--- a/src/HOL/Word/Bit_Representation.thy Mon Apr 03 21:17:47 2017 +0200
+++ b/src/HOL/Word/Bit_Representation.thy Mon Apr 03 23:12:16 2017 +0200
@@ -1,27 +1,24 @@
-(*
- Author: Jeremy Dawson, NICTA
-*)
+(* Title: HOL/Word/Bit_Representation.thy
+ Author: Jeremy Dawson, NICTA
+*)
section \<open>Integers as implict bit strings\<close>
theory Bit_Representation
-imports Misc_Numeric
+ imports Misc_Numeric
begin
subsection \<open>Constructors and destructors for binary integers\<close>
-definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int" (infixl "BIT" 90)
-where
- "k BIT b = (if b then 1 else 0) + k + k"
+definition Bit :: "int \<Rightarrow> bool \<Rightarrow> int" (infixl "BIT" 90)
+ where "k BIT b = (if b then 1 else 0) + k + k"
-lemma Bit_B0:
- "k BIT False = k + k"
- by (unfold Bit_def) simp
+lemma Bit_B0: "k BIT False = k + k"
+ by (simp add: Bit_def)
-lemma Bit_B1:
- "k BIT True = k + k + 1"
- by (unfold Bit_def) simp
-
+lemma Bit_B1: "k BIT True = k + k + 1"
+ by (simp add: Bit_def)
+
lemma Bit_B0_2t: "k BIT False = 2 * k"
by (rule trans, rule Bit_B0) simp
@@ -29,36 +26,28 @@
by (rule trans, rule Bit_B1) simp
definition bin_last :: "int \<Rightarrow> bool"
-where
- "bin_last w \<longleftrightarrow> w mod 2 = 1"
+ where "bin_last w \<longleftrightarrow> w mod 2 = 1"
-lemma bin_last_odd:
- "bin_last = odd"
+lemma bin_last_odd: "bin_last = odd"
by (rule ext) (simp add: bin_last_def even_iff_mod_2_eq_zero)
definition bin_rest :: "int \<Rightarrow> int"
-where
- "bin_rest w = w div 2"
+ where "bin_rest w = w div 2"
-lemma bin_rl_simp [simp]:
- "bin_rest w BIT bin_last w = w"
+lemma bin_rl_simp [simp]: "bin_rest w BIT bin_last w = w"
unfolding bin_rest_def bin_last_def Bit_def
- using div_mult_mod_eq [of w 2]
- by (cases "w mod 2 = 0", simp_all)
+ by (cases "w mod 2 = 0") (use div_mult_mod_eq [of w 2] in simp_all)
lemma bin_rest_BIT [simp]: "bin_rest (x BIT b) = x"
unfolding bin_rest_def Bit_def
- by (cases b, simp_all)
+ by (cases b) simp_all
lemma bin_last_BIT [simp]: "bin_last (x BIT b) = b"
unfolding bin_last_def Bit_def
by (cases b) simp_all
lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \<longleftrightarrow> u = v \<and> b = c"
- apply (auto simp add: Bit_def)
- apply arith
- apply arith
- done
+ by (auto simp: Bit_def) arith+
lemma BIT_bin_simps [simp]:
"numeral k BIT False = numeral (Num.Bit0 k)"
@@ -66,34 +55,32 @@
"(- numeral k) BIT False = - numeral (Num.Bit0 k)"
"(- numeral k) BIT True = - numeral (Num.BitM k)"
unfolding numeral.simps numeral_BitM
- unfolding Bit_def
- by (simp_all del: arith_simps add_numeral_special diff_numeral_special)
+ by (simp_all add: Bit_def del: arith_simps add_numeral_special diff_numeral_special)
lemma BIT_special_simps [simp]:
- shows "0 BIT False = 0" and "0 BIT True = 1"
- and "1 BIT False = 2" and "1 BIT True = 3"
- and "(- 1) BIT False = - 2" and "(- 1) BIT True = - 1"
- unfolding Bit_def by simp_all
+ shows "0 BIT False = 0"
+ and "0 BIT True = 1"
+ and "1 BIT False = 2"
+ and "1 BIT True = 3"
+ and "(- 1) BIT False = - 2"
+ and "(- 1) BIT True = - 1"
+ by (simp_all add: Bit_def)
lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> \<not> b"
- apply (auto simp add: Bit_def)
- apply arith
- done
+ by (auto simp: Bit_def) arith
lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b"
- apply (auto simp add: Bit_def)
- apply arith
- done
+ by (auto simp: Bit_def) arith
lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w"
- by (induct w, simp_all)
+ by (induct w) simp_all
lemma expand_BIT:
"numeral (Num.Bit0 w) = numeral w BIT False"
