author | wenzelm |
Sun, 03 Dec 2017 19:00:55 +0100 | |
changeset 67121 | 116968454d70 |
parent 67120 | 491fd7f0b5df |
child 67399 | eab6ce8368fa |
permissions | -rw-r--r-- |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
1 |
(* Title: HOL/Word/WordBitwise.thy |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
2 |
Authors: Thomas Sewell, NICTA and Sascha Boehme, TU Muenchen |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
3 |
*) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
4 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
5 |
theory WordBitwise |
65363 | 6 |
imports Word |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
7 |
begin |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
8 |
|
61799 | 9 |
text \<open>Helper constants used in defining addition\<close> |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
10 |
|
65363 | 11 |
definition xor3 :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool" |
12 |
where "xor3 a b c = (a = (b = c))" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
13 |
|
65363 | 14 |
definition carry :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool" |
15 |
where "carry a b c = ((a \<and> (b \<or> c)) \<or> (b \<and> c))" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
16 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
17 |
lemma carry_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
18 |
"carry True a b = (a \<or> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
19 |
"carry a True b = (a \<or> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
20 |
"carry a b True = (a \<or> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
21 |
"carry False a b = (a \<and> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
22 |
"carry a False b = (a \<and> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
23 |
"carry a b False = (a \<and> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
24 |
by (auto simp add: carry_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
25 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
26 |
lemma xor3_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
27 |
"xor3 True a b = (a = b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
28 |
"xor3 a True b = (a = b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
29 |
"xor3 a b True = (a = b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
30 |
"xor3 False a b = (a \<noteq> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
31 |
"xor3 a False b = (a \<noteq> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
32 |
"xor3 a b False = (a \<noteq> b)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
33 |
by (simp_all add: xor3_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
34 |
|
61799 | 35 |
text \<open>Breaking up word equalities into equalities on their |
65363 | 36 |
bit lists. Equalities are generated and manipulated in the |
66446 | 37 |
reverse order to @{const to_bl}.\<close> |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
38 |
|
65363 | 39 |
lemma word_eq_rbl_eq: "x = y \<longleftrightarrow> rev (to_bl x) = rev (to_bl y)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
40 |
by simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
41 |
|
65363 | 42 |
lemma rbl_word_or: "rev (to_bl (x OR y)) = map2 op \<or> (rev (to_bl x)) (rev (to_bl y))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
43 |
by (simp add: map2_def zip_rev bl_word_or rev_map) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
44 |
|
65363 | 45 |
lemma rbl_word_and: "rev (to_bl (x AND y)) = map2 op \<and> (rev (to_bl x)) (rev (to_bl y))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
46 |
by (simp add: map2_def zip_rev bl_word_and rev_map) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
47 |
|
65363 | 48 |
lemma rbl_word_xor: "rev (to_bl (x XOR y)) = map2 op \<noteq> (rev (to_bl x)) (rev (to_bl y))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
49 |
by (simp add: map2_def zip_rev bl_word_xor rev_map) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
50 |
|
65363 | 51 |
lemma rbl_word_not: "rev (to_bl (NOT x)) = map Not (rev (to_bl x))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
52 |
by (simp add: bl_word_not rev_map) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
53 |
|
65363 | 54 |
lemma bl_word_sub: "to_bl (x - y) = to_bl (x + (- y))" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
51717
diff
changeset
|
55 |
by simp |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
56 |
|
65363 | 57 |
lemma rbl_word_1: "rev (to_bl (1 :: 'a::len0 word)) = takefill False (len_of TYPE('a)) [True]" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
58 |
apply (rule_tac s="rev (to_bl (word_succ (0 :: 'a word)))" in trans) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
59 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
60 |
apply (simp only: rtb_rbl_ariths(1)[OF refl]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
61 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
62 |
apply (case_tac "len_of TYPE('a)") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
63 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
64 |
apply (simp add: takefill_alt) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
65 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
66 |
|
65363 | 67 |
