--- a/src/HOL/Word/Bit_Representation.thy Tue Apr 16 19:50:03 2019 +0000
+++ b/src/HOL/Word/Bit_Representation.thy Tue Apr 16 19:50:05 2019 +0000
@@ -5,9 +5,37 @@
section \<open>Integers as implicit bit strings\<close>
theory Bit_Representation
- imports Misc_Numeric
+ imports Main
begin
+lemma int_mod_lem: "0 < n \<Longrightarrow> 0 \<le> b \<and> b < n \<longleftrightarrow> b mod n = b"
+ for b n :: int
+ apply safe
+ apply (erule (1) mod_pos_pos_trivial)
+ apply (erule_tac [!] subst)
+ apply auto
+ done
+
+lemma int_mod_ge: "a < n \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a mod n"
+ for a n :: int
+ by (metis dual_order.trans le_cases mod_pos_pos_trivial pos_mod_conj)
+
+lemma int_mod_ge': "b < 0 \<Longrightarrow> 0 < n \<Longrightarrow> b + n \<le> b mod n"
+ for b n :: int
+ by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)
+
+lemma int_mod_le': "0 \<le> b - n \<Longrightarrow> b mod n \<le> b - n"
+ for b n :: int
+ by (metis minus_mod_self2 zmod_le_nonneg_dividend)
+
+lemma emep1: "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (n + 1) mod d = (n mod d) + 1"
+ for n d :: int
+ by (auto simp add: pos_zmod_mult_2 add.commute dvd_def)
+
+lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
+ by (rule zmod_minus1) simp
+
+
subsection \<open>Constructors and destructors for binary integers\<close>
definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int" (infixl "BIT" 90)
@@ -563,7 +591,9 @@
apply (unfold no_bintr_alt1)
apply (auto simp add: image_iff)
apply (rule exI)
- apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
+ apply (rule sym)
+ using int_mod_lem [symmetric, of "2 ^ n"]
+ apply auto
done
lemma no_sbintr_alt2: "sbintrunc n = (\<lambda>w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
@@ -572,15 +602,15 @@
lemma range_sbintrunc: "range (sbintrunc n) = {i. - (2 ^ n) \<le> i \<and> i < 2 ^ n}"
apply (unfold no_sbintr_alt2)
apply (auto simp add: image_iff eq_diff_eq)
+
apply (rule exI)
apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
done
lemma sb_inc_lem: "a + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
for a :: int
- apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p])
- apply (rule TrueI)
- done
+ using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
+ by simp
lemma sb_inc_lem': "a < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) \<le> (a + 2^k) mod 2^(Suc k)"
for a :: int
@@ -600,8 +630,6 @@
lemma sbintrunc_dec: "x \<ge> (2 ^ n) \<Longrightarrow> x - 2 ^ (Suc n) >= sbintrunc n x"
unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp
-lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]
-
lemma bintr_ge0: "0 \<le> bintrunc n w"
by (simp add: bintrunc_mod2p)
@@ -815,7 +843,7 @@
apply (simp add: bin_rest_def zdiv_zmult2_eq)
apply (case_tac b rule: bin_exhaust)
apply simp
- apply (simp add: Bit_def mod_mult_mult1 p1mod22k)
+ apply (simp add: Bit_def mod_mult_mult1 pos_zmod_mult_2 add.commute)
done
end
--- a/src/HOL/Word/Bits_Int.thy Tue Apr 16 19:50:03 2019 +0000
+++ b/src/HOL/Word/Bits_Int.thy Tue Apr 16 19:50:05 2019 +0000
@@ -370,7 +370,7 @@
have "2 * (bin mod 2 ^ n) + 1 = (2 * bin mod 2 ^ Suc n) + 1"
by (simp add: mod_mult_mult1)
also have "\<dots> = ((2 * bin mod 2 ^ Suc n) + 1) mod 2 ^ Suc n"
- by (simp add: ac_simps p1mod22k')
+ by (simp add: ac_simps pos_zmod_mult_2)
also have "\<dots> = (2 * bin + 1) mod 2 ^ Suc n"
by (simp only: mod_simps)
finally show ?thesis
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Word/Misc_Arithmetic.thy Tue Apr 16 19:50:05 2019 +0000
@@ -0,0 +1,433 @@
+(* Title: HOL/Word/Misc_Arithmetic.thy *)
+
+section \<open>Miscellaneous lemmas, mostly for arithmetic\<close>
+
+theory Misc_Arithmetic
+ imports "HOL-Library.Bit" Bit_Representation
+begin
+
+lemma one_mod_exp_eq_one [simp]:
+ "1 mod (2 * 2 ^ n) = (1::int)"
+ using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)
+
+lemma mod_2_neq_1_eq_eq_0: "k mod 2 \<noteq> 1 \<longleftrightarrow> k mod 2 = 0"
+ for k :: int
+ by (fact not_mod_2_eq_1_eq_0)
+
+lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
+ for b :: int
+ by arith
+
+lemma diff_le_eq': "a - b \<le> c \<longleftrightarrow> a \<le> b + c"
+ for a b c :: int
+ by arith
+
+lemma zless2: "0 < (2 :: int)"
+ by (fact zero_less_numeral)
+
+lemma zless2p: "0 < (2 ^ n :: int)"
+ by arith
+
+lemma zle2p: "0 \<le> (2 ^ n :: int)"
+ by arith
+
+lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
+ for b :: int
+ using zle2p by (rule pos_zmod_mult_2)
+
+lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
+ for b :: int
+ by (simp add: p1mod22k' add.commute)
+
+lemma ex_eq_or: "(\<exists>m. n = Suc m \<and> (m = k \<or> P m)) \<longleftrightarrow> n = Suc k \<or> (\<exists>m. n = Suc m \<and> P m)"
+ by auto
+
+lemma power_minus_simp: "0 < n \<Longrightarrow> a ^ n = a * a ^ (n - 1)"
+ by (auto dest: gr0_implies_Suc)
+
+lemma funpow_minus_simp: "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
+ by (auto dest: gr0_implies_Suc)
+
+lemma power_numeral: "a ^ numeral k = a * a ^ (pred_numeral k)"
+ by (simp add: numeral_eq_Suc)
+
+lemma funpow_numeral [simp]: "f ^^ numeral k = f \<circ> f ^^ (pred_numeral k)"
+ by (simp add: numeral_eq_Suc)
+
+lemma replicate_numeral [simp]: "replicate (numeral k) x = x # replicate (pred_numeral k) x"
+ by (simp add: numeral_eq_Suc)
+
+lemma rco_alt: "(f \<circ> g) ^^ n \<circ> f = f \<circ> (g \<circ> f) ^^ n"
+ apply (rule ext)
+ apply (induct n)
+ apply (simp_all add: o_def)
+ done
+
+lemma list_exhaust_size_gt0:
+ assumes "\<And>a list. y = a # list \<Longrightarrow> P"
+ shows "0 < length y \<Longrightarrow> P"
+ apply (cases y)
+ apply simp
+ apply (rule assms)
+ apply fastforce
+ done
+
+lemma list_exhaust_size_eq0:
+ assumes "y = [] \<Longrightarrow> P"
+ shows "length y = 0 \<Longrightarrow> P"
+ apply (cases y)
+ apply (rule assms)
+ apply simp
+ apply simp
+ done
+
+lemma size_Cons_lem_eq: "y = xa # list \<Longrightarrow> size y = Suc k \<Longrightarrow> size list = k"
+ by auto
+
+lemmas ls_splits = prod.split prod.split_asm if_split_asm
+
+\<comment> \<open>simplifications for specific word lengths\<close>
+lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'
+
+lemmas s2n_ths = n2s_ths [symmetric]
+
+lemma and_len: "xs = ys \<Longrightarrow> xs = ys \<and> length xs = length ys"
+ by auto
+
+lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
+ by auto
+
+lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
+ by auto
+
+lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
+ by auto
+
+lemma if_Not_x: "(if p then \<not> x else x) = (p = (\<not> x))"
+ by auto
+
+lemma if_x_Not: "(if p then x else \<not> x) = (p = x)"
+ by auto
+
+lemma if_same_and: "(If p x y \<and> If p u v) = (if p then x \<and> u else y \<and> v)"
+ by auto
+
+lemma if_same_eq: "(If p x y = (If p u v)) = (if p then x = u else y = v)"
+ by auto
+
+lemma if_same_eq_not: "(If p x y = (\<not> If p u v)) = (if p then x = (\<not> u) else y = (\<not> v))"
+ by auto
+
+\<comment> \<open>note -- \<open>if_Cons\<close> can cause blowup in the size, if \<open>p\<close> is complex, so make a simproc\<close>
+lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
+ by auto
+
+lemma if_single: "(if xc then [xab] else [an]) = [if xc then xab else an]"
+ by auto
+
+lemma if_bool_simps:
+ "If p True y = (p \<or> y) \<and> If p False y = (\<not> p \<and> y) \<and>
+ If p y True = (p \<longrightarrow> y) \<and> If p y False = (p \<and> y)"
+ by auto
+
+lemmas if_simps =
+ if_x_Not if_Not_x if_cancel if_True if_False if_bool_simps
+
+lemmas seqr = eq_reflection [where x = "size w"] for w (* FIXME: delete *)
