author | haftmann |
Fri, 01 Nov 2013 18:51:14 +0100 | |
changeset 54230 | b1d955791529 |
parent 53438 | 6301ed01e34d |
child 54489 | 03ff4d1e6784 |
permissions | -rw-r--r-- |
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(* |
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Author: Jeremy Dawson, NICTA |
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*) |
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header {* Integers as implict bit strings *} |
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theory Bit_Representation |
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imports "~~/src/HOL/Library/Bit" Misc_Numeric |
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begin |
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subsection {* Constructors and destructors for binary integers *} |
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definition bitval :: "bit \<Rightarrow> 'a\<Colon>zero_neq_one" where |
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"bitval = bit_case 0 1" |
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lemma bitval_simps [simp]: |
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"bitval 0 = 0" |
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"bitval 1 = 1" |
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by (simp_all add: bitval_def) |
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definition Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where |
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"k BIT b = bitval b + k + k" |
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lemma Bit_B0: |
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"k BIT (0::bit) = k + k" |
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by (unfold Bit_def) simp |
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|
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lemma Bit_B1: |
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"k BIT (1::bit) = k + k + 1" |
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by (unfold Bit_def) simp |
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lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k" |
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by (rule trans, rule Bit_B0) simp |
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|
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lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1" |
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by (rule trans, rule Bit_B1) simp |
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definition bin_last :: "int \<Rightarrow> bit" where |
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"bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))" |
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definition bin_rest :: "int \<Rightarrow> int" where |
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"bin_rest w = w div 2" |
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lemma bin_rl_simp [simp]: |
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"bin_rest w BIT bin_last w = w" |
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unfolding bin_rest_def bin_last_def Bit_def |
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using mod_div_equality [of w 2] |
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by (cases "w mod 2 = 0", simp_all) |
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lemma bin_rest_BIT [simp]: "bin_rest (x BIT b) = x" |
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unfolding bin_rest_def Bit_def |
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by (cases b, simp_all) |
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lemma bin_last_BIT [simp]: "bin_last (x BIT b) = b" |
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unfolding bin_last_def Bit_def |
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by (cases b, simp_all add: z1pmod2) |
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lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \<longleftrightarrow> u = v \<and> b = c" |
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by (metis bin_rest_BIT bin_last_BIT) |
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lemma BIT_bin_simps [simp]: |
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"numeral k BIT 0 = numeral (Num.Bit0 k)" |
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"numeral k BIT 1 = numeral (Num.Bit1 k)" |
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"neg_numeral k BIT 0 = neg_numeral (Num.Bit0 k)" |
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"neg_numeral k BIT 1 = neg_numeral (Num.BitM k)" |
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unfolding neg_numeral_def numeral.simps numeral_BitM |
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unfolding Bit_def bitval_simps |
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by (simp_all del: arith_simps add_numeral_special diff_numeral_special) |
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lemma BIT_special_simps [simp]: |
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shows "0 BIT 0 = 0" and "0 BIT 1 = 1" and "1 BIT 0 = 2" and "1 BIT 1 = 3" |
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unfolding Bit_def by simp_all |
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lemma Bit_eq_0_iff: "w BIT b = 0 \<longleftrightarrow> w = 0 \<and> b = 0" |
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by (subst BIT_eq_iff [symmetric], simp) |
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lemma Bit_eq_m1_iff: "w BIT b = -1 \<longleftrightarrow> w = -1 \<and> b = 1" |