lemma rbl_word_if: "rev (to_bl (if P then x else y)) = map2 (If P) (rev (to_bl x)) (rev (to_bl y))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
68 |
by (simp add: map2_def split_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
69 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
70 |
lemma rbl_add_carry_Cons: |
65363 | 71 |
"(if car then rbl_succ else id) (rbl_add (x # xs) (y # ys)) = |
72 |
xor3 x y car # (if carry x y car then rbl_succ else id) (rbl_add xs ys)" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
73 |
by (simp add: carry_def xor3_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
74 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
75 |
lemma rbl_add_suc_carry_fold: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
76 |
"length xs = length ys \<Longrightarrow> |
65363 | 77 |
\<forall>car. (if car then rbl_succ else id) (rbl_add xs ys) = |
78 |
(foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. [])) car" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
79 |
apply (erule list_induct2) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
80 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
81 |
apply (simp only: rbl_add_carry_Cons) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
82 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
83 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
84 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
85 |
lemma to_bl_plus_carry: |
65363 | 86 |
"to_bl (x + y) = |
87 |
rev (foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) |
|
88 |
(rev (zip (to_bl x) (to_bl y))) (\<lambda>_. []) False)" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
89 |
using rbl_add_suc_carry_fold[where xs="rev (to_bl x)" and ys="rev (to_bl y)"] |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
90 |
apply (simp add: word_add_rbl[OF refl refl]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
91 |
apply (drule_tac x=False in spec) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
92 |
apply (simp add: zip_rev) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
93 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
94 |
|
65363 | 95 |
definition "rbl_plus cin xs ys = |
96 |
foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (zip xs ys) (\<lambda>_. []) cin" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
97 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
98 |
lemma rbl_plus_simps: |
65363 | 99 |
"rbl_plus cin (x # xs) (y # ys) = xor3 x y cin # rbl_plus (carry x y cin) xs ys" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
100 |
"rbl_plus cin [] ys = []" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
101 |
"rbl_plus cin xs [] = []" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
102 |
by (simp_all add: rbl_plus_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
103 |
|
65363 | 104 |
lemma rbl_word_plus: "rev (to_bl (x + y)) = rbl_plus False (rev (to_bl x)) (rev (to_bl y))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
105 |
by (simp add: rbl_plus_def to_bl_plus_carry zip_rev) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
106 |
|
65363 | 107 |
definition "rbl_succ2 b xs = (if b then rbl_succ xs else xs)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
108 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
109 |
lemma rbl_succ2_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
110 |
"rbl_succ2 b [] = []" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
111 |
"rbl_succ2 b (x # xs) = (b \<noteq> x) # rbl_succ2 (x \<and> b) xs" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
112 |
by (simp_all add: rbl_succ2_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
113 |
|
65363 | 114 |
lemma twos_complement: "- x = word_succ (NOT x)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
115 |
using arg_cong[OF word_add_not[where x=x], where f="\<lambda>a. a - x + 1"] |
65363 | 116 |
by (simp add: word_succ_p1 word_sp_01[unfolded word_succ_p1] del: word_add_not) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
117 |
|
65363 | 118 |
lemma rbl_word_neg: "rev (to_bl (- x)) = rbl_succ2 True (map Not (rev (to_bl x)))" |
119 |
by (simp add: twos_complement word_succ_rbl[OF refl] bl_word_not rev_map rbl_succ2_def) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
120 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
121 |
lemma rbl_word_cat: |
65363 | 122 |
"rev (to_bl (word_cat x y :: 'a::len0 word)) = |
123 |
takefill False (len_of TYPE('a)) (rev (to_bl y) @ rev (to_bl x))" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
124 |
by (simp add: word_cat_bl word_rev_tf) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
125 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
126 |
lemma rbl_word_slice: |
65363 | 127 |
"rev (to_bl (slice n w :: 'a::len0 word)) = |
128 |
takefill False (len_of TYPE('a)) (drop n (rev (to_bl w)))" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
129 |
apply (simp add: slice_take word_rev_tf rev_take) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
130 |
apply (cases "n < len_of TYPE('b)", simp_all) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
131 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
132 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
133 |
lemma rbl_word_ucast: |
65363 | 134 |
"rev (to_bl (ucast x :: 'a::len0 word)) = takefill False (len_of TYPE('a)) (rev (to_bl x))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