"numeral (Num.Bit1 w) = numeral w BIT True"
"- numeral (Num.Bit0 w) = (- numeral w) BIT False"
"- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT True"
- unfolding add_One by (simp_all add: BitM_inc)
+ by (simp_all add: add_One BitM_inc)
lemma bin_last_numeral_simps [simp]:
"\<not> bin_last 0"
@@ -117,13 +104,11 @@
"bin_rest (- numeral (Num.Bit1 w)) = - numeral (w + Num.One)"
by (simp_all add: bin_rest_def zdiv_zminus1_eq_if) (auto simp add: divmod_def)
-lemma less_Bits:
- "v BIT b < w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> \<not> b \<and> c"
- unfolding Bit_def by auto
+lemma less_Bits: "v BIT b < w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> \<not> b \<and> c"
+ by (auto simp: Bit_def)
-lemma le_Bits:
- "v BIT b \<le> w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> (\<not> b \<or> c)"
- unfolding Bit_def by auto
+lemma le_Bits: "v BIT b \<le> w BIT c \<longleftrightarrow> v < w \<or> v \<le> w \<and> (\<not> b \<or> c)"
+ by (auto simp: Bit_def)
lemma pred_BIT_simps [simp]:
"x BIT False - 1 = (x - 1) BIT True"
@@ -148,31 +133,27 @@
"x BIT True * y = (x * y) BIT False + y"
by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps)
-lemma B_mod_2':
- "X = 2 ==> (w BIT True) mod X = 1 & (w BIT False) mod X = 0"
- apply (simp (no_asm) only: Bit_B0 Bit_B1)
- apply simp
- done
+lemma B_mod_2': "X = 2 \<Longrightarrow> (w BIT True) mod X = 1 \<and> (w BIT False) mod X = 0"
+ by (simp add: Bit_B0 Bit_B1)
-lemma bin_ex_rl: "EX w b. w BIT b = bin"
+lemma bin_ex_rl: "\<exists>w b. w BIT b = bin"
by (metis bin_rl_simp)
lemma bin_exhaust:
- assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
+ assumes that: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
shows "Q"
- apply (insert bin_ex_rl [of bin])
+ apply (insert bin_ex_rl [of bin])
apply (erule exE)+
- apply (rule Q)
+ apply (rule that)
apply force
done
-primrec bin_nth where
- Z: "bin_nth w 0 \<longleftrightarrow> bin_last w"
+primrec bin_nth
+ where
+ Z: "bin_nth w 0 \<longleftrightarrow> bin_last w"
| Suc: "bin_nth w (Suc n) \<longleftrightarrow> bin_nth (bin_rest w) n"
-lemma bin_abs_lem:
- "bin = (w BIT b) ==> bin ~= -1 --> bin ~= 0 -->
- nat \<bar>w\<bar> < nat \<bar>bin\<bar>"
+lemma bin_abs_lem: "bin = (w BIT b) \<Longrightarrow> bin \<noteq> -1 \<longrightarrow> bin \<noteq> 0 \<longrightarrow> nat \<bar>w\<bar> < nat \<bar>bin\<bar>"
apply clarsimp
apply (unfold Bit_def)
apply (cases b)
@@ -183,10 +164,9 @@
lemma bin_induct:
assumes PPls: "P 0"
and PMin: "P (- 1)"
- and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"
+ and PBit: "\<And>bin bit. P bin \<Longrightarrow> P (bin BIT bit)"
shows "P bin"
- apply (rule_tac P=P and a=bin and f1="nat o abs"
- in wf_measure [THEN wf_induct])
+ apply (rule_tac P=P and a=bin and f1="nat \<circ> abs" in wf_measure [THEN wf_induct])
apply (simp add: measure_def inv_image_def)
apply (case_tac x rule: bin_exhaust)
apply (frule bin_abs_lem)
@@ -196,26 +176,23 @@
lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
unfolding bin_rest_def [symmetric] by (rule bin_rest_BIT)
-lemma bin_nth_eq_iff:
- "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
+lemma bin_nth_eq_iff: "bin_nth x = bin_nth y \<longleftrightarrow> x = y"
proof -
- have bin_nth_lem [rule_format]: "ALL y. bin_nth x = bin_nth y --> x = y"
+ have bin_nth_lem [rule_format]: "\<forall>y. bin_nth x = bin_nth y \<longrightarrow> x = y"
apply (induct x rule: bin_induct)
apply safe
apply (erule rev_mp)
apply (induct_tac y rule: bin_induct)
apply safe