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by (subst BIT_eq_iff [symmetric], simp) |
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lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w" |
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by (induct w, simp_all) |
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lemma expand_BIT: |
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"numeral (Num.Bit0 w) = numeral w BIT 0" |
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"numeral (Num.Bit1 w) = numeral w BIT 1" |
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"neg_numeral (Num.Bit0 w) = neg_numeral w BIT 0" |
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"neg_numeral (Num.Bit1 w) = neg_numeral (w + Num.One) BIT 1" |
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unfolding add_One by (simp_all add: BitM_inc) |
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lemma bin_last_numeral_simps [simp]: |
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"bin_last 0 = 0" |
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"bin_last 1 = 1" |
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"bin_last -1 = 1" |
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"bin_last Numeral1 = 1" |
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"bin_last (numeral (Num.Bit0 w)) = 0" |
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"bin_last (numeral (Num.Bit1 w)) = 1" |
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"bin_last (neg_numeral (Num.Bit0 w)) = 0" |
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"bin_last (neg_numeral (Num.Bit1 w)) = 1" |
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unfolding expand_BIT bin_last_BIT by (simp_all add: bin_last_def) |
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lemma bin_rest_numeral_simps [simp]: |
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"bin_rest 0 = 0" |
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"bin_rest 1 = 0" |
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"bin_rest -1 = -1" |
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"bin_rest Numeral1 = 0" |
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"bin_rest (numeral (Num.Bit0 w)) = numeral w" |
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"bin_rest (numeral (Num.Bit1 w)) = numeral w" |
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"bin_rest (neg_numeral (Num.Bit0 w)) = neg_numeral w" |
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"bin_rest (neg_numeral (Num.Bit1 w)) = neg_numeral (w + Num.One)" |
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unfolding expand_BIT bin_rest_BIT by (simp_all add: bin_rest_def) |
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|
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lemma less_Bits: |
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"(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))" |
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unfolding Bit_def by (auto simp add: bitval_def split: bit.split) |
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lemma le_Bits: |
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"(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))" |
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unfolding Bit_def by (auto simp add: bitval_def split: bit.split) |
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lemma pred_BIT_simps [simp]: |
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"x BIT 0 - 1 = (x - 1) BIT 1" |
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"x BIT 1 - 1 = x BIT 0" |
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by (simp_all add: Bit_B0_2t Bit_B1_2t) |
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|
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lemma succ_BIT_simps [simp]: |
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"x BIT 0 + 1 = x BIT 1" |
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"x BIT 1 + 1 = (x + 1) BIT 0" |
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by (simp_all add: Bit_B0_2t Bit_B1_2t) |
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lemma add_BIT_simps [simp]: |
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"x BIT 0 + y BIT 0 = (x + y) BIT 0" |
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"x BIT 0 + y BIT 1 = (x + y) BIT 1" |
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"x BIT 1 + y BIT 0 = (x + y) BIT 1" |
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"x BIT 1 + y BIT 1 = (x + y + 1) BIT 0" |
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by (simp_all add: Bit_B0_2t Bit_B1_2t) |
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lemma mult_BIT_simps [simp]: |
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"x BIT 0 * y = (x * y) BIT 0" |
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"x * y BIT 0 = (x * y) BIT 0" |
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"x BIT 1 * y = (x * y) BIT 0 + y" |
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by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps) |
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|
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lemma B_mod_2': |
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"X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0" |
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apply (simp (no_asm) only: Bit_B0 Bit_B1) |
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apply (simp add: z1pmod2) |
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done |
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lemma neB1E [elim!]: |
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assumes ne: "y \<noteq> (1::bit)" |