135 |
apply (simp add: to_bl_ucast takefill_alt) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
136 |
apply (simp add: rev_drop) |
65363 | 137 |
apply (cases "len_of TYPE('a) < len_of TYPE('b)") |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
138 |
apply simp_all |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
139 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
140 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
141 |
lemma rbl_shiftl: |
65363 | 142 |
"rev (to_bl (w << n)) = takefill False (size w) (replicate n False @ rev (to_bl w))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
143 |
by (simp add: bl_shiftl takefill_alt word_size rev_drop) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
144 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
145 |
lemma rbl_shiftr: |
65363 | 146 |
"rev (to_bl (w >> n)) = takefill False (size w) (drop n (rev (to_bl w)))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
147 |
by (simp add: shiftr_slice rbl_word_slice word_size) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
148 |
|
65363 | 149 |
definition "drop_nonempty v n xs = (if n < length xs then drop n xs else [last (v # xs)])" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
150 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
151 |
lemma drop_nonempty_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
152 |
"drop_nonempty v (Suc n) (x # xs) = drop_nonempty x n xs" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
153 |
"drop_nonempty v 0 (x # xs) = (x # xs)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
154 |
"drop_nonempty v n [] = [v]" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
155 |
by (simp_all add: drop_nonempty_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
156 |
|
65363 | 157 |
definition "takefill_last x n xs = takefill (last (x # xs)) n xs" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
158 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
159 |
lemma takefill_last_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
160 |
"takefill_last z (Suc n) (x # xs) = x # takefill_last x n xs" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
161 |
"takefill_last z 0 xs = []" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
162 |
"takefill_last z n [] = replicate n z" |
65363 | 163 |
by (simp_all add: takefill_last_def) (simp_all add: takefill_alt) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
164 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
165 |
lemma rbl_sshiftr: |
65363 | 166 |
"rev (to_bl (w >>> n)) = takefill_last False (size w) (drop_nonempty False n (rev (to_bl w)))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
167 |
apply (cases "n < size w") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
168 |
apply (simp add: bl_sshiftr takefill_last_def word_size |
65363 | 169 |
takefill_alt rev_take last_rev |
170 |
drop_nonempty_def) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
171 |
apply (subgoal_tac "(w >>> n) = of_bl (replicate (size w) (msb w))") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
172 |
apply (simp add: word_size takefill_last_def takefill_alt |
65363 | 173 |
last_rev word_msb_alt word_rev_tf |
174 |
drop_nonempty_def take_Cons') |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
175 |
apply (case_tac "len_of TYPE('a)", simp_all) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
176 |
apply (rule word_eqI) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
177 |
apply (simp add: nth_sshiftr word_size test_bit_of_bl |
65363 | 178 |
msb_nth) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
179 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
180 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
181 |
lemma nth_word_of_int: |
65363 | 182 |
"(word_of_int x :: 'a::len0 word) !! n = (n < len_of TYPE('a) \<and> bin_nth x n)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
183 |
apply (simp add: test_bit_bl word_size to_bl_of_bin) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
184 |
apply (subst conj_cong[OF refl], erule bin_nth_bl) |
65363 | 185 |
apply auto |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
186 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
187 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
188 |
lemma nth_scast: |
65363 | 189 |
"(scast (x :: 'a::len word) :: 'b::len word) !! n = |
190 |
(n < len_of TYPE('b) \<and> |
|
191 |
(if n < len_of TYPE('a) - 1 then x !! n |
|
192 |
else x !! (len_of TYPE('a) - 1)))" |
|
193 |
by (simp add: scast_def nth_sint) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
194 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
195 |
lemma rbl_word_scast: |
65363 | 196 |
"rev (to_bl (scast x :: 'a::len word)) = takefill_last False (len_of TYPE('a)) (rev (to_bl x))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
197 |
apply (rule nth_equalityI) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
198 |
apply (simp add: word_size takefill_last_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
199 |
apply (clarsimp simp: nth_scast takefill_last_def |
65363 | 200 |
nth_takefill word_size nth_rev to_bl_nth) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
201 |
apply (cases "len_of TYPE('b)") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
202 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
203 |