+
+lemma the_elemI: "y = {x} \<Longrightarrow> the_elem y = x"
+ by simp
+
+lemma nonemptyE: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> R) \<Longrightarrow> R"
+ by auto
+
+lemma gt_or_eq_0: "0 < y \<or> 0 = y"
+ for y :: nat
+ by arith
+
+lemmas xtr1 = xtrans(1)
+lemmas xtr2 = xtrans(2)
+lemmas xtr3 = xtrans(3)
+lemmas xtr4 = xtrans(4)
+lemmas xtr5 = xtrans(5)
+lemmas xtr6 = xtrans(6)
+lemmas xtr7 = xtrans(7)
+lemmas xtr8 = xtrans(8)
+
+lemmas nat_simps = diff_add_inverse2 diff_add_inverse
+lemmas nat_iffs = le_add1 le_add2
+
+lemma sum_imp_diff: "j = k + i \<Longrightarrow> j - i = k"
+ for k :: nat
+ by arith
+
+lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]]
+lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]]
+
+lemma nmod2: "n mod 2 = 0 \<or> n mod 2 = 1"
+ for n :: int
+ by arith
+
+lemmas eme1p = emep1 [simplified add.commute]
+
+lemma le_diff_eq': "a \<le> c - b \<longleftrightarrow> b + a \<le> c"
+ for a b c :: int
+ by arith
+
+lemma less_diff_eq': "a < c - b \<longleftrightarrow> b + a < c"
+ for a b c :: int
+ by arith
+
+lemma diff_less_eq': "a - b < c \<longleftrightarrow> a < b + c"
+ for a b c :: int
+ by arith
+
+lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]
+
+lemma z1pdiv2: "(2 * b + 1) div 2 = b"
+ for b :: int
+ by arith
+
+lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
+ simplified int_one_le_iff_zero_less, simplified]
+
+lemma axxbyy: "a + m + m = b + n + n \<Longrightarrow> a = 0 \<or> a = 1 \<Longrightarrow> b = 0 \<or> b = 1 \<Longrightarrow> a = b \<and> m = n"
+ for a b m n :: int
+ by arith
+
+lemma axxmod2: "(1 + x + x) mod 2 = 1 \<and> (0 + x + x) mod 2 = 0"
+ for x :: int
+ by arith
+
+lemma axxdiv2: "(1 + x + x) div 2 = x \<and> (0 + x + x) div 2 = x"
+ for x :: int
+ by arith
+
+lemmas iszero_minus =
+ trans [THEN trans, OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric]]
+
+lemmas zadd_diff_inverse =
+ trans [OF diff_add_cancel [symmetric] add.commute]
+
+lemmas add_diff_cancel2 =
+ add.commute [THEN diff_eq_eq [THEN iffD2]]
+
+lemmas rdmods [symmetric] = mod_minus_eq
+ mod_diff_left_eq mod_diff_right_eq mod_add_left_eq
+ mod_add_right_eq mod_mult_right_eq mod_mult_left_eq
+
+lemma mod_plus_right: "(a + x) mod m = (b + x) mod m \<longleftrightarrow> a mod m = b mod m"
+ for a b m x :: nat
+ by (induct x) (simp_all add: mod_Suc, arith)
+
+lemma nat_minus_mod: "(n - n mod m) mod m = 0"
+ for m n :: nat
+ by (induct n) (simp_all add: mod_Suc)
+
+lemmas nat_minus_mod_plus_right =
+ trans [OF nat_minus_mod mod_0 [symmetric],
+ THEN mod_plus_right [THEN iffD2], simplified]
+
+lemmas push_mods' = mod_add_eq
+ mod_mult_eq mod_diff_eq
+ mod_minus_eq
+
+lemmas push_mods = push_mods' [THEN eq_reflection]
+lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]
+
+lemma nat_mod_eq: "b < n \<Longrightarrow> a mod n = b mod n \<Longrightarrow> a mod n = b"
+ for a b n :: nat
+ by (induct a) auto
+
+lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
+
+lemma nat_mod_lem: "0 < n \<Longrightarrow> b < n \<longleftrightarrow> b mod n = b"
+ for b n :: nat
+ apply safe
+ apply (erule nat_mod_eq')
+ apply (erule subst)
+ apply (erule mod_less_divisor)
+ done
+
+lemma mod_nat_add: "x < z \<Longrightarrow> y < z \<Longrightarrow> (x + y) mod z = (if x + y < z then x + y else x + y - z)"
+ for x y z :: nat
+ apply (rule nat_mod_eq)
+ apply auto
+ apply (rule trans)
+ apply (rule le_mod_geq)
+ apply simp
+ apply (rule nat_mod_eq')
+ apply arith
+ done
+
+lemma mod_nat_sub: "x < z \<Longrightarrow> (x - y) mod z = x - y"
+ for x y :: nat
+ by (rule nat_mod_eq') arith
+
+lemma int_mod_eq: "0 \<le> b \<Longrightarrow> b < n \<Longrightarrow> a mod n = b mod n \<Longrightarrow> a mod n = b"
+ for a b n :: int
+ by (metis mod_pos_pos_trivial)