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assumes y: "y = (0::bit) \<Longrightarrow> P" |
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shows "P" |
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apply (rule y) |
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apply (cases y rule: bit.exhaust, simp) |
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apply (simp add: ne) |
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done |
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|
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lemma bin_ex_rl: "EX w b. w BIT b = bin" |
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by (metis bin_rl_simp) |
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160 |
|
26557 | 161 |
lemma bin_exhaust: |
162 |
assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q" |
|
163 |
shows "Q" |
|
164 |
apply (insert bin_ex_rl [of bin]) |
|
165 |
apply (erule exE)+ |
|
166 |
apply (rule Q) |
|
167 |
apply force |
|
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168 |
done |
24333 | 169 |
|
26514 | 170 |
primrec bin_nth where |
37654 | 171 |
Z: "bin_nth w 0 = (bin_last w = (1::bit))" |
26557 | 172 |
| Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n" |
24364 | 173 |
|
26557 | 174 |
lemma bin_abs_lem: |
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"bin = (w BIT b) ==> bin ~= -1 --> bin ~= 0 --> |
26557 | 176 |
nat (abs w) < nat (abs bin)" |
46598 | 177 |
apply clarsimp |
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apply (unfold Bit_def) |
26557 | 179 |
apply (cases b) |
180 |
apply (clarsimp, arith) |
|
181 |
apply (clarsimp, arith) |
|
182 |
done |
|
183 |
||
184 |
lemma bin_induct: |
|
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assumes PPls: "P 0" |
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and PMin: "P -1" |
26557 | 187 |
and PBit: "!!bin bit. P bin ==> P (bin BIT bit)" |
188 |
shows "P bin" |
|
189 |
apply (rule_tac P=P and a=bin and f1="nat o abs" |
|
190 |
in wf_measure [THEN wf_induct]) |
|
191 |
apply (simp add: measure_def inv_image_def) |
|
192 |
apply (case_tac x rule: bin_exhaust) |
|
193 |
apply (frule bin_abs_lem) |
|
194 |
apply (auto simp add : PPls PMin PBit) |
|
195 |
done |
|
196 |
||
24333 | 197 |
lemma Bit_div2 [simp]: "(w BIT b) div 2 = w" |
46600 | 198 |
unfolding bin_rest_def [symmetric] by (rule bin_rest_BIT) |
24333 | 199 |
|
200 |
lemma bin_nth_lem [rule_format]: |
|
201 |
"ALL y. bin_nth x = bin_nth y --> x = y" |
|
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apply (induct x rule: bin_induct) |
24333 | 203 |
apply safe |
204 |
apply (erule rev_mp) |
|
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apply (induct_tac y rule: bin_induct) |
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apply safe |
24333 | 207 |
apply (drule_tac x=0 in fun_cong, force) |
208 |
apply (erule notE, rule ext, |
|
209 |
drule_tac x="Suc x" in fun_cong, force) |
|
46600 | 210 |
apply (drule_tac x=0 in fun_cong, force) |
24333 | 211 |
apply (erule rev_mp) |
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apply (induct_tac y rule: bin_induct) |
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213 |
apply safe |
24333 | 214 |
apply (drule_tac x=0 in fun_cong, force) |
215 |
apply (erule notE, rule ext, |
|
216 |
drule_tac x="Suc x" in fun_cong, force) |
|
46600 | 217 |
apply (drule_tac x=0 in fun_cong, force) |
24333 | 218 |
apply (case_tac y rule: bin_exhaust) |
219 |
apply clarify |
|
220 |
apply (erule allE) |
|
221 |
apply (erule impE) |
|
222 |
prefer 2 |
|
45848 | 223 |
apply (erule conjI) |
24333 | 224 |
apply (drule_tac x=0 in fun_cong, force) |
225 |
apply (rule ext) |
|
226 |
apply (drule_tac x="Suc ?x" in fun_cong, force) |
|
227 |
done |
|
228 |
||
229 |
lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)" |
|
230 |
by (auto elim: bin_nth_lem) |
|
231 |
||
45604 | 232 |
lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1]] |
24333 | 233 |
|
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lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)" |
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by (auto intro!: bin_nth_lem del: equalityI) |
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236 |
|
45853 | 237 |
lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n" |
238 |
by (induct n) auto |
|
239 |
||
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lemma bin_nth_1 [simp]: "bin_nth 1 n \<longleftrightarrow> n = 0" |
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by (cases n) simp_all |
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242 |
|
45952 | 243 |
lemma bin_nth_minus1 [simp]: "bin_nth -1 n" |
244 |
by (induct n) auto |
|
245 |
||
37654 | 246 |
lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))" |
24333 | 247 |
by auto |
248 |
||
249 |
lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n" |
|
250 |
by auto |
|
251 |
||
252 |
lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)" |
|
253 |
by (cases n) auto |
|
254 |
||
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lemma bin_nth_numeral: |
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"bin_rest x = y \<Longrightarrow> bin_nth x (numeral n) = bin_nth y (pred_numeral n)" |
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by (simp add: numeral_eq_Suc) |
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258 |
|
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lemmas bin_nth_numeral_simps [simp] = |
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260 |
bin_nth_numeral [OF bin_rest_numeral_simps(2)] |
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bin_nth_numeral [OF bin_rest_numeral_simps(5)] |