apply (clarsimp simp: less_Suc_eq_le linorder_not_less |
65363 | 204 |
last_rev word_msb_alt[symmetric] |
205 |
msb_nth) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
206 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
207 |
|
65363 | 208 |
definition rbl_mul :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list" |
209 |
where "rbl_mul xs ys = foldr (\<lambda>x sm. rbl_plus False (map (op \<and> x) ys) (False # sm)) xs []" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
210 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
211 |
lemma rbl_mul_simps: |
65363 | 212 |
"rbl_mul (x # xs) ys = rbl_plus False (map (op \<and> x) ys) (False # rbl_mul xs ys)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
213 |
"rbl_mul [] ys = []" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
214 |
by (simp_all add: rbl_mul_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
215 |
|
65363 | 216 |
lemma takefill_le2: "length xs \<le> n \<Longrightarrow> takefill x m (takefill x n xs) = takefill x m xs" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
217 |
by (simp add: takefill_alt replicate_add[symmetric]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
218 |
|
65363 | 219 |
lemma take_rbl_plus: "\<forall>n b. take n (rbl_plus b xs ys) = rbl_plus b (take n xs) (take n ys)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
220 |
apply (simp add: rbl_plus_def take_zip[symmetric]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
221 |
apply (rule_tac list="zip xs ys" in list.induct) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
222 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
223 |
apply (clarsimp simp: split_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
224 |
apply (case_tac n, simp_all) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
225 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
226 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
227 |
lemma word_rbl_mul_induct: |
65363 | 228 |
"length xs \<le> size y \<Longrightarrow> |
229 |
rbl_mul xs (rev (to_bl y)) = take (length xs) (rev (to_bl (of_bl (rev xs) * y)))" |
|
230 |
for y :: "'a::len word" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
231 |
proof (induct xs) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
232 |
case Nil |
65363 | 233 |
show ?case by (simp add: rbl_mul_simps) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
234 |
next |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
235 |
case (Cons z zs) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
236 |
|
65363 | 237 |
have rbl_word_plus': "to_bl (x + y) = rev (rbl_plus False (rev (to_bl x)) (rev (to_bl y)))" |
238 |
for x y :: "'a word" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
239 |
by (simp add: rbl_word_plus[symmetric]) |
65363 | 240 |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
241 |
have mult_bit: "to_bl (of_bl [z] * y) = map (op \<and> z) (to_bl y)" |
65363 | 242 |
by (cases z) (simp cong: map_cong, simp add: map_replicate_const cong: map_cong) |
243 |
||
244 |
have shiftl: "of_bl xs * 2 * y = (of_bl xs * y) << 1" for xs |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
245 |
by (simp add: shiftl_t2n) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
246 |
|
65363 | 247 |
have zip_take_triv: "\<And>xs ys n. n = length ys \<Longrightarrow> zip (take n xs) ys = zip xs ys" |
248 |
by (rule nth_equalityI) simp_all |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
249 |
|
65363 | 250 |
from Cons show ?case |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
55465
diff
changeset
|
251 |
apply (simp add: trans [OF of_bl_append add.commute] |
65363 | 252 |
rbl_mul_simps rbl_word_plus' distrib_right mult_bit shiftl rbl_shiftl) |
253 |
apply (simp add: takefill_alt word_size rev_map take_rbl_plus min_def) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
254 |
apply (simp add: rbl_plus_def zip_take_triv) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
255 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
256 |
qed |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
257 |
|
65363 | 258 |
lemma rbl_word_mul: "rev (to_bl (x * y)) = rbl_mul (rev (to_bl x)) (rev (to_bl y))" |
259 |
for x :: "'a::len word" |
|
260 |
using word_rbl_mul_induct[where xs="rev (to_bl x)" and y=y] by (simp add: word_size) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
261 |
|
61799 | 262 |
text \<open>Breaking up inequalities into bitlist properties.\<close> |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
263 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
264 |
definition |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
265 |
"rev_bl_order F xs ys = |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
266 |
(length xs = length ys \<and> |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
267 |
((xs = ys \<and> F) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
268 |
\<or> (\<exists>n < length xs. drop (Suc n) xs = drop (Suc n) ys |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
269 |
\<and> \<not> xs ! n \<and> ys ! n)))" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
270 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
271 |
lemma rev_bl_order_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
272 |
"rev_bl_order F [] [] = F" |
65363 | 273 |
"rev_bl_order F (x # xs) (y # ys) = rev_bl_order ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> F)) xs ys" |