+
+lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)
+
+lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)
+
+lemma mod_add_if_z:
+ "x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
+ (x + y) mod z = (if x + y < z then x + y else x + y - z)"
+ for x y z :: int
+ by (auto intro: int_mod_eq)
+
+lemma mod_sub_if_z:
+ "x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
+ (x - y) mod z = (if y \<le> x then x - y else x - y + z)"
+ for x y z :: int
+ by (auto intro: int_mod_eq)
+
+lemmas zmde = mult_div_mod_eq [symmetric, THEN diff_eq_eq [THEN iffD2], symmetric]
+lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
+
+(* already have this for naturals, div_mult_self1/2, but not for ints *)
+lemma zdiv_mult_self: "m \<noteq> 0 \<Longrightarrow> (a + m * n) div m = a div m + n"
+ for a m n :: int
+ apply (rule mcl)
+ prefer 2
+ apply (erule asm_rl)
+ apply (simp add: zmde ring_distribs)
+ done
+
+lemma mod_power_lem: "a > 1 \<Longrightarrow> a ^ n mod a ^ m = (if m \<le> n then 0 else a ^ n)"
+ for a :: int
+ by (simp add: mod_eq_0_iff_dvd le_imp_power_dvd)
+
+lemma pl_pl_rels: "a + b = c + d \<Longrightarrow> a \<ge> c \<and> b \<le> d \<or> a \<le> c \<and> b \<ge> d"
+ for a b c d :: nat
+ by arith
+
+lemmas pl_pl_rels' = add.commute [THEN [2] trans, THEN pl_pl_rels]
+
+lemma minus_eq: "m - k = m \<longleftrightarrow> k = 0 \<or> m = 0"
+ for k m :: nat
+ by arith
+
+lemma pl_pl_mm: "a + b = c + d \<Longrightarrow> a - c = d - b"
+ for a b c d :: nat
+ by arith
+
+lemmas pl_pl_mm' = add.commute [THEN [2] trans, THEN pl_pl_mm]
+
+lemmas dme = div_mult_mod_eq
+lemmas dtle = div_times_less_eq_dividend
+lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] div_times_less_eq_dividend]
+
+lemma td_gal: "0 < c \<Longrightarrow> a \<ge> b * c \<longleftrightarrow> a div c \<ge> b"
+ for a b c :: nat
+ apply safe
+ apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m])
+ apply (erule th2)
+ done
+
+lemmas td_gal_lt = td_gal [simplified not_less [symmetric], simplified]
+
+lemmas div_mult_le = div_times_less_eq_dividend
+
+lemmas sdl = div_nat_eqI
+
+lemma given_quot: "f > 0 \<Longrightarrow> (f * l + (f - 1)) div f = l"
+ for f l :: nat
+ by (rule div_nat_eqI) (simp_all)
+
+lemma given_quot_alt: "f > 0 \<Longrightarrow> (l * f + f - Suc 0) div f = l"
+ for f l :: nat
+ apply (frule given_quot)
+ apply (rule trans)
+ prefer 2
+ apply (erule asm_rl)
+ apply (rule_tac f="\<lambda>n. n div f" in arg_cong)
+ apply (simp add : ac_simps)
+ done
+
+lemma diff_mod_le: "a < d \<Longrightarrow> b dvd d \<Longrightarrow> a - a mod b \<le> d - b"
+ for a b d :: nat
+ apply (unfold dvd_def)
+ apply clarify
+ apply (case_tac k)
+ apply clarsimp
+ apply clarify
+ apply (cases "b > 0")
+ apply (drule mult.commute [THEN xtr1])
+ apply (frule (1) td_gal_lt [THEN iffD1])
+ apply (clarsimp simp: le_simps)
+ apply (rule minus_mod_eq_mult_div [symmetric, THEN [2] xtr4])
+ apply (rule mult_mono)
+ apply auto
+ done
+
+lemma less_le_mult': "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> (w + 1) * c \<le> b * c"
+ for b c w :: int
+ apply (rule mult_right_mono)
+ apply (rule zless_imp_add1_zle)
+ apply (erule (1) mult_right_less_imp_less)
+ apply assumption
+ done
+
+lemma less_le_mult: "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> w * c + c \<le> b * c"
+ for b c w :: int
+ using less_le_mult' [of w c b] by (simp add: algebra_simps)
+
+lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
+ simplified left_diff_distrib]
+
+lemma gen_minus: "0 < n \<Longrightarrow> f n = f (Suc (n - 1))"
+ by auto
+
+lemma mpl_lem: "j \<le> i \<Longrightarrow> k < j \<Longrightarrow> i - j + k < i"
+ for i j k :: nat
+ by arith
+
+lemma nonneg_mod_div: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> (a mod b) \<and> 0 \<le> a div b"