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|
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bin_nth_numeral [OF bin_rest_numeral_simps(6)] |
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bin_nth_numeral [OF bin_rest_numeral_simps(7)] |
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|
264 |
bin_nth_numeral [OF bin_rest_numeral_simps(8)] |
24333 | 265 |
|
266 |
lemmas bin_nth_simps = |
|
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267 |
bin_nth.Z bin_nth.Suc bin_nth_zero bin_nth_minus1 |
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|
268 |
bin_nth_numeral_simps |
24333 | 269 |
|
26557 | 270 |
|
271 |
subsection {* Truncating binary integers *} |
|
272 |
||
45846 | 273 |
definition bin_sign :: "int \<Rightarrow> int" where |
37667 | 274 |
bin_sign_def: "bin_sign k = (if k \<ge> 0 then 0 else - 1)" |
26557 | 275 |
|
276 |
lemma bin_sign_simps [simp]: |
|
45850 | 277 |
"bin_sign 0 = 0" |
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|
278 |
"bin_sign 1 = 0" |
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279 |
"bin_sign (numeral k) = 0" |
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|
280 |
"bin_sign (neg_numeral k) = -1" |
26557 | 281 |
"bin_sign (w BIT b) = bin_sign w" |
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282 |
unfolding bin_sign_def Bit_def bitval_def |
45850 | 283 |
by (simp_all split: bit.split) |
26557 | 284 |
|
24364 | 285 |
lemma bin_sign_rest [simp]: |
37667 | 286 |
"bin_sign (bin_rest w) = bin_sign w" |
26557 | 287 |
by (cases w rule: bin_exhaust) auto |
24364 | 288 |
|
37667 | 289 |
primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int" where |
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290 |
Z : "bintrunc 0 bin = 0" |
37667 | 291 |
| Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)" |
24364 | 292 |
|
37667 | 293 |
primrec sbintrunc :: "nat => int => int" where |
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|
294 |
Z : "sbintrunc 0 bin = (case bin_last bin of (1::bit) \<Rightarrow> -1 | (0::bit) \<Rightarrow> 0)" |
37667 | 295 |
| Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)" |
296 |
||
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lemma sign_bintr: "bin_sign (bintrunc n w) = 0" |
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|
298 |
by (induct n arbitrary: w) auto |
24333 | 299 |
|
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|
300 |
lemma bintrunc_mod2p: "bintrunc n w = (w mod 2 ^ n)" |
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|
301 |
apply (induct n arbitrary: w, clarsimp) |
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|
302 |
apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq) |
24333 | 303 |
done |
304 |
||
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|
305 |
lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n" |
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|
306 |
apply (induct n arbitrary: w) |
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|
307 |
apply simp |
30034 | 308 |
apply (subst mod_add_left_eq) |
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|
309 |
apply (simp add: bin_last_def) |
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|
310 |
apply (simp add: bin_last_def bin_rest_def Bit_def) |
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|
311 |
apply (clarsimp simp: mod_mult_mult1 [symmetric] |
24333 | 312 |
zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]]) |
313 |
apply (rule trans [symmetric, OF _ emep1]) |
|
314 |
apply auto |
|
315 |
apply (auto simp: even_def) |
|
316 |
done |
|
317 |
||
24465 | 318 |
subsection "Simplifications for (s)bintrunc" |
319 |
||
45852 | 320 |
lemma bintrunc_n_0 [simp]: "bintrunc n 0 = 0" |
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|
321 |
by (induct n) auto |
45852 | 322 |
|
45855 | 323 |
lemma sbintrunc_n_0 [simp]: "sbintrunc n 0 = 0" |
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|
324 |
by (induct n) auto |
45855 | 325 |
|
45856 | 326 |
lemma sbintrunc_n_minus1 [simp]: "sbintrunc n -1 = -1" |
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|
327 |
by (induct n) auto |
45856 | 328 |
|
45852 | 329 |
lemma bintrunc_Suc_numeral: |
330 |
"bintrunc (Suc n) 1 = 1" |
|
331 |
"bintrunc (Suc n) -1 = bintrunc n -1 BIT 1" |
|
47108
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|
332 |
"bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT 0" |
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|
333 |
"bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT 1" |
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|
334 |
"bintrunc (Suc n) (neg_numeral (Num.Bit0 w)) = |
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|
335 |
bintrunc n (neg_numeral w) BIT 0" |
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|
336 |
"bintrunc (Suc n) (neg_numeral (Num.Bit1 w)) = |
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|
337 |
bintrunc n (neg_numeral (w + Num.One)) BIT 1" |
45852 | 338 |
by simp_all |
339 |
||
45856 | 340 |
lemma sbintrunc_0_numeral [simp]: |
341 |
"sbintrunc 0 1 = -1" |
|
47108
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|
342 |
"sbintrunc 0 (numeral (Num.Bit0 w)) = 0" |
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changeset
|
343 |
"sbintrunc 0 (numeral (Num.Bit1 w)) = -1" |
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changeset
|
344 |
"sbintrunc 0 (neg_numeral (Num.Bit0 w)) = 0" |
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|
345 |
"sbintrunc 0 (neg_numeral (Num.Bit1 w)) = -1" |
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|
346 |
by simp_all |
45856 | 347 |
|
45855 | 348 |