apply (drule_tac x=0 in fun_cong, force)
- apply (erule notE, rule ext,
- drule_tac x="Suc x" in fun_cong, force)
+ apply (erule notE, rule ext, drule_tac x="Suc x" in fun_cong, force)
apply (drule_tac x=0 in fun_cong, force)
apply (erule rev_mp)
apply (induct_tac y rule: bin_induct)
apply safe
apply (drule_tac x=0 in fun_cong, force)
- apply (erule notE, rule ext,
- drule_tac x="Suc x" in fun_cong, force)
- apply (metis Bit_eq_m1_iff Z bin_last_BIT)
+ apply (erule notE, rule ext, drule_tac x="Suc x" in fun_cong, force)
+ apply (metis Bit_eq_m1_iff Z bin_last_BIT)
apply (case_tac y rule: bin_exhaust)
apply clarify
apply (erule allE)
@@ -227,13 +204,12 @@
apply (drule_tac x="Suc x" for x in fun_cong, force)
done
show ?thesis
- by (auto elim: bin_nth_lem)
+ by (auto elim: bin_nth_lem)
qed
lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1]]
-lemma bin_eq_iff:
- "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
+lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
using bin_nth_eq_iff by auto
lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n"
@@ -251,11 +227,10 @@
lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
by auto
-lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)"
+lemma bin_nth_minus [simp]: "0 < n \<Longrightarrow> bin_nth (w BIT b) n = bin_nth w (n - 1)"
by (cases n) auto
-lemma bin_nth_numeral:
- "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)"
+lemma bin_nth_numeral: "bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)"
by (simp add: numeral_eq_Suc)
lemmas bin_nth_numeral_simps [simp] =
@@ -265,7 +240,7 @@
bin_nth_numeral [OF bin_rest_numeral_simps(7)]
bin_nth_numeral [OF bin_rest_numeral_simps(8)]
-lemmas bin_nth_simps =
+lemmas bin_nth_simps =
bin_nth.Z bin_nth.Suc bin_nth_zero bin_nth_minus1
bin_nth_numeral_simps
@@ -273,8 +248,7 @@
subsection \<open>Truncating binary integers\<close>
definition bin_sign :: "int \<Rightarrow> int"
-where
- bin_sign_def: "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
+ where "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
lemma bin_sign_simps [simp]:
"bin_sign 0 = 0"
@@ -283,28 +257,26 @@
"bin_sign (numeral k) = 0"
"bin_sign (- numeral k) = -1"
"bin_sign (w BIT b) = bin_sign w"
- unfolding bin_sign_def Bit_def
- by simp_all
+ by (simp_all add: bin_sign_def Bit_def)
-lemma bin_sign_rest [simp]:
- "bin_sign (bin_rest w) = bin_sign w"
+lemma bin_sign_rest [simp]: "bin_sign (bin_rest w) = bin_sign w"
by (cases w rule: bin_exhaust) auto
-primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int" where
- Z : "bintrunc 0 bin = 0"
-| Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
+primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int"
+ where
+ Z : "bintrunc 0 bin = 0"
+ | Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
-primrec sbintrunc :: "nat => int => int" where
- Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)"
-| Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
+primrec sbintrunc :: "nat \<Rightarrow> int \<Rightarrow> int"
+ where
+ Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)"
+ | Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
by (induct n arbitrary: w) auto
-lemma bintrunc_mod2p: "bintrunc n w = (w mod 2 ^ n)"
- apply (induct n arbitrary: w, clarsimp)
- apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)
- done
+lemma bintrunc_mod2p: "bintrunc n w = w mod 2 ^ n"
+ by (induct n arbitrary: w) (auto simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)
lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n"
apply (induct n arbitrary: w)