+ for a b :: int
+ by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
+
+declare iszero_0 [intro]
+
+lemma min_pm [simp]: "min a b + (a - b) = a"
+ for a b :: nat
+ by arith
+
+lemma min_pm1 [simp]: "a - b + min a b = a"
+ for a b :: nat
+ by arith
+
+lemma rev_min_pm [simp]: "min b a + (a - b) = a"
+ for a b :: nat
+ by arith
+
+lemma rev_min_pm1 [simp]: "a - b + min b a = a"
+ for a b :: nat
+ by arith
+
+lemma min_minus [simp]: "min m (m - k) = m - k"
+ for m k :: nat
+ by arith
+
+lemma min_minus' [simp]: "min (m - k) m = m - k"
+ for m k :: nat
+ by arith
+
+lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]
+
+lemma not_B1_is_B0: "y \<noteq> 1 \<Longrightarrow> y = 0"
+ for y :: bit
+ by (cases y) auto
+
+lemma B1_ass_B0:
+ fixes y :: bit
+ assumes y: "y = 0 \<Longrightarrow> y = 1"
+ shows "y = 1"
+ apply (rule classical)
+ apply (drule not_B1_is_B0)
+ apply (erule y)
+ done
+
+end
--- a/src/HOL/Word/Misc_Numeric.thy Tue Apr 16 19:50:03 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,71 +0,0 @@
-(* Title: HOL/Word/Misc_Numeric.thy
- Author: Jeremy Dawson, NICTA
-*)
-
-section \<open>Useful Numerical Lemmas\<close>
-
-theory Misc_Numeric
- imports Main
-begin
-
-lemma one_mod_exp_eq_one [simp]:
- "1 mod (2 * 2 ^ n) = (1::int)"
- using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)
-
-lemma mod_2_neq_1_eq_eq_0: "k mod 2 \<noteq> 1 \<longleftrightarrow> k mod 2 = 0"
- for k :: int
- by (fact not_mod_2_eq_1_eq_0)
-
-lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
- for b :: int
- by arith
-
-lemma diff_le_eq': "a - b \<le> c \<longleftrightarrow> a \<le> b + c"
- for a b c :: int
- by arith
-
-lemma emep1: "even n \<Longrightarrow> even d \<Longrightarrow> 0 \<le> d \<Longrightarrow> (n + 1) mod d = (n mod d) + 1"
- for n d :: int
- by (auto simp add: pos_zmod_mult_2 add.commute dvd_def)
-
-lemma int_mod_ge: "a < n \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a mod n"
- for a n :: int
- by (metis dual_order.trans le_cases mod_pos_pos_trivial pos_mod_conj)
-
-lemma int_mod_ge': "b < 0 \<Longrightarrow> 0 < n \<Longrightarrow> b + n \<le> b mod n"
- for b n :: int
- by (metis add_less_same_cancel2 int_mod_ge mod_add_self2)
-
-lemma int_mod_le': "0 \<le> b - n \<Longrightarrow> b mod n \<le> b - n"
- for b n :: int
- by (metis minus_mod_self2 zmod_le_nonneg_dividend)
-
-lemma zless2: "0 < (2 :: int)"
- by (fact zero_less_numeral)
-
-lemma zless2p: "0 < (2 ^ n :: int)"
- by arith
-
-lemma zle2p: "0 \<le> (2 ^ n :: int)"
- by arith
-
-lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
- using zless2p by (rule zmod_minus1)
-
-lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
- for b :: int
- using zle2p by (rule pos_zmod_mult_2)
-
-lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
- for b :: int
- by (simp add: p1mod22k' add.commute)
-
-lemma int_mod_lem: "0 < n \<Longrightarrow> 0 \<le> b \<and> b < n \<longleftrightarrow> b mod n = b"
- for b n :: int
- apply safe
- apply (erule (1) mod_pos_pos_trivial)
- apply (erule_tac [!] subst)
- apply auto
- done
-
-end
--- a/src/HOL/Word/Word.thy Tue Apr 16 19:50:03 2019 +0000
+++ b/src/HOL/Word/Word.thy Tue Apr 16 19:50:05 2019 +0000
@@ -11,7 +11,7 @@
Bits_Bit
Bits_Int
Misc_Typedef
- Word_Miscellaneous
+ Misc_Arithmetic
begin
text \<open>See \<^file>\<open>WordExamples.thy\<close> for examples.\<close>
--- a/src/HOL/Word/Word_Miscellaneous.thy Tue Apr 16 19:50:03 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,398 +0,0 @@
-(* Title: HOL/Word/Word_Miscellaneous.thy *)
-
-section \<open>Miscellaneous lemmas, of at least doubtful value\<close>
-
-theory Word_Miscellaneous
- imports "HOL-Library.Bit" Misc_Numeric
-begin
-
-lemma ex_eq_or: "(\<exists>m. n = Suc m \<and> (m = k \<or> P m)) \<longleftrightarrow> n = Suc k \<or> (\<exists>m. n = Suc m \<and> P m)"