lemma sbintrunc_Suc_numeral: |
349 |
"sbintrunc (Suc n) 1 = 1" |
|
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|
350 |
"sbintrunc (Suc n) (numeral (Num.Bit0 w)) = |
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changeset
|
351 |
sbintrunc n (numeral w) BIT 0" |
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changeset
|
352 |
"sbintrunc (Suc n) (numeral (Num.Bit1 w)) = |
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changeset
|
353 |
sbintrunc n (numeral w) BIT 1" |
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changeset
|
354 |
"sbintrunc (Suc n) (neg_numeral (Num.Bit0 w)) = |
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changeset
|
355 |
sbintrunc n (neg_numeral w) BIT 0" |
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changeset
|
356 |
"sbintrunc (Suc n) (neg_numeral (Num.Bit1 w)) = |
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|
357 |
sbintrunc n (neg_numeral (w + Num.One)) BIT 1" |
45855 | 358 |
by simp_all |
359 |
||
24465 | 360 |
lemma bit_bool: |
37654 | 361 |
"(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))" |
24465 | 362 |
by (cases b') auto |
363 |
||
364 |
lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric] |
|
24333 | 365 |
|
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|
366 |
lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n" |
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|
367 |
apply (induct n arbitrary: bin) |
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diff
changeset
|
368 |
apply (case_tac bin rule: bin_exhaust, case_tac b, auto) |
24333 | 369 |
done |
370 |
||
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
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diff
changeset
|
371 |
lemma nth_bintr: "bin_nth (bintrunc m w) n = (n < m & bin_nth w n)" |
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
372 |
apply (induct n arbitrary: w m) |
24333 | 373 |
apply (case_tac m, auto)[1] |
374 |
apply (case_tac m, auto)[1] |
|
375 |
done |
|
376 |
||
377 |
lemma nth_sbintr: |
|
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
378 |
"bin_nth (sbintrunc m w) n = |
24333 | 379 |
(if n < m then bin_nth w n else bin_nth w m)" |
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
380 |
apply (induct n arbitrary: w m) |
24333 | 381 |
apply (case_tac m, simp_all split: bit.splits)[1] |
382 |
apply (case_tac m, simp_all split: bit.splits)[1] |
|
383 |
done |
|
384 |
||
385 |
lemma bin_nth_Bit: |
|
37654 | 386 |
"bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))" |
24333 | 387 |
by (cases n) auto |
388 |
||
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
389 |
lemma bin_nth_Bit0: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46607
diff
changeset
|
390 |
"bin_nth (numeral (Num.Bit0 w)) n \<longleftrightarrow> |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
391 |
(\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
392 |
using bin_nth_Bit [where w="numeral w" and b="(0::bit)"] by simp |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
393 |
|
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
394 |
lemma bin_nth_Bit1: |
47108
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merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
395 |
"bin_nth (numeral (Num.Bit1 w)) n \<longleftrightarrow> |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
396 |
n = 0 \<or> (\<exists>m. n = Suc m \<and> bin_nth (numeral w) m)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
397 |
using bin_nth_Bit [where w="numeral w" and b="(1::bit)"] by simp |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
398 |
|
24333 | 399 |
lemma bintrunc_bintrunc_l: |
400 |
"n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)" |
|
401 |
by (rule bin_eqI) (auto simp add : nth_bintr) |
|
402 |
||
403 |
lemma sbintrunc_sbintrunc_l: |
|
404 |
"n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)" |
|
32439 | 405 |
by (rule bin_eqI) (auto simp: nth_sbintr) |
24333 | 406 |
|
407 |
lemma bintrunc_bintrunc_ge: |
|
408 |
"n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)" |
|
409 |
by (rule bin_eqI) (auto simp: nth_bintr) |
|
410 |
||
411 |
lemma bintrunc_bintrunc_min [simp]: |
|
412 |
"bintrunc m (bintrunc n w) = bintrunc (min m n) w" |
|
413 |
apply (rule bin_eqI) |
|
414 |
apply (auto simp: nth_bintr) |
|
415 |
done |
|
416 |
||
417 |
lemma sbintrunc_sbintrunc_min [simp]: |
|
418 |
"sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w" |
|
419 |
apply (rule bin_eqI) |
|
32642
026e7c6a6d08
be more cautious wrt. simp rules: inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer
haftmann
parents:
32439
diff
changeset
|
420 |
apply (auto simp: nth_sbintr min_max.inf_absorb1 min_max.inf_absorb2) |
24333 | 421 |
done |
422 |
||
423 |
lemmas bintrunc_Pls = |
|
47108
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merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
424 |
bintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps] |
24333 | 425 |
|
426 |
lemmas bintrunc_Min [simp] = |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
427 |
bintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps] |
24333 | 428 |
|
429 |
lemmas bintrunc_BIT [simp] = |
|
46600 | 430 |
bintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b |
24333 | 431 |
|
432 |
lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT |
|
45852 | 433 |
bintrunc_Suc_numeral |
24333 | 434 |
|
435 |
lemmas sbintrunc_Suc_Pls = |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
436 |
sbintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps] |
24333 | 437 |
|
438 |
lemmas sbintrunc_Suc_Min = |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