274 |
apply (simp_all add: rev_bl_order_def) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
275 |
apply (rule conj_cong[OF refl]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
276 |
apply (cases "xs = ys") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
277 |
apply (simp add: nth_Cons') |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
278 |
apply blast |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
279 |
apply (simp add: nth_Cons') |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
280 |
apply safe |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
281 |
apply (rule_tac x="n - 1" in exI) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
282 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
283 |
apply (rule_tac x="Suc n" in exI) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
284 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
285 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
286 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
287 |
lemma rev_bl_order_rev_simp: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
288 |
"length xs = length ys \<Longrightarrow> |
65363 | 289 |
rev_bl_order F (xs @ [x]) (ys @ [y]) = ((y \<and> \<not> x) \<or> ((y \<or> \<not> x) \<and> rev_bl_order F xs ys))" |
290 |
by (induct arbitrary: F rule: list_induct2) (auto simp: rev_bl_order_simps) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
291 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
292 |
lemma rev_bl_order_bl_to_bin: |
65363 | 293 |
"length xs = length ys \<Longrightarrow> |
294 |
rev_bl_order True xs ys = (bl_to_bin (rev xs) \<le> bl_to_bin (rev ys)) \<and> |
|
295 |
rev_bl_order False xs ys = (bl_to_bin (rev xs) < bl_to_bin (rev ys))" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
296 |
apply (induct xs ys rule: list_induct2) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
297 |
apply (simp_all add: rev_bl_order_simps bl_to_bin_app_cat) |
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54489
diff
changeset
|
298 |
apply (auto simp add: bl_to_bin_def Bit_B0 Bit_B1 add1_zle_eq Bit_def) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
299 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
300 |
|
65363 | 301 |
lemma word_le_rbl: "x \<le> y \<longleftrightarrow> rev_bl_order True (rev (to_bl x)) (rev (to_bl y))" |
302 |
for x y :: "'a::len0 word" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
303 |
by (simp add: rev_bl_order_bl_to_bin word_le_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
304 |
|
65363 | 305 |
lemma word_less_rbl: "x < y \<longleftrightarrow> rev_bl_order False (rev (to_bl x)) (rev (to_bl y))" |
306 |
for x y :: "'a::len0 word" |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
307 |
by (simp add: word_less_alt rev_bl_order_bl_to_bin) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
308 |
|
65363 | 309 |
lemma word_sint_msb_eq: "sint x = uint x - (if msb x then 2 ^ size x else 0)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
310 |
apply (cases "msb x") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
311 |
apply (rule word_sint.Abs_eqD[where 'a='a], simp_all) |
65363 | 312 |
apply (simp add: word_size wi_hom_syms word_of_int_2p_len) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
313 |
apply (simp add: sints_num word_size) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
314 |
apply (rule conjI) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
315 |
apply (simp add: le_diff_eq') |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
316 |
apply (rule order_trans[where y="2 ^ (len_of TYPE('a) - 1)"]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
317 |
apply (simp add: power_Suc[symmetric]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
318 |
apply (simp add: linorder_not_less[symmetric] mask_eq_iff[symmetric]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
319 |
apply (rule notI, drule word_eqD[where x="size x - 1"]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
320 |
apply (simp add: msb_nth word_ops_nth_size word_size) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
321 |
apply (simp add: order_less_le_trans[where y=0]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
322 |
apply (rule word_uint.Abs_eqD[where 'a='a], simp_all) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
323 |
apply (simp add: linorder_not_less uints_num word_msb_sint) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
324 |
apply (rule order_less_le_trans[OF sint_lt]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
325 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
326 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
327 |
|
65363 | 328 |
lemma word_sle_msb_le: "x <=s y \<longleftrightarrow> (msb y \<longrightarrow> msb x) \<and> ((msb x \<and> \<not> msb y) \<or> x \<le> y)" |
329 |
apply (simp add: word_sle_def word_sint_msb_eq word_size word_le_def) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
330 |
apply safe |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
331 |
apply (rule order_trans[OF _ uint_ge_0]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
332 |
apply (simp add: order_less_imp_le) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
333 |
apply (erule notE[OF leD]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
334 |
