--- a/src/HOL/Word/BinGeneral.thy Wed Jun 30 16:28:13 2010 +0200
+++ b/src/HOL/Word/BinGeneral.thy Wed Jun 30 16:28:13 2010 +0200
@@ -9,27 +9,22 @@
header {* Basic Definitions for Binary Integers *}
theory BinGeneral
-imports Num_Lemmas
+imports Misc_Numeric Bit
begin
subsection {* Further properties of numerals *}
-datatype bit = B0 | B1
+definition Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
+ "k BIT b = bit_case 0 1 b + k + k"
-definition
- Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
- "k BIT b = (case b of B0 \<Rightarrow> 0 | B1 \<Rightarrow> 1) + k + k"
-
-lemma BIT_B0_eq_Bit0 [simp]: "w BIT B0 = Int.Bit0 w"
+lemma BIT_B0_eq_Bit0 [simp]: "w BIT 0 = Int.Bit0 w"
unfolding Bit_def Bit0_def by simp
-lemma BIT_B1_eq_Bit1 [simp]: "w BIT B1 = Int.Bit1 w"
+lemma BIT_B1_eq_Bit1 [simp]: "w BIT 1 = Int.Bit1 w"
unfolding Bit_def Bit1_def by simp
lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1
-hide_const (open) B0 B1
-
lemma Min_ne_Pls [iff]:
"Int.Min ~= Int.Pls"
unfolding Min_def Pls_def by auto
@@ -63,32 +58,32 @@
lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]]
lemma less_Bits:
- "(v BIT b < w BIT c) = (v < w | v <= w & b = bit.B0 & c = bit.B1)"
+ "(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))"
unfolding Bit_def by (auto split: bit.split)
lemma le_Bits:
- "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= bit.B1 | c ~= bit.B0))"
+ "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))"
unfolding Bit_def by (auto split: bit.split)
lemma no_no [simp]: "number_of (number_of i) = i"
unfolding number_of_eq by simp
lemma Bit_B0:
- "k BIT bit.B0 = k + k"
+ "k BIT (0::bit) = k + k"
by (unfold Bit_def) simp
lemma Bit_B1:
- "k BIT bit.B1 = k + k + 1"
+ "k BIT (1::bit) = k + k + 1"
by (unfold Bit_def) simp
-lemma Bit_B0_2t: "k BIT bit.B0 = 2 * k"
+lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k"
by (rule trans, rule Bit_B0) simp
-lemma Bit_B1_2t: "k BIT bit.B1 = 2 * k + 1"
+lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1"
by (rule trans, rule Bit_B1) simp
lemma B_mod_2':
- "X = 2 ==> (w BIT bit.B1) mod X = 1 & (w BIT bit.B0) mod X = 0"
+ "X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0"
apply (simp (no_asm) only: Bit_B0 Bit_B1)
apply (simp add: z1pmod2)
done
@@ -100,8 +95,8 @@
unfolding numeral_simps number_of_is_id by simp
lemma neB1E [elim!]:
- assumes ne: "y \<noteq> bit.B1"
- assumes y: "y = bit.B0 \<Longrightarrow> P"
+ assumes ne: "y \<noteq> (1::bit)"
+ assumes y: "y = (0::bit) \<Longrightarrow> P"
shows "P"
apply (rule y)
apply (cases y rule: bit.exhaust, simp)
@@ -128,7 +123,7 @@
subsection {* Destructors for binary integers *}
definition bin_last :: "int \<Rightarrow> bit" where
- "bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)"
+ "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"
definition bin_rest :: "int \<Rightarrow> int" where
"bin_rest w = w div 2"
@@ -144,21 +139,17 @@
done
primrec bin_nth where
- Z: "bin_nth w 0 = (bin_last w = bit.B1)"
+ Z: "bin_nth w 0 = (bin_last w = (1::bit))"
| Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"
lemma bin_rl_simps [simp]:
- "bin_rl Int.Pls = (Int.Pls, bit.B0)"
- "bin_rl Int.Min = (Int.Min, bit.B1)"
- "bin_rl (Int.Bit0 r) = (r, bit.B0)"
- "bin_rl (Int.Bit1 r) = (r, bit.B1)"
+ "bin_rl Int.Pls = (Int.Pls, (0::bit))"
+ "bin_rl Int.Min = (Int.Min, (1::bit))"
+ "bin_rl (Int.Bit0 r) = (r, (0::bit))"
+ "bin_rl (Int.Bit1 r) = (r, (1::bit))"
"bin_rl (r BIT b) = (r, b)"