439 |
sbintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps] |
24333 | 440 |
|
441 |
lemmas sbintrunc_Suc_BIT [simp] = |
|
46600 | 442 |
sbintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b |
24333 | 443 |
|
444 |
lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT |
|
45855 | 445 |
sbintrunc_Suc_numeral |
24333 | 446 |
|
447 |
lemmas sbintrunc_Pls = |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
448 |
sbintrunc.Z [where bin="0", |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
449 |
simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] |
24333 | 450 |
|
451 |
lemmas sbintrunc_Min = |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
452 |
sbintrunc.Z [where bin="-1", |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
453 |
simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] |
24333 | 454 |
|
455 |
lemmas sbintrunc_0_BIT_B0 [simp] = |
|
37654 | 456 |
sbintrunc.Z [where bin="w BIT (0::bit)", |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
457 |
simplified bin_last_numeral_simps bin_rest_numeral_simps bit.simps] for w |
24333 | 458 |
|
459 |
lemmas sbintrunc_0_BIT_B1 [simp] = |
|
37654 | 460 |
sbintrunc.Z [where bin="w BIT (1::bit)", |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
461 |
simplified bin_last_BIT bin_rest_numeral_simps bit.simps] for w |
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
25919
diff
changeset
|
462 |
|
24333 | 463 |
lemmas sbintrunc_0_simps = |
464 |
sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1 |
|
465 |
||
466 |
lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs |
|
467 |
lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs |
|
468 |
||
469 |
lemma bintrunc_minus: |
|
470 |
"0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w" |
|
471 |
by auto |
|
472 |
||
473 |
lemma sbintrunc_minus: |
|
474 |
"0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w" |
|
475 |
by auto |
|
476 |
||
477 |
lemmas bintrunc_minus_simps = |
|
45604 | 478 |
bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans]] |
24333 | 479 |
lemmas sbintrunc_minus_simps = |
45604 | 480 |
sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]] |
24333 | 481 |
|
45604 | 482 |
lemmas thobini1 = arg_cong [where f = "%w. w BIT b"] for b |
24333 | 483 |
|
484 |
lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1] |
|
485 |
lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1] |
|
486 |
||
45604 | 487 |
lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]] |
24333 | 488 |
lemmas bintrunc_Pls_minus_I = bmsts(1) |
489 |
lemmas bintrunc_Min_minus_I = bmsts(2) |
|
490 |
lemmas bintrunc_BIT_minus_I = bmsts(3) |
|
491 |
||
492 |
lemma bintrunc_Suc_lem: |
|
493 |
"bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y" |
|
494 |
by auto |
|
495 |
||
496 |
lemmas bintrunc_Suc_Ialts = |
|
45604 | 497 |
bintrunc_Min_I [THEN bintrunc_Suc_lem] |
498 |
bintrunc_BIT_I [THEN bintrunc_Suc_lem] |
|
24333 | 499 |
|
500 |
lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1] |
|
501 |
||
502 |
lemmas sbintrunc_Suc_Is = |
|
45604 | 503 |
sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans]] |
24333 | 504 |
|
505 |
lemmas sbintrunc_Suc_minus_Is = |
|
45604 | 506 |
sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]] |
24333 | 507 |
|
508 |
lemma sbintrunc_Suc_lem: |
|
509 |
"sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y" |
|
510 |
by auto |
|
511 |
||
512 |
lemmas sbintrunc_Suc_Ialts = |
|
45604 | 513 |
sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem] |
24333 | 514 |
|
515 |
lemma sbintrunc_bintrunc_lt: |
|
516 |
"m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w" |
|
517 |
by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr) |
|
518 |
||
519 |
lemma bintrunc_sbintrunc_le: |
|
520 |
"m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w" |
|
521 |
apply (rule bin_eqI) |
|
522 |
apply (auto simp: nth_sbintr nth_bintr) |
|
523 |
apply (subgoal_tac "x=n", safe, arith+)[1] |
|
524 |
apply (subgoal_tac "x=n", safe, arith+)[1] |
|
525 |
done |
|
526 |
||
527 |
lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le] |
|
528 |
lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt] |
|
529 |
lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l] |
|
530 |
lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l] |
|
531 |
||
532 |
lemma bintrunc_sbintrunc' [simp]: |
|
533 |
"0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w" |
|
534 |
by (cases n) (auto simp del: bintrunc.Suc) |
|
535 |
||
536 |
lemma sbintrunc_bintrunc' [simp]: |
|
537 |
"0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w" |
|
538 |
by (cases n) (auto simp del: bintrunc.Suc) |
|
539 |
||
540 |
lemma bin_sbin_eq_iff: |
|
541 |
"bintrunc (Suc n) x = bintrunc (Suc n) y <-> |
|
542 |
sbintrunc n x = sbintrunc n y" |
|
543 |
apply (rule iffI) |
|
544 |
apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc]) |
|
545 |
apply simp |
|
546 |
apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc]) |
|
547 |
apply simp |
|
548 |
done |
|
549 |
||
550 |
lemma bin_sbin_eq_iff': |
|
551 |
"0 < n \<Longrightarrow> bintrunc n x = bintrunc n y <-> |
|
552 |
sbintrunc (n - 1) x = sbintrunc (n - 1) y" |
|
553 |
by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc) |
|
554 |
||
555 |
lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def] |
|
556 |
lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def] |
|
557 |
||
558 |
lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l] |
|
559 |
lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l] |
|
560 |
||
561 |
(* although bintrunc_minus_simps, if added to default simpset, |
|
562 |
tends to get applied where it's not wanted in developing the theories, |