@@ -330,10 +302,8 @@
"bintrunc (Suc n) (- 1) = bintrunc n (- 1) BIT True"
"bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT False"
"bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT True"
- "bintrunc (Suc n) (- numeral (Num.Bit0 w)) =
- bintrunc n (- numeral w) BIT False"
- "bintrunc (Suc n) (- numeral (Num.Bit1 w)) =
- bintrunc n (- numeral (w + Num.One)) BIT True"
+ "bintrunc (Suc n) (- numeral (Num.Bit0 w)) = bintrunc n (- numeral w) BIT False"
+ "bintrunc (Suc n) (- numeral (Num.Bit1 w)) = bintrunc n (- numeral (w + Num.One)) BIT True"
by simp_all
lemma sbintrunc_0_numeral [simp]:
@@ -346,14 +316,10 @@
lemma sbintrunc_Suc_numeral:
"sbintrunc (Suc n) 1 = 1"
- "sbintrunc (Suc n) (numeral (Num.Bit0 w)) =
- sbintrunc n (numeral w) BIT False"
- "sbintrunc (Suc n) (numeral (Num.Bit1 w)) =
- sbintrunc n (numeral w) BIT True"
- "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) =
- sbintrunc n (- numeral w) BIT False"
- "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) =
- sbintrunc n (- numeral (w + Num.One)) BIT True"
+ "sbintrunc (Suc n) (numeral (Num.Bit0 w)) = sbintrunc n (numeral w) BIT False"
+ "sbintrunc (Suc n) (numeral (Num.Bit1 w)) = sbintrunc n (numeral w) BIT True"
+ "sbintrunc (Suc n) (- numeral (Num.Bit0 w)) = sbintrunc n (- numeral w) BIT False"
+ "sbintrunc (Suc n) (- numeral (Num.Bit1 w)) = sbintrunc n (- numeral (w + Num.One)) BIT True"
by simp_all
lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n"
@@ -361,24 +327,21 @@
apply (case_tac bin rule: bin_exhaust, case_tac b, auto)
done
-lemma nth_bintr: "bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
+lemma nth_bintr: "bin_nth (bintrunc m w) n \<longleftrightarrow> n < m \<and> bin_nth w n"
apply (induct n arbitrary: w m)
apply (case_tac m, auto)[1]
apply (case_tac m, auto)[1]
done
-lemma nth_sbintr:
- "bin_nth (sbintrunc m w) n =
- (if n < m then bin_nth w n else bin_nth w m)"
+lemma nth_sbintr: "bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)"
apply (induct n arbitrary: w m)
+ apply (case_tac m)
+ apply simp_all
apply (case_tac m)
- apply simp_all
- apply (case_tac m)
- apply simp_all
+ apply simp_all
done
-lemma bin_nth_Bit:
- "bin_nth (w BIT b) n = (n = 0 & b | (EX m. n = Suc m & bin_nth w m))"
+lemma bin_nth_Bit: "bin_nth (w BIT b) n \<longleftrightarrow> n = 0 \<and> b \<or> (\<exists>m. n = Suc m \<and> bin_nth w m)"
by (cases n) auto
lemma bin_nth_Bit0:
@@ -391,69 +354,58 @@
n = 0 \<or> (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)"
using bin_nth_Bit [where w="numeral w" and b="True"] by simp
-lemma bintrunc_bintrunc_l:
- "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
- by (rule bin_eqI) (auto simp add : nth_bintr)
+lemma bintrunc_bintrunc_l: "n \<le> m \<Longrightarrow> bintrunc m (bintrunc n w) = bintrunc n w"
+ by (rule bin_eqI) (auto simp: nth_bintr)
-lemma sbintrunc_sbintrunc_l:
- "n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)"
+lemma sbintrunc_sbintrunc_l: "n \<le> m \<Longrightarrow> sbintrunc m (sbintrunc n w) = sbintrunc n w"
by (rule bin_eqI) (auto simp: nth_sbintr)
-lemma bintrunc_bintrunc_ge:
- "n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)"
+lemma bintrunc_bintrunc_ge: "n \<le> m \<Longrightarrow> bintrunc n (bintrunc m w) = bintrunc n w"
by (rule bin_eqI) (auto simp: nth_bintr)
-lemma bintrunc_bintrunc_min [simp]:
- "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
- apply (rule bin_eqI)
- apply (auto simp: nth_bintr)
- done
+lemma bintrunc_bintrunc_min [simp]: "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