apply (rule order_less_le_trans[OF _ uint_ge_0]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
335 |
apply simp |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
336 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
337 |
|
65363 | 338 |
lemma word_sless_msb_less: "x <s y \<longleftrightarrow> (msb y \<longrightarrow> msb x) \<and> ((msb x \<and> \<not> msb y) \<or> x < y)" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
339 |
by (auto simp add: word_sless_def word_sle_msb_le) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
340 |
|
65363 | 341 |
definition "map_last f xs = (if xs = [] then [] else butlast xs @ [f (last xs)])" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
342 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
343 |
lemma map_last_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
344 |
"map_last f [] = []" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
345 |
"map_last f [x] = [f x]" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
346 |
"map_last f (x # y # zs) = x # map_last f (y # zs)" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
347 |
by (simp_all add: map_last_def) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
348 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
349 |
lemma word_sle_rbl: |
65363 | 350 |
"x <=s y \<longleftrightarrow> rev_bl_order True (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
351 |
using word_msb_alt[where w=x] word_msb_alt[where w=y] |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
352 |
apply (simp add: word_sle_msb_le word_le_rbl) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
353 |
apply (subgoal_tac "length (to_bl x) = length (to_bl y)") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
354 |
apply (cases "to_bl x", simp) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
355 |
apply (cases "to_bl y", simp) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
356 |
apply (clarsimp simp: map_last_def rev_bl_order_rev_simp) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
357 |
apply auto |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
358 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
359 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
360 |
lemma word_sless_rbl: |
65363 | 361 |
"x <s y \<longleftrightarrow> rev_bl_order False (map_last Not (rev (to_bl x))) (map_last Not (rev (to_bl y)))" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
362 |
using word_msb_alt[where w=x] word_msb_alt[where w=y] |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
363 |
apply (simp add: word_sless_msb_less word_less_rbl) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
364 |
apply (subgoal_tac "length (to_bl x) = length (to_bl y)") |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
365 |
apply (cases "to_bl x", simp) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
366 |
apply (cases "to_bl y", simp) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
367 |
apply (clarsimp simp: map_last_def rev_bl_order_rev_simp) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
368 |
apply auto |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
369 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
370 |
|
66446 | 371 |
text \<open>Lemmas for unpacking @{term "rev (to_bl n)"} for numerals n and also |
65363 | 372 |
for irreducible values and expressions.\<close> |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
373 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
374 |
lemma rev_bin_to_bl_simps: |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
375 |
"rev (bin_to_bl 0 x) = []" |
65363 | 376 |
"rev (bin_to_bl (Suc n) (numeral (num.Bit0 nm))) = False # rev (bin_to_bl n (numeral nm))" |
377 |
"rev (bin_to_bl (Suc n) (numeral (num.Bit1 nm))) = True # rev (bin_to_bl n (numeral nm))" |
|
378 |
"rev (bin_to_bl (Suc n) (numeral (num.One))) = True # replicate n False" |
|
379 |
"rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm))) = False # rev (bin_to_bl n (- numeral nm))" |
|
380 |
"rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm))) = |
|
381 |
True # rev (bin_to_bl n (- numeral (nm + num.One)))" |
|
382 |
"rev (bin_to_bl (Suc n) (- numeral (num.One))) = True # replicate n True" |
|
383 |
"rev (bin_to_bl (Suc n) (- numeral (num.Bit0 nm + num.One))) = |
|
384 |
True # rev (bin_to_bl n (- numeral (nm + num.One)))" |
|
385 |
"rev (bin_to_bl (Suc n) (- numeral (num.Bit1 nm + num.One))) = |
|
386 |
False # rev (bin_to_bl n (- numeral (nm + num.One)))" |
|
387 |
"rev (bin_to_bl (Suc n) (- numeral (num.One + num.One))) = |
|
388 |
False # rev (bin_to_bl n (- numeral num.One))" |
|
389 |
apply simp_all |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
390 |
apply (simp_all only: bin_to_bl_aux_alt) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
391 |
apply (simp_all) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
392 |
apply (simp_all add: bin_to_bl_zero_aux bin_to_bl_minus1_aux) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
393 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
394 |
|
65363 | 395 |
lemma to_bl_upt: "to_bl x = rev (map (op !! x) [0 ..< size x])" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
396 |
apply (rule nth_equalityI) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
397 |
apply (simp add: word_size) |
65363 | 398 |