|
563 |
we get a version for when the word length is given literally *) |
|
564 |
||
565 |
lemmas nat_non0_gr = |
|
45604 | 566 |
trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl] |
24333 | 567 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
568 |
lemma bintrunc_numeral: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
569 |
"bintrunc (numeral k) x = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
570 |
bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x" |
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
571 |
by (simp add: numeral_eq_Suc) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
572 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
573 |
lemma sbintrunc_numeral: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
574 |
"sbintrunc (numeral k) x = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
575 |
sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x" |
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
576 |
by (simp add: numeral_eq_Suc) |
24333 | 577 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
578 |
lemma bintrunc_numeral_simps [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
579 |
"bintrunc (numeral k) (numeral (Num.Bit0 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
580 |
bintrunc (pred_numeral k) (numeral w) BIT 0" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
581 |
"bintrunc (numeral k) (numeral (Num.Bit1 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
582 |
bintrunc (pred_numeral k) (numeral w) BIT 1" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
583 |
"bintrunc (numeral k) (neg_numeral (Num.Bit0 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
584 |
bintrunc (pred_numeral k) (neg_numeral w) BIT 0" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
585 |
"bintrunc (numeral k) (neg_numeral (Num.Bit1 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
586 |
bintrunc (pred_numeral k) (neg_numeral (w + Num.One)) BIT 1" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
587 |
"bintrunc (numeral k) 1 = 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
588 |
by (simp_all add: bintrunc_numeral) |
24333 | 589 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
590 |
lemma sbintrunc_numeral_simps [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
591 |
"sbintrunc (numeral k) (numeral (Num.Bit0 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
592 |
sbintrunc (pred_numeral k) (numeral w) BIT 0" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
593 |
"sbintrunc (numeral k) (numeral (Num.Bit1 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
594 |
sbintrunc (pred_numeral k) (numeral w) BIT 1" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
595 |
"sbintrunc (numeral k) (neg_numeral (Num.Bit0 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
596 |
sbintrunc (pred_numeral k) (neg_numeral w) BIT 0" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
597 |
"sbintrunc (numeral k) (neg_numeral (Num.Bit1 w)) = |
47219
172c031ad743
restate various simp rules for word operations using pred_numeral
huffman
parents:
47170
diff
changeset
|
598 |
sbintrunc (pred_numeral k) (neg_numeral (w + Num.One)) BIT 1" |
47108
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huffman
parents:
46607
diff
changeset
|
599 |
"sbintrunc (numeral k) 1 = 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
600 |
by (simp_all add: sbintrunc_numeral) |
24333 | 601 |
|
53062
3af1a6020014
some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents:
47219
diff
changeset
|
602 |
lemma no_bintr_alt1: "bintrunc n = (\<lambda>w. w mod 2 ^ n :: int)" |
24333 | 603 |
by (rule ext) (rule bintrunc_mod2p) |
604 |
||
605 |
lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}" |
|
606 |
apply (unfold no_bintr_alt1) |
|
607 |
apply (auto simp add: image_iff) |
|
608 |
apply (rule exI) |
|
609 |
apply (auto intro: int_mod_lem [THEN iffD1, symmetric]) |
|
610 |
done |
|
611 |
||
612 |
lemma no_sbintr_alt2: |
|
613 |
"sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)" |
|
614 |
by (rule ext) (simp add : sbintrunc_mod2p) |
|
615 |
||
616 |
lemma range_sbintrunc: |
|
617 |
"range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}" |
|
618 |
apply (unfold no_sbintr_alt2) |
|
619 |
apply (auto simp add: image_iff eq_diff_eq) |
|
620 |
apply (rule exI) |
|
621 |
apply (auto intro: int_mod_lem [THEN iffD1, symmetric]) |
|
622 |
done |
|
623 |
||
25349
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
624 |
lemma sb_inc_lem: |
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
625 |
"(a::int) + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)" |
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
626 |
apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p]) |
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
627 |
apply (rule TrueI) |
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
628 |
done |
24333 | 629 |
|
25349
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
630 |
lemma sb_inc_lem': |
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
631 |
"(a::int) < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)" |
35048 | 632 |
by (rule sb_inc_lem) simp |
24333 | 633 |
|
634 |
lemma sbintrunc_inc: |
|
25349
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
635 |
"x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x" |
24333 | 636 |
unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp |
637 |
||
25349
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
638 |
lemma sb_dec_lem: |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53438
diff
changeset
|
639 |
"(0::int) \<le> - (2 ^ k) + a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53438