+ by (rule bin_eqI) (auto simp: nth_bintr)
-lemma sbintrunc_sbintrunc_min [simp]:
- "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
- apply (rule bin_eqI)
- apply (auto simp: nth_sbintr min.absorb1 min.absorb2)
- done
+lemma sbintrunc_sbintrunc_min [simp]: "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
+ by (rule bin_eqI) (auto simp: nth_sbintr min.absorb1 min.absorb2)
-lemmas bintrunc_Pls =
+lemmas bintrunc_Pls =
bintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
-lemmas bintrunc_Min [simp] =
+lemmas bintrunc_Min [simp] =
bintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
-lemmas bintrunc_BIT [simp] =
+lemmas bintrunc_BIT [simp] =
bintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b
lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
bintrunc_Suc_numeral
-lemmas sbintrunc_Suc_Pls =
+lemmas sbintrunc_Suc_Pls =
sbintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
-lemmas sbintrunc_Suc_Min =
+lemmas sbintrunc_Suc_Min =
sbintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
-lemmas sbintrunc_Suc_BIT [simp] =
+lemmas sbintrunc_Suc_BIT [simp] =
sbintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b
lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
sbintrunc_Suc_numeral
-lemmas sbintrunc_Pls =
- sbintrunc.Z [where bin="0",
- simplified bin_last_numeral_simps bin_rest_numeral_simps]
+lemmas sbintrunc_Pls =
+ sbintrunc.Z [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]
-lemmas sbintrunc_Min =
- sbintrunc.Z [where bin="-1",
- simplified bin_last_numeral_simps bin_rest_numeral_simps]
+lemmas sbintrunc_Min =
+ sbintrunc.Z [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]
-lemmas sbintrunc_0_BIT_B0 [simp] =
- sbintrunc.Z [where bin="w BIT False",
- simplified bin_last_numeral_simps bin_rest_numeral_simps] for w
+lemmas sbintrunc_0_BIT_B0 [simp] =
+ sbintrunc.Z [where bin="w BIT False", simplified bin_last_numeral_simps bin_rest_numeral_simps]
+ for w
-lemmas sbintrunc_0_BIT_B1 [simp] =
- sbintrunc.Z [where bin="w BIT True",
- simplified bin_last_BIT bin_rest_numeral_simps] for w
+lemmas sbintrunc_0_BIT_B1 [simp] =
+ sbintrunc.Z [where bin="w BIT True", simplified bin_last_BIT bin_rest_numeral_simps]
+ for w
lemmas sbintrunc_0_simps =
sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
@@ -461,20 +413,18 @@
lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs
lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs
-lemma bintrunc_minus:
- "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"
+lemma bintrunc_minus: "0 < n \<Longrightarrow> bintrunc (Suc (n - 1)) w = bintrunc n w"
+ by auto
+
+lemma sbintrunc_minus: "0 < n \<Longrightarrow> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
by auto
-lemma sbintrunc_minus:
- "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
- by auto
-
-lemmas bintrunc_minus_simps =
+lemmas bintrunc_minus_simps =
bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans]]
-lemmas sbintrunc_minus_simps =
+lemmas sbintrunc_minus_simps =
sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]
-lemmas thobini1 = arg_cong [where f = "%w. w BIT b"] for b
+lemmas thobini1 = arg_cong [where f = "\<lambda>w. w BIT b"] for b
lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]
lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]
@@ -484,35 +434,31 @@
lemmas bintrunc_Min_minus_I = bmsts(2)
lemmas bintrunc_BIT_minus_I = bmsts(3)
-lemma bintrunc_Suc_lem:
- "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"
+lemma bintrunc_Suc_lem: "bintrunc (Suc n) x = y \<Longrightarrow> m = Suc n \<Longrightarrow> bintrunc m x = y"
