp
- then show ?thesis
- by (simp add: of_nat_mod)
-qed
-
-lemma mask_mod_exp:
- \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - 1\<close>
-proof -
- have \<open>(2 ^ n - 1) mod 2 ^ m = 2 ^ min m n - (1::nat)\<close> (is \<open>?lhs = ?rhs\<close>)
- proof (cases \<open>n \<le> m\<close>)
- case True
- then show ?thesis
- by (simp add: Suc_le_lessD)
- next
- case False
- then have \<open>m < n\<close>
- by simp
- then obtain q where n: \<open>n = Suc q + m\<close>
- by (auto dest: less_imp_Suc_add)
- then have \<open>min m n = m\<close>
- by simp
- moreover have \<open>(2::nat) ^ m \<le> 2 * 2 ^ q * 2 ^ m\<close>
- using mult_le_mono1 [of 1 \<open>2 * 2 ^ q\<close> \<open>2 ^ m\<close>] by simp
- with n have \<open>2 ^ n - 1 = (2 ^ Suc q - 1) * 2 ^ m + (2 ^ m - (1::nat))\<close>
- by (simp add: monoid_mult_class.power_add algebra_simps)
- ultimately show ?thesis
- by (simp only: euclidean_semiring_cancel_class.mod_mult_self3) simp
- qed
- then have \<open>of_nat ?lhs = of_nat ?rhs\<close>
- by simp
- then show ?thesis
- by (simp add: of_nat_mod of_nat_diff)
-qed
-
-lemma of_bool_half_eq_0 [simp]:
- \<open>of_bool b div 2 = 0\<close>
- by simp
-
-end
-
-class unique_euclidean_ring_with_nat = ring + unique_euclidean_semiring_with_nat
-
-instance nat :: unique_euclidean_semiring_with_nat
- by standard (simp_all add: dvd_eq_mod_eq_0)
-
-instance int :: unique_euclidean_ring_with_nat
- by standard (auto simp add: divide_int_def division_segment_int_def elim: contrapos_np)
-
-
-subsection \<open>More on euclidean division on \<^typ>\<open>int\<close>\<close>
-
subsubsection \<open>Trivial reduction steps\<close>
lemma div_pos_pos_trivial [simp]:
@@ -2478,16 +2198,54 @@
subsubsection \<open>Algebraic rewrites\<close>
-lemma zdiv_zmult2_eq:
- \<open>a div (b * c) = (a div b) div c\<close> if \<open>c \<ge> 0\<close> for a b c :: int
-proof (cases \<open>b \<ge> 0\<close>)
- case True
- with that show ?thesis
- using div_mult2_eq' [of a \<open>nat b\<close> \<open>nat c\<close>] by simp
-next
- case False
- with that show ?thesis
- using div_mult2_eq' [of \<open>- a\<close> \<open>nat (- b)\<close> \<open>nat c\<close>] by simp
+lemma zdiv_zmult2_eq: \<open>a div (b * c) = (a div b) div c\<close> (is ?Q)
+ and zmod_zmult2_eq: \<open>a mod (b * c) = b * (a div b mod c) + a mod b\<close> (is ?P)
+ if \<open>c \<ge> 0\<close> for a b c :: int
+proof -
+ have *: \<open>(a div (b * c), a mod (b * c)) = ((a div b) div c, b * (a div b mod c) + a mod b)\<close>
+ if \<open>b > 0\<close> for a b
+ proof (induction rule: euclidean_relationI)
+ case by0
+ then show ?case by auto
+ next
+ case divides
+ then obtain d where \<open>a = b * c * d\<close>
+ by blast
+ with divides that show ?case
+ by (simp add: ac_simps)
+ next
+ case euclidean_relation
+ with \<open>b > 0\<close> \<open>c \<ge> 0\<close> have \<open>0 < c\<close> \<open>b > 0\<close>
+ by simp_all
+ then have \<open>a mod b < b\<close>
+ by simp
+ moreover have \<open>1 \<le> c - a div b mod c\<close>
+ using \<open>c > 0\<close> by (simp add: int_one_le_iff_zero_less)
+ ultimately have \<open>a mod b * 1 < b * (c - a div b mod c)\<close>
+ by (rule mult_less_le_imp_less) (use \<open>b > 0\<close> in simp_all)
+ with \<open>0 < b\<close> \<open>0 < c\<close> show ?case
+ by (simp add: division_segment_int_def algebra_simps flip: minus_mod_eq_mult_div)
+ qed
+ show ?Q
+ proof (cases \<open>b \<ge> 0\<close>)
+ case True
+ with * [of b a] show ?thesis
+ by (cases \<open>b = 0\<close>) simp_all
+ next
+ case False
+ with * [of \<open>- b\<close> \<open>- a\<close>] show ?thesis
+ by simp
+ qed
+ show ?P
+ proof (cases \<open>b \<ge> 0\<close>)
+ case True
+ with * [of b a] show ?thesis
+ by (cases \<open>b = 0\<close>) simp_all
+ next
+ case False
+ with * [of \<open>- b\<close> \<open>- a\<close>] show ?thesis
+ by simp
+ qed
qed
lemma zdiv_zmult2_eq':
@@ -2502,18 +2260,6 @@
finally show ?thesis .
qed
-lemma zmod_zmult2_eq:
- \<open>a mod (b * c) = b * (a div b mod c) + a mod b\<close> if \<open>c \<ge> 0\<close> for a b c :: int
-proof (cases \<open>b \<ge> 0\<close>)
- case True
- with that show ?thesis
- using mod_mult2_eq' [of a \<open>nat b\<close> \<open>nat c\<close>] by simp
-next
- case False
- with that show ?thesis
- using mod_mult2_eq' [of \<open>- a\<close> \<open>nat (- b)\<close> \<open>nat c\<close>] by simp
-qed
-
lemma half_nonnegative_int_iff [simp]:
\<open>k div 2 \<ge> 0 \<longleftrightarrow> k \<ge> 0\<close> for k :: int
by auto
@@ -2526,12 +2272,12 @@
subsubsection \<open>Distributive laws for conversions.\<close>
lemma zdiv_int:
- "int (a div b) = int a div int b"
- by (fact of_nat_div)
+ \<open>int (m div n) = int m div int n\<close>
+ by (cases \<open>m = 0\<close>) (auto simp add: divide_int_def)
lemma zmod_int:
- "int (a mod b) = int a mod int b"
- by (fact of_nat_mod)
+ \<open>int (m mod n) = int m mod int n\<close>
+ by (cases \<open>m = 0\<close>) (auto simp add: modulo_int_def)
lemma nat_div_distrib:
\<open>nat (x div y) = nat x div nat y\<close> if \<open>0 \<le> x\<close>
@@ -2961,384 +2707,7 @@
by (rule pos_zmod_mult_2) simp
-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>
+subsection \<open>Code generation\<close>
context
begin