unfolding bin_rl_char by simp_all
-declare bin_rl_simps(1-4) [code]
-
-thm iffD1 [OF bin_rl_char bin_rl_def]
-
lemma bin_rl_simp [simp]:
"bin_rest w BIT bin_last w = w"
by (simp add: iffD1 [OF bin_rl_char bin_rl_def])
@@ -212,35 +203,27 @@
"bin_rest (w BIT b) = w"
using bin_rl_simps bin_rl_def by auto
-declare bin_rest_simps(1-4) [code]
-
lemma bin_last_simps [simp]:
- "bin_last Int.Pls = bit.B0"
- "bin_last Int.Min = bit.B1"
- "bin_last (Int.Bit0 w) = bit.B0"
- "bin_last (Int.Bit1 w) = bit.B1"
+ "bin_last Int.Pls = (0::bit)"
+ "bin_last Int.Min = (1::bit)"
+ "bin_last (Int.Bit0 w) = (0::bit)"
+ "bin_last (Int.Bit1 w) = (1::bit)"
"bin_last (w BIT b) = b"
using bin_rl_simps bin_rl_def by auto
-declare bin_last_simps(1-4) [code]
-
lemma bin_r_l_extras [simp]:
- "bin_last 0 = bit.B0"
- "bin_last (- 1) = bit.B1"
- "bin_last -1 = bit.B1"
- "bin_last 1 = bit.B1"
+ "bin_last 0 = (0::bit)"
+ "bin_last (- 1) = (1::bit)"
+ "bin_last -1 = (1::bit)"
+ "bin_last 1 = (1::bit)"
"bin_rest 1 = 0"
"bin_rest 0 = 0"
"bin_rest (- 1) = - 1"
"bin_rest -1 = -1"
- apply (unfold number_of_Min)
- apply (unfold Pls_def [symmetric] Min_def [symmetric])
- apply (unfold numeral_1_eq_1 [symmetric])
- apply (auto simp: number_of_eq)
- done
+ by (simp_all add: bin_last_def bin_rest_def)
lemma bin_last_mod:
- "bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)"
+ "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"
apply (case_tac w rule: bin_exhaust)
apply (case_tac b)
apply auto
@@ -261,10 +244,10 @@
unfolding bin_rest_div [symmetric] by auto
lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w"
- using Bit_div2 [where b=bit.B0] by simp
+ using Bit_div2 [where b="(0::bit)"] by simp
lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w"
- using Bit_div2 [where b=bit.B1] by simp
+ using Bit_div2 [where b="(1::bit)"] by simp
lemma bin_nth_lem [rule_format]:
"ALL y. bin_nth x = bin_nth y --> x = y"
@@ -306,7 +289,7 @@
lemma bin_nth_Min [simp]: "bin_nth Int.Min n"
by (induct n) auto
-lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = bit.B1)"
+lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))"
by auto
lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
@@ -317,11 +300,11 @@
lemma bin_nth_minus_Bit0 [simp]:
"0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)"
- using bin_nth_minus [where b=bit.B0] by simp
+ using bin_nth_minus [where b="(0::bit)"] by simp
lemma bin_nth_minus_Bit1 [simp]:
"0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)"
- using bin_nth_minus [where b=bit.B1] by simp
+ using bin_nth_minus [where b="(1::bit)"] by simp
lemmas bin_nth_0 = bin_nth.simps(1)
lemmas bin_nth_Suc = bin_nth.simps(2)
@@ -364,18 +347,18 @@
by (simp add: bin_rec.simps)
lemma bin_rec_Bit0:
- "f3 Int.Pls bit.B0 f1 = f1 \<Longrightarrow>
- bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w bit.B0 (bin_rec f1 f2 f3 w)"
+ "f3 Int.Pls (0::bit) f1 = f1 \<Longrightarrow>
+ bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w (0::bit) (bin_rec f1 f2 f3 w)"
by (simp add: bin_rec_Pls bin_rec.simps [of _ _ _ "Int.Bit0 w"])
lemma bin_rec_Bit1:
- "f3 Int.Min bit.B1 f2 = f2 \<Longrightarrow>
- bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w bit.B1 (bin_rec f1 f2 f3 w)"
+ "f3 Int.Min (1::bit) f2 = f2 \<Longrightarrow>
+ bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w (1::bit) (bin_rec f1 f2 f3 w)"
by (simp add: bin_rec_Min bin_rec.simps [of _ _ _ "Int.Bit1 w"])
lemma bin_rec_Bit:
- "f = bin_rec f1 f2 f3 ==> f3 Int.Pls bit.B0 f1 = f1 ==>
- f3 Int.Min bit.B1 f2 = f2 ==> f (w BIT b) = f3 w b (f w)"
+ "f = bin_rec f1 f2 f3 ==> f3 Int.Pls (0::bit) f1 = f1 ==>
+ f3 Int.Min (1::bit) f2 = f2 ==> f (w BIT b) = f3 w b (f w)"
by (cases b, simp add: bin_rec_Bit0, simp add: bin_rec_Bit1)
lemmas bin_rec_simps = refl [THEN bin_rec_Bit] bin_rec_Pls bin_rec_Min
@@ -411,7 +394,7 @@
sbintrunc :: "nat => int => int"
primrec
Z : "sbintrunc 0 bin =
- (case bin_last bin of bit.B1 => Int.Min | bit.B0 => Int.Pls)"
+ (case bin_last bin of (1::bit) => Int.Min | (0::bit) => Int.Pls)"
Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
lemma sign_bintr:
@@ -445,7 +428,7 @@
subsection "Simplifications for (s)bintrunc"
lemma bit_bool:
- "(b = (b' = bit.B1)) = (b' = (if b then bit.B1 else bit.B0))"
+ "(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))"
by (cases b') auto
lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]
@@ -472,16 +455,16 @@
done
lemma bin_nth_Bit:
- "bin_nth (w BIT b) n = (n = 0 & b = bit.B1 | (EX m. n = Suc m & bin_nth w m))"
+ "bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))"
by (cases n) auto
lemma bin_nth_Bit0:
"bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)"
- using bin_nth_Bit [where b=bit.B0] by simp
+ using bin_nth_Bit [where b="(0::bit)"] by simp
lemma bin_nth_Bit1:
"bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))"
- using bin_nth_Bit [where b=bit.B1] by simp
+ using bin_nth_Bit [where b="(1::bit)"] by simp
lemma bintrunc_bintrunc_l:
"n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
@@ -518,11 +501,11 @@
lemma bintrunc_Bit0 [simp]:
"bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)"
- using bintrunc_BIT [where b=bit.B0] by simp
+ using bintrunc_BIT [where b="(0::bit)"] by simp
lemma bintrunc_Bit1 [simp]:
"bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)"
- using bintrunc_BIT [where b=bit.B1] by simp
+ using bintrunc_BIT [where b="(1::bit)"] by simp
lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
bintrunc_Bit0 bintrunc_Bit1
@@ -538,11 +521,11 @@
lemma sbintrunc_Suc_Bit0 [simp]:
"sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)"
- using sbintrunc_Suc_BIT [where b=bit.B0] by simp
+ using sbintrunc_Suc_BIT [where b="(0::bit)"] by simp
lemma sbintrunc_Suc_Bit1 [simp]:
"sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)"
- using sbintrunc_Suc_BIT [where b=bit.B1] by simp
+ using sbintrunc_Suc_BIT [where b="(1::bit)"] by simp
lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1
@@ -556,11 +539,11 @@
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemmas sbintrunc_0_BIT_B0 [simp] =
- sbintrunc.Z [where bin="w BIT bit.B0",
+ sbintrunc.Z [where bin="w BIT (0::bit)",
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemmas sbintrunc_0_BIT_B1 [simp] =
- sbintrunc.Z [where bin="w BIT bit.B1",
+ sbintrunc.Z [where bin="w BIT (1::bit)",
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = Int.Pls"
@@ -936,12 +919,12 @@
lemmas ls_splits =
prod.split split_split prod.split_asm split_split_asm split_if_asm
-lemma not_B1_is_B0: "y \<noteq> bit.B1 \<Longrightarrow> y = bit.B0"
+lemma not_B1_is_B0: "y \<noteq> (1::bit) \<Longrightarrow> y = (0::bit)"
by (cases y) auto
lemma B1_ass_B0:
- assumes y: "y = bit.B0 \<Longrightarrow> y = bit.B1"
- shows "y = bit.B1"
+ assumes y: "y = (0::bit) \<Longrightarrow> y = (1::bit)"
+ shows "y = (1::bit)"
apply (rule classical)
apply (drule not_B1_is_B0)
apply (erule y)