by auto
-lemmas bintrunc_Suc_Ialts =
+lemmas bintrunc_Suc_Ialts =
bintrunc_Min_I [THEN bintrunc_Suc_lem]
bintrunc_BIT_I [THEN bintrunc_Suc_lem]
lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]
-lemmas sbintrunc_Suc_Is =
+lemmas sbintrunc_Suc_Is =
sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans]]
-lemmas sbintrunc_Suc_minus_Is =
+lemmas sbintrunc_Suc_minus_Is =
sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]
-lemma sbintrunc_Suc_lem:
- "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"
+lemma sbintrunc_Suc_lem: "sbintrunc (Suc n) x = y \<Longrightarrow> m = Suc n \<Longrightarrow> sbintrunc m x = y"
by auto
-lemmas sbintrunc_Suc_Ialts =
+lemmas sbintrunc_Suc_Ialts =
sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]
-lemma sbintrunc_bintrunc_lt:
- "m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w"
+lemma sbintrunc_bintrunc_lt: "m > n \<Longrightarrow> sbintrunc n (bintrunc m w) = sbintrunc n w"
by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)
-lemma bintrunc_sbintrunc_le:
- "m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w"
+lemma bintrunc_sbintrunc_le: "m \<le> Suc n \<Longrightarrow> bintrunc m (sbintrunc n w) = bintrunc m w"
apply (rule bin_eqI)
apply (auto simp: nth_sbintr nth_bintr)
apply (subgoal_tac "x=n", safe, arith+)[1]
@@ -522,19 +468,15 @@
lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]
lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]
lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]
-lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]
+lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]
-lemma bintrunc_sbintrunc' [simp]:
- "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
+lemma bintrunc_sbintrunc' [simp]: "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
by (cases n) (auto simp del: bintrunc.Suc)
-lemma sbintrunc_bintrunc' [simp]:
- "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
+lemma sbintrunc_bintrunc' [simp]: "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
by (cases n) (auto simp del: bintrunc.Suc)
-lemma bin_sbin_eq_iff:
- "bintrunc (Suc n) x = bintrunc (Suc n) y \<longleftrightarrow>
- sbintrunc n x = sbintrunc n y"
+lemma bin_sbin_eq_iff: "bintrunc (Suc n) x = bintrunc (Suc n) y \<longleftrightarrow> sbintrunc n x = sbintrunc n y"
apply (rule iffI)
apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])
apply simp
@@ -543,8 +485,7 @@
done
lemma bin_sbin_eq_iff':
- "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y \<longleftrightarrow>
- sbintrunc (n - 1) x = sbintrunc (n - 1) y"
+ "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y \<longleftrightarrow> sbintrunc (n - 1) x = sbintrunc (n - 1) y"
by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc)
lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]
@@ -557,36 +498,29 @@
tends to get applied where it's not wanted in developing the theories,
we get a version for when the word length is given literally *)
-lemmas nat_non0_gr =
+lemmas nat_non0_gr =
trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl]
lemma bintrunc_numeral:
- "bintrunc (numeral k) x =
- bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
+ "bintrunc (numeral k) x = bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
by (simp add: numeral_eq_Suc)
lemma sbintrunc_numeral:
- "sbintrunc (numeral k) x =
- sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
+ "sbintrunc (numeral k) x = sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"
by (simp add: numeral_eq_Suc)