apply (auto simp: to_bl_nth word_size nth_rev) |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
399 |
done |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
400 |
|
65363 | 401 |
lemma rev_to_bl_upt: "rev (to_bl x) = map (op !! x) [0 ..< size x]" |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
402 |
by (simp add: to_bl_upt) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
403 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
404 |
lemma upt_eq_list_intros: |
65363 | 405 |
"j \<le> i \<Longrightarrow> [i ..< j] = []" |
406 |
"i = x \<Longrightarrow> x < j \<Longrightarrow> [x + 1 ..< j] = xs \<Longrightarrow> [i ..< j] = (x # xs)" |
|
407 |
by (simp_all add: upt_eq_Cons_conv) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
408 |
|
65363 | 409 |
|
410 |
subsection \<open>Tactic definition\<close> |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
411 |
|
61799 | 412 |
ML \<open> |
61144 | 413 |
structure Word_Bitwise_Tac = |
414 |
struct |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
415 |
|
51686 | 416 |
val word_ss = simpset_of @{theory_context Word}; |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
417 |
|
67120 | 418 |
fun mk_nat_clist ns = |
67121 | 419 |
fold_rev (Thm.mk_binop @{cterm "Cons :: nat \<Rightarrow> _"}) |
420 |
ns @{cterm "[] :: nat list"}; |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
421 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
422 |
fun upt_conv ctxt ct = |
59582 | 423 |
case Thm.term_of ct of |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
424 |
(@{const upt} $ n $ m) => |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
425 |
let |
59058
a78612c67ec0
renamed "pairself" to "apply2", in accordance to @{apply 2};
wenzelm
parents:
57512
diff
changeset
|
426 |
val (i, j) = apply2 (snd o HOLogic.dest_number) (n, m); |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
427 |
val ns = map (Numeral.mk_cnumber @{ctyp nat}) (i upto (j - 1)) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
428 |
|> mk_nat_clist; |
67120 | 429 |
val prop = |
430 |
Thm.mk_binop @{cterm "op = :: nat list \<Rightarrow> _"} ct ns |
|
431 |
|> Thm.apply @{cterm Trueprop}; |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
432 |
in |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
433 |
try (fn () => |
65363 | 434 |
Goal.prove_internal ctxt [] prop |
59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59058
diff
changeset
|
435 |
(K (REPEAT_DETERM (resolve_tac ctxt @{thms upt_eq_list_intros} 1 |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
436 |
ORELSE simp_tac (put_simpset word_ss ctxt) 1))) |> mk_meta_eq) () |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
437 |
end |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
438 |
| _ => NONE; |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
439 |
|
61144 | 440 |
val expand_upt_simproc = |
441 |
Simplifier.make_simproc @{context} "expand_upt" |
|
62913 | 442 |
{lhss = [@{term "upt x y"}], proc = K upt_conv}; |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
443 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
444 |
fun word_len_simproc_fn ctxt ct = |
67120 | 445 |
(case Thm.term_of ct of |
446 |
Const (@{const_name len_of}, _) $ t => |
|
447 |
(let |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
448 |
val T = fastype_of t |> dest_Type |> snd |> the_single |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
449 |
val n = Numeral.mk_cnumber @{ctyp nat} (Word_Lib.dest_binT T); |
67120 | 450 |
val prop = |
451 |
Thm.mk_binop @{cterm "op = :: nat \<Rightarrow> _"} ct n |
|
452 |
|> Thm.apply @{cterm Trueprop}; |
|
453 |
in |
|
454 |
Goal.prove_internal ctxt [] prop (K (simp_tac (put_simpset word_ss ctxt) 1)) |
|
455 |
|> mk_meta_eq |> SOME |
|
456 |
end handle TERM _ => NONE | TYPE _ => NONE) |
|
457 |
| _ => NONE); |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
458 |
|
61144 | 459 |
val word_len_simproc = |
460 |
Simplifier.make_simproc @{context} "word_len" |
|
62913 | 461 |
{lhss = [@{term "len_of x"}], proc = K word_len_simproc_fn}; |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
462 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
463 |
(* convert 5 or nat 5 to Suc 4 when n_sucs = 1, Suc (Suc 4) when n_sucs = 2, |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
464 |
or just 5 (discarding nat) when n_sucs = 0 *) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
465 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
466 |
fun nat_get_Suc_simproc_fn n_sucs ctxt ct = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
467 |
let |
59582 | 468 |
val (f $ arg) = Thm.term_of ct; |
67120 | 469 |
val n = |
470 |
(case arg of @{term nat} $ n => n | n => n) |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
471 |
|> HOLogic.dest_number |> snd; |
67120 | 472 |
val (i, j) = if n > n_sucs then (n_sucs, n - n_sucs) else (n, 0); |
67121 | 473 |
val arg' = funpow i HOLogic.mk_Suc (HOLogic.mk_number @{typ nat} j); |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
474 |
val _ = if arg = arg' then raise TERM ("", []) else (); |
67120 | 475 |
fun propfn g = |
476 |
HOLogic.mk_eq (g arg, g arg') |
|
59643 | 477 |
|> HOLogic.mk_Trueprop |> Thm.cterm_of ctxt; |
67120 | 478 |
val eq1 = |
479 |
Goal.prove_internal ctxt [] (propfn I) |
|
480 |
(K (simp_tac (put_simpset word_ss ctxt) 1)); |
|
481 |
in |
|
482 |