- by auto
-
-lemma power_minus_simp: "0 < n \<Longrightarrow> a ^ n = a * a ^ (n - 1)"
- by (auto dest: gr0_implies_Suc)
-
-lemma funpow_minus_simp: "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
- by (auto dest: gr0_implies_Suc)
-
-lemma power_numeral: "a ^ numeral k = a * a ^ (pred_numeral k)"
- by (simp add: numeral_eq_Suc)
-
-lemma funpow_numeral [simp]: "f ^^ numeral k = f \<circ> f ^^ (pred_numeral k)"
- by (simp add: numeral_eq_Suc)
-
-lemma replicate_numeral [simp]: "replicate (numeral k) x = x # replicate (pred_numeral k) x"
- by (simp add: numeral_eq_Suc)
-
-lemma rco_alt: "(f \<circ> g) ^^ n \<circ> f = f \<circ> (g \<circ> f) ^^ n"
- apply (rule ext)
- apply (induct n)
- apply (simp_all add: o_def)
- done
-
-lemma list_exhaust_size_gt0:
- assumes "\<And>a list. y = a # list \<Longrightarrow> P"
- shows "0 < length y \<Longrightarrow> P"
- apply (cases y)
- apply simp
- apply (rule assms)
- apply fastforce
- done
-
-lemma list_exhaust_size_eq0:
- assumes "y = [] \<Longrightarrow> P"
- shows "length y = 0 \<Longrightarrow> P"
- apply (cases y)
- apply (rule assms)
- apply simp
- apply simp
- done
-
-lemma size_Cons_lem_eq: "y = xa # list \<Longrightarrow> size y = Suc k \<Longrightarrow> size list = k"
- by auto
-
-lemmas ls_splits = prod.split prod.split_asm if_split_asm
-
-lemma not_B1_is_B0: "y \<noteq> 1 \<Longrightarrow> y = 0"
- for y :: bit
- by (cases y) auto
-
-lemma B1_ass_B0:
- fixes y :: bit
- assumes y: "y = 0 \<Longrightarrow> y = 1"
- shows "y = 1"
- apply (rule classical)
- apply (drule not_B1_is_B0)
- apply (erule y)
- done
-
-\<comment> \<open>simplifications for specific word lengths\<close>
-lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'
-
-lemmas s2n_ths = n2s_ths [symmetric]
-
-lemma and_len: "xs = ys \<Longrightarrow> xs = ys \<and> length xs = length ys"
- by auto
-
-lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
- by auto
-
-lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
- by auto
-
-lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
- by auto
-
-lemma if_Not_x: "(if p then \<not> x else x) = (p = (\<not> x))"
- by auto
-
-lemma if_x_Not: "(if p then x else \<not> x) = (p = x)"
- by auto
-
-lemma if_same_and: "(If p x y \<and> If p u v) = (if p then x \<and> u else y \<and> v)"
- by auto
-
-lemma if_same_eq: "(If p x y = (If p u v)) = (if p then x = u else y = v)"
- by auto
-
-lemma if_same_eq_not: "(If p x y = (\<not> If p u v)) = (if p then x = (\<not> u) else y = (\<not> v))"
- by auto
-
-\<comment> \<open>note -- \<open>if_Cons\<close> can cause blowup in the size, if \<open>p\<close> is complex, so make a simproc\<close>
-lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
- by auto
-
-lemma if_single: "(if xc then [xab] else [an]) = [if xc then xab else an]"
- by auto
-
-lemma if_bool_simps:
- "If p True y = (p \<or> y) \<and> If p False y = (\<not> p \<and> y) \<and>
- If p y True = (p \<longrightarrow> y) \<and> If p y False = (p \<and> y)"
- by auto
-
-lemmas if_simps =
- if_x_Not if_Not_x if_cancel if_True if_False if_bool_simps
-
-lemmas seqr = eq_reflection [where x = "size w"] for w (* FIXME: delete *)
-
-lemma the_elemI: "y = {x} \<Longrightarrow> the_elem y = x"
- by simp
-
-lemma nonemptyE: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> R) \<Longrightarrow> R"
- by auto
-
-lemma gt_or_eq_0: "0 < y \<or> 0 = y"
- for y :: nat
- by arith
-
-lemmas xtr1 = xtrans(1)
-lemmas xtr2 = xtrans(2)
-lemmas xtr3 = xtrans(3)
-lemmas xtr4 = xtrans(4)
-lemmas xtr5 = xtrans(5)
-lemmas xtr6 = xtrans(6)
-lemmas xtr7 = xtrans(7)
-lemmas xtr8 = xtrans(8)
-
-lemmas nat_simps = diff_add_inverse2 diff_add_inverse
-lemmas nat_iffs = le_add1 le_add2
-
-lemma sum_imp_diff: "j = k + i \<Longrightarrow> j - i = k"