lemma bintrunc_numeral_simps [simp]:
- "bintrunc (numeral k) (numeral (Num.Bit0 w)) =
- bintrunc (pred_numeral k) (numeral w) BIT False"
- "bintrunc (numeral k) (numeral (Num.Bit1 w)) =
- bintrunc (pred_numeral k) (numeral w) BIT True"
- "bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
- bintrunc (pred_numeral k) (- numeral w) BIT False"
+ "bintrunc (numeral k) (numeral (Num.Bit0 w)) = bintrunc (pred_numeral k) (numeral w) BIT False"
+ "bintrunc (numeral k) (numeral (Num.Bit1 w)) = bintrunc (pred_numeral k) (numeral w) BIT True"
+ "bintrunc (numeral k) (- numeral (Num.Bit0 w)) = bintrunc (pred_numeral k) (- numeral w) BIT False"
"bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
"bintrunc (numeral k) 1 = 1"
by (simp_all add: bintrunc_numeral)
lemma sbintrunc_numeral_simps [simp]:
- "sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
- sbintrunc (pred_numeral k) (numeral w) BIT False"
- "sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
- sbintrunc (pred_numeral k) (numeral w) BIT True"
+ "sbintrunc (numeral k) (numeral (Num.Bit0 w)) = sbintrunc (pred_numeral k) (numeral w) BIT False"
+ "sbintrunc (numeral k) (numeral (Num.Bit1 w)) = sbintrunc (pred_numeral k) (numeral w) BIT True"
"sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
sbintrunc (pred_numeral k) (- numeral w) BIT False"
"sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
@@ -597,49 +531,45 @@
lemma no_bintr_alt1: "bintrunc n = (\<lambda>w. w mod 2 ^ n :: int)"
by (rule ext) (rule bintrunc_mod2p)
-lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}"
+lemma range_bintrunc: "range (bintrunc n) = {i. 0 \<le> i \<and> i < 2 ^ n}"
apply (unfold no_bintr_alt1)
apply (auto simp add: image_iff)
apply (rule exI)
apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
done
-lemma no_sbintr_alt2:
- "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
+lemma no_sbintr_alt2: "sbintrunc n = (\<lambda>w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
by (rule ext) (simp add : sbintrunc_mod2p)
-lemma range_sbintrunc:
- "range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}"
+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::int) + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
+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
-lemma sb_inc_lem':
- "(a::int) < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
+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
by (rule sb_inc_lem) simp
-lemma sbintrunc_inc:
- "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"
+lemma sbintrunc_inc: "x < - (2^n) \<Longrightarrow> x + 2^(Suc n) \<le> sbintrunc n x"
unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp
-lemma sb_dec_lem:
- "(0::int) \<le> - (2 ^ k) + a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+lemma sb_dec_lem: "0 \<le> - (2 ^ k) + a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+ for a :: int
using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp
-lemma sb_dec_lem':
- "(2::int) ^ k \<le> a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+lemma sb_dec_lem': "2 ^ k \<le> a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
+ for a :: int
by (rule sb_dec_lem) simp
-lemma sbintrunc_dec:
- "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"
+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]
@@ -659,55 +589,48 @@
lemma sbintr_lt: "sbintrunc n w < 2 ^ n"
by (simp add: sbintrunc_mod2p)
-lemma sign_Pls_ge_0:
- "(bin_sign bin = 0) = (bin >= (0 :: int))"
- unfolding bin_sign_def by simp
+lemma sign_Pls_ge_0: "bin_sign bin = 0 \<longleftrightarrow> bin \<ge> 0"