Goal.prove_internal ctxt [] (propfn (curry (op $) f)) |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
483 |
(K (simp_tac (put_simpset HOL_ss ctxt addsimps [eq1]) 1)) |
67120 | 484 |
|> mk_meta_eq |> SOME |
485 |
end handle TERM _ => NONE; |
|
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
486 |
|
61144 | 487 |
fun nat_get_Suc_simproc n_sucs ts = |
488 |
Simplifier.make_simproc @{context} "nat_get_Suc" |
|
489 |
{lhss = map (fn t => t $ @{term "n :: nat"}) ts, |
|
62913 | 490 |
proc = K (nat_get_Suc_simproc_fn n_sucs)}; |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
491 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
492 |
val no_split_ss = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
493 |
simpset_of (put_simpset HOL_ss @{context} |
62390 | 494 |
|> Splitter.del_split @{thm if_split}); |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
495 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
496 |
val expand_word_eq_sss = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
497 |
(simpset_of (put_simpset HOL_basic_ss @{context} addsimps |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
498 |
@{thms word_eq_rbl_eq word_le_rbl word_less_rbl word_sle_rbl word_sless_rbl}), |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
499 |
map simpset_of [ |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
500 |
put_simpset no_split_ss @{context} addsimps |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
501 |
@{thms rbl_word_plus rbl_word_and rbl_word_or rbl_word_not |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
502 |
rbl_word_neg bl_word_sub rbl_word_xor |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
503 |
rbl_word_cat rbl_word_slice rbl_word_scast |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
504 |
rbl_word_ucast rbl_shiftl rbl_shiftr rbl_sshiftr |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
505 |
rbl_word_if}, |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
506 |
put_simpset no_split_ss @{context} addsimps |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
507 |
@{thms to_bl_numeral to_bl_neg_numeral to_bl_0 rbl_word_1}, |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
508 |
put_simpset no_split_ss @{context} addsimps |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
509 |
@{thms rev_rev_ident rev_replicate rev_map to_bl_upt word_size} |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
510 |
addsimprocs [word_len_simproc], |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
511 |
put_simpset no_split_ss @{context} addsimps |
55465 | 512 |
@{thms list.simps split_conv replicate.simps list.map |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
513 |
zip_Cons_Cons zip_Nil drop_Suc_Cons drop_0 drop_Nil |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
514 |
foldr.simps map2_Cons map2_Nil takefill_Suc_Cons |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
515 |
takefill_Suc_Nil takefill.Z rbl_succ2_simps |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
516 |
rbl_plus_simps rev_bin_to_bl_simps append.simps |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
517 |
takefill_last_simps drop_nonempty_simps |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
518 |
rev_bl_order_simps} |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
519 |
addsimprocs [expand_upt_simproc, |
54883
dd04a8b654fc
proper context for norm_hhf and derived operations;
wenzelm
parents:
54847
diff
changeset
|
520 |
nat_get_Suc_simproc 4 |
61144 | 521 |
[@{term replicate}, @{term "takefill x"}, |
522 |
@{term drop}, @{term "bin_to_bl"}, |
|
523 |
@{term "takefill_last x"}, |
|
524 |
@{term "drop_nonempty x"}]], |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
525 |
put_simpset no_split_ss @{context} addsimps @{thms xor3_simps carry_simps if_bool_simps} |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
526 |
]) |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
527 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
528 |
fun tac ctxt = |
50107 | 529 |
let |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
530 |
val (ss, sss) = expand_word_eq_sss; |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
531 |
in |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
532 |
foldr1 (op THEN_ALL_NEW) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
533 |
((CHANGED o safe_full_simp_tac (put_simpset ss ctxt)) :: |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
534 |
map (fn ss => safe_full_simp_tac (put_simpset ss ctxt)) sss) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51686
diff
changeset
|
535 |
end; |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
536 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
537 |
end |
61799 | 538 |
\<close> |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
539 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
540 |
method_setup word_bitwise = |
61799 | 541 |
\<open>Scan.succeed (fn ctxt => Method.SIMPLE_METHOD (Word_Bitwise_Tac.tac ctxt 1))\<close> |
47567
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
542 |
"decomposer for word equalities and inequalities into bit propositions" |
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
543 |
|
407cabf66f21
New tactic "word_bitwise" expands word equalities/inequalities into logic.
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
diff
changeset
|
544 |
end |