- for k :: nat
- by arith
-
-lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]]
-lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]]
-
-lemma nmod2: "n mod 2 = 0 \<or> n mod 2 = 1"
- for n :: int
- by arith
-
-lemmas eme1p = emep1 [simplified add.commute]
-
-lemma le_diff_eq': "a \<le> c - b \<longleftrightarrow> b + a \<le> c"
- for a b c :: int
- by arith
-
-lemma less_diff_eq': "a < c - b \<longleftrightarrow> b + a < c"
- for a b c :: int
- by arith
-
-lemma diff_less_eq': "a - b < c \<longleftrightarrow> a < b + c"
- for a b c :: int
- by arith
-
-lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]
-
-lemma z1pdiv2: "(2 * b + 1) div 2 = b"
- for b :: int
- by arith
-
-lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
- simplified int_one_le_iff_zero_less, simplified]
-
-lemma axxbyy: "a + m + m = b + n + n \<Longrightarrow> a = 0 \<or> a = 1 \<Longrightarrow> b = 0 \<or> b = 1 \<Longrightarrow> a = b \<and> m = n"
- for a b m n :: int
- by arith
-
-lemma axxmod2: "(1 + x + x) mod 2 = 1 \<and> (0 + x + x) mod 2 = 0"
- for x :: int
- by arith
-
-lemma axxdiv2: "(1 + x + x) div 2 = x \<and> (0 + x + x) div 2 = x"
- for x :: int
- by arith
-
-lemmas iszero_minus =
- trans [THEN trans, OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric]]
-
-lemmas zadd_diff_inverse =
- trans [OF diff_add_cancel [symmetric] add.commute]
-
-lemmas add_diff_cancel2 =
- add.commute [THEN diff_eq_eq [THEN iffD2]]
-
-lemmas rdmods [symmetric] = mod_minus_eq
- mod_diff_left_eq mod_diff_right_eq mod_add_left_eq
- mod_add_right_eq mod_mult_right_eq mod_mult_left_eq
-
-lemma mod_plus_right: "(a + x) mod m = (b + x) mod m \<longleftrightarrow> a mod m = b mod m"
- for a b m x :: nat
- by (induct x) (simp_all add: mod_Suc, arith)
-
-lemma nat_minus_mod: "(n - n mod m) mod m = 0"
- for m n :: nat
- by (induct n) (simp_all add: mod_Suc)
-
-lemmas nat_minus_mod_plus_right =
- trans [OF nat_minus_mod mod_0 [symmetric],
- THEN mod_plus_right [THEN iffD2], simplified]
-
-lemmas push_mods' = mod_add_eq
- mod_mult_eq mod_diff_eq
- mod_minus_eq
-
-lemmas push_mods = push_mods' [THEN eq_reflection]
-lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]
-
-lemma nat_mod_eq: "b < n \<Longrightarrow> a mod n = b mod n \<Longrightarrow> a mod n = b"
- for a b n :: nat
- by (induct a) auto
-
-lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]
-
-lemma nat_mod_lem: "0 < n \<Longrightarrow> b < n \<longleftrightarrow> b mod n = b"
- for b n :: nat
- apply safe
- apply (erule nat_mod_eq')
- apply (erule subst)
- apply (erule mod_less_divisor)
- done
-
-lemma mod_nat_add: "x < z \<Longrightarrow> y < z \<Longrightarrow> (x + y) mod z = (if x + y < z then x + y else x + y - z)"
- for x y z :: nat
- apply (rule nat_mod_eq)
- apply auto
- apply (rule trans)
- apply (rule le_mod_geq)
- apply simp
- apply (rule nat_mod_eq')
- apply arith
- done
-
-lemma mod_nat_sub: "x < z \<Longrightarrow> (x - y) mod z = x - y"
- for x y :: nat
- by (rule nat_mod_eq') arith
-
-lemma int_mod_eq: "0 \<le> b \<Longrightarrow> b < n \<Longrightarrow> a mod n = b mod n \<Longrightarrow> a mod n = b"
- for a b n :: int
- by (metis mod_pos_pos_trivial)
-
-lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)
-
-lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)
-
-lemma mod_add_if_z:
- "x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
- (x + y) mod z = (if x + y < z then x + y else x + y - z)"
- for x y z :: int
- by (auto intro: int_mod_eq)
-
-lemma mod_sub_if_z:
- "x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow>
- (x - y) mod z = (if y \<le> x then x - y else x - y + z)"
- for x y z :: int
- by (auto intro: int_mod_eq)
-
-lemmas zmde = mult_div_mod_eq [symmetric, THEN diff_eq_eq [THEN iffD2], symmetric]
-lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule]
-
-(* already have this for naturals, div_mult_self1/2, but not for ints *)
-lemma zdiv_mult_self: "m \<noteq> 0 \<Longrightarrow> (a + m * n) div m = a div m + n"
- for a m n :: int
- apply (rule mcl)
- prefer 2
- apply (erule asm_rl)
- apply (simp add: zmde ring_distribs)
- done
-
-lemma mod_power_lem: "a > 1 \<Longrightarrow> a ^ n mod a ^ m = (if m \<le> n then 0 else a ^ n)"
- for a :: int
- by (simp add: mod_eq_0_iff_dvd le_imp_power_dvd)
-
-lemma pl_pl_rels: "a + b = c + d \<Longrightarrow> a \<ge> c \<and> b \<le> d \<or> a \<le> c \<and> b \<ge> d"
- for a b c d :: nat
- by arith
-
-lemmas pl_pl_rels' = add.commute [THEN [2] trans, THEN pl_pl_rels]
-
-lemma minus_eq: "m - k = m \<longleftrightarrow> k = 0 \<or> m = 0"
- for k m :: nat
- by arith
-
-lemma pl_pl_mm: "a + b = c + d \<Longrightarrow> a - c = d - b"
- for a b c d :: nat
- by arith
-
-lemmas pl_pl_mm' = add.commute [THEN [2] trans, THEN pl_pl_mm]
-
-lemmas dme = div_mult_mod_eq
-lemmas dtle = div_times_less_eq_dividend
-lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] div_times_less_eq_dividend]
-
-lemma td_gal: "0 < c \<Longrightarrow> a \<ge> b * c \<longleftrightarrow> a div c \<ge> b"
- for a b c :: nat
- apply safe
- apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m])
- apply (erule th2)
- done
-
-lemmas td_gal_lt = td_gal [simplified not_less [symmetric], simplified]
-
-lemmas div_mult_le = div_times_less_eq_dividend
-
-lemmas sdl = div_nat_eqI
-
-lemma given_quot: "f > 0 \<Longrightarrow> (f * l + (f - 1)) div f = l"
- for f l :: nat
- by (rule div_nat_eqI) (simp_all)
-
-lemma given_quot_alt: "f > 0 \<Longrightarrow> (l * f + f - Suc 0) div f = l"
- for f l :: nat
- apply (frule given_quot)
- apply (rule trans)
- prefer 2
- apply (erule asm_rl)
- apply (rule_tac f="\<lambda>n. n div f" in arg_cong)
- apply (simp add : ac_simps)
- done
-
-lemma diff_mod_le: "a < d \<Longrightarrow> b dvd d \<Longrightarrow> a - a mod b \<le> d - b"
- for a b d :: nat
- apply (unfold dvd_def)
- apply clarify
- apply (case_tac k)
- apply clarsimp
- apply clarify
- apply (cases "b > 0")
- apply (drule mult.commute [THEN xtr1])
- apply (frule (1) td_gal_lt [THEN iffD1])
- apply (clarsimp simp: le_simps)
- apply (rule minus_mod_eq_mult_div [symmetric, THEN [2] xtr4])
- apply (rule mult_mono)
- apply auto
- done
-
-lemma less_le_mult': "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> (w + 1) * c \<le> b * c"
- for b c w :: int
- apply (rule mult_right_mono)
- apply (rule zless_imp_add1_zle)
- apply (erule (1) mult_right_less_imp_less)
- apply assumption
- done
-
-lemma less_le_mult: "w * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> w * c + c \<le> b * c"
- for b c w :: int
- using less_le_mult' [of w c b] by (simp add: algebra_simps)
-
-lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
- simplified left_diff_distrib]
-
-lemma gen_minus: "0 < n \<Longrightarrow> f n = f (Suc (n - 1))"
- by auto
-
-lemma mpl_lem: "j \<le> i \<Longrightarrow> k < j \<Longrightarrow> i - j + k < i"
- for i j k :: nat
- by arith
-
-lemma nonneg_mod_div: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> (a mod b) \<and> 0 \<le> a div b"
- for a b :: int
- by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])
-
-declare iszero_0 [intro]
-
-lemma min_pm [simp]: "min a b + (a - b) = a"
- for a b :: nat
- by arith
-
-lemma min_pm1 [simp]: "a - b + min a b = a"
- for a b :: nat
- by arith
-
-lemma rev_min_pm [simp]: "min b a + (a - b) = a"
- for a b :: nat
- by arith
-
-lemma rev_min_pm1 [simp]: "a - b + min b a = a"
- for a b :: nat
- by arith
-
-lemma min_minus [simp]: "min m (m - k) = m - k"
- for m k :: nat
- by arith
-
-lemma min_minus' [simp]: "min (m - k) m = m - k"
- for m k :: nat
- by arith
-
-end