+ for bin :: int
+ by (simp add: bin_sign_def)
-lemma sign_Min_lt_0:
- "(bin_sign bin = -1) = (bin < (0 :: int))"
- unfolding bin_sign_def by simp
+lemma sign_Min_lt_0: "bin_sign bin = -1 \<longleftrightarrow> bin < 0"
+ for bin :: int
+ by (simp add: bin_sign_def)
-lemma bin_rest_trunc:
- "(bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
+lemma bin_rest_trunc: "bin_rest (bintrunc n bin) = bintrunc (n - 1) (bin_rest bin)"
by (induct n arbitrary: bin) auto
lemma bin_rest_power_trunc:
- "(bin_rest ^^ k) (bintrunc n bin) =
- bintrunc (n - k) ((bin_rest ^^ k) bin)"
+ "(bin_rest ^^ k) (bintrunc n bin) = bintrunc (n - k) ((bin_rest ^^ k) bin)"
by (induct k) (auto simp: bin_rest_trunc)
-lemma bin_rest_trunc_i:
- "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
+lemma bin_rest_trunc_i: "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
by auto
-lemma bin_rest_strunc:
- "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
+lemma bin_rest_strunc: "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
by (induct n arbitrary: bin) auto
-lemma bintrunc_rest [simp]:
- "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
- apply (induct n arbitrary: bin, simp)
+lemma bintrunc_rest [simp]: "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
+ apply (induct n arbitrary: bin)
+ apply simp
apply (case_tac bin rule: bin_exhaust)
apply (auto simp: bintrunc_bintrunc_l)
done
-lemma sbintrunc_rest [simp]:
- "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
- apply (induct n arbitrary: bin, simp)
+lemma sbintrunc_rest [simp]: "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
+ apply (induct n arbitrary: bin)
+ apply simp
apply (case_tac bin rule: bin_exhaust)
apply (auto simp: bintrunc_bintrunc_l split: bool.splits)
done
-lemma bintrunc_rest':
- "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"
+lemma bintrunc_rest': "bintrunc n \<circ> bin_rest \<circ> bintrunc n = bin_rest \<circ> bintrunc n"
by (rule ext) auto
-lemma sbintrunc_rest' :
- "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"
+lemma sbintrunc_rest': "sbintrunc n \<circ> bin_rest \<circ> sbintrunc n = bin_rest \<circ> sbintrunc n"
by (rule ext) auto
-lemma rco_lem:
- "f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f"
+lemma rco_lem: "f \<circ> g \<circ> f = g \<circ> f \<Longrightarrow> f \<circ> (g \<circ> f) ^^ n = g ^^ n \<circ> f"
apply (rule ext)
apply (induct_tac n)
apply (simp_all (no_asm))
@@ -717,27 +640,29 @@
apply simp
done
-lemmas rco_bintr = bintrunc_rest'
+lemmas rco_bintr = bintrunc_rest'
[THEN rco_lem [THEN fun_cong], unfolded o_def]
-lemmas rco_sbintr = sbintrunc_rest'
+lemmas rco_sbintr = sbintrunc_rest'
[THEN rco_lem [THEN fun_cong], unfolded o_def]
-
+
subsection \<open>Splitting and concatenation\<close>
-primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where
- Z: "bin_split 0 w = (w, 0)"
- | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
- in (w1, w2 BIT bin_last w))"
+primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int"
+ where
+ Z: "bin_split 0 w = (w, 0)"
+ | Suc: "bin_split (Suc n) w =
+ (let (w1, w2) = bin_split n (bin_rest w)
+ in (w1, w2 BIT bin_last w))"
lemma [code]:
"bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))"
"bin_split 0 w = (w, 0)"
by simp_all
-primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where
- Z: "bin_cat w 0 v = w"
+primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int"
+ where
+ Z: "bin_cat w 0 v = w"
| Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"
end
-