diff
changeset
|
640 |
using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp |
24333 | 641 |
|
25349
0d46bea01741
eliminated illegal schematic variables in where/of;
wenzelm
parents:
25134
diff
changeset
|
642 |
lemma sb_dec_lem': |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53438
diff
changeset
|
643 |
"(2::int) ^ k \<le> a \<Longrightarrow> (a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a" |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53438
diff
changeset
|
644 |
by (rule sb_dec_lem) simp |
24333 | 645 |
|
646 |
lemma sbintrunc_dec: |
|
647 |
"x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x" |
|
648 |
unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp |
|
649 |
||
47168 | 650 |
lemmas zmod_uminus' = zminus_zmod [where m=c] for c |
47164 | 651 |
lemmas zpower_zmod' = power_mod [where b=c and n=k] for c k |
24333 | 652 |
|
47168 | 653 |
lemmas brdmod1s' [symmetric] = |
654 |
mod_add_left_eq mod_add_right_eq |
|
655 |
mod_diff_left_eq mod_diff_right_eq |
|
656 |
mod_mult_left_eq mod_mult_right_eq |
|
24333 | 657 |
|
658 |
lemmas brdmods' [symmetric] = |
|
659 |
zpower_zmod' [symmetric] |
|
30034 | 660 |
trans [OF mod_add_left_eq mod_add_right_eq] |
47168 | 661 |
trans [OF mod_diff_left_eq mod_diff_right_eq] |
662 |
trans [OF mod_mult_right_eq mod_mult_left_eq] |
|
24333 | 663 |
zmod_uminus' [symmetric] |
30034 | 664 |
mod_add_left_eq [where b = "1::int"] |
47168 | 665 |
mod_diff_left_eq [where b = "1::int"] |
24333 | 666 |
|
667 |
lemmas bintr_arith1s = |
|
46000 | 668 |
brdmod1s' [where c="2^n::int", folded bintrunc_mod2p] for n |
24333 | 669 |
lemmas bintr_ariths = |
46000 | 670 |
brdmods' [where c="2^n::int", folded bintrunc_mod2p] for n |
24333 | 671 |
|
45604 | 672 |
lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p] |
24364 | 673 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
674 |
lemma bintr_ge0: "0 \<le> bintrunc n w" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
675 |
by (simp add: bintrunc_mod2p) |
24333 | 676 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
677 |
lemma bintr_lt2p: "bintrunc n w < 2 ^ n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
678 |
by (simp add: bintrunc_mod2p) |
24333 | 679 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
680 |
lemma bintr_Min: "bintrunc n -1 = 2 ^ n - 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
681 |
by (simp add: bintrunc_mod2p m1mod2k) |
24333 | 682 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
683 |
lemma sbintr_ge: "- (2 ^ n) \<le> sbintrunc n w" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
684 |
by (simp add: sbintrunc_mod2p) |
24333 | 685 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
686 |
lemma sbintr_lt: "sbintrunc n w < 2 ^ n" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
687 |
by (simp add: sbintrunc_mod2p) |
24333 | 688 |
|
689 |
lemma sign_Pls_ge_0: |
|
46604
9f9e85264e4d
make uses of bin_sign respect int/bin distinction
huffman
parents:
46601
diff
changeset
|
690 |
"(bin_sign bin = 0) = (bin >= (0 :: int))" |
9f9e85264e4d
make uses of bin_sign respect int/bin distinction
huffman
parents:
46601
diff
changeset
|
691 |
unfolding bin_sign_def by simp |
24333 | 692 |
|
693 |
lemma sign_Min_lt_0: |
|
46604
9f9e85264e4d
make uses of bin_sign respect int/bin distinction
huffman
parents:
46601
diff
changeset
|
694 |
"(bin_sign bin = -1) = (bin < (0 :: int))" |
9f9e85264e4d
make uses of bin_sign respect int/bin distinction
huffman
parents:
46601
diff
changeset
|
695 |
unfolding bin_sign_def by simp |
24333 | 696 |
|
697 |
lemma bin_rest_trunc: |
|
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
698 |
"(bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)" |
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
699 |
by (induct n arbitrary: bin) auto |
24333 | 700 |
|
53062
3af1a6020014
some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents:
47219
diff
changeset
|
701 |
lemma bin_rest_power_trunc: |
30971 | 702 |
"(bin_rest ^^ k) (bintrunc n bin) = |
703 |
bintrunc (n - k) ((bin_rest ^^ k) bin)" |
|
24333 | 704 |
by (induct k) (auto simp: bin_rest_trunc) |
705 |
||
706 |
lemma bin_rest_trunc_i: |
|
707 |
"bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)" |
|
708 |
by auto |
|
709 |
||
710 |
lemma bin_rest_strunc: |
|
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
711 |
"bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)" |
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
712 |
by (induct n arbitrary: bin) auto |
24333 | 713 |
|
714 |
lemma bintrunc_rest [simp]: |
|
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
715 |
"bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)" |
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
716 |
apply (induct n arbitrary: bin, simp) |
24333 | 717 |
apply (case_tac bin rule: bin_exhaust) |
718 |
apply (auto simp: bintrunc_bintrunc_l) |
|
719 |
done |
|
720 |
||
721 |
lemma sbintrunc_rest [simp]: |
|
45954
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
722 |
"sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)" |
f67d3bb5f09c
use 'induct arbitrary' instead of universal quantifiers
huffman
parents:
45953
diff
changeset
|
723 |
apply (induct n arbitrary: bin, simp) |
24333 | 724 |
apply (case_tac bin rule: bin_exhaust) |
725 |
apply (auto simp: bintrunc_bintrunc_l split: bit.splits) |
|
726 |
done |
|
727 |
||
728 |
lemma bintrunc_rest': |
|
729 |
"bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n" |
|
730 |
by (rule ext) auto |
|
731 |
||
732 |
lemma sbintrunc_rest' : |
|
733 |
"sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n" |
|
734 |
by (rule ext) auto |
|
735 |
||
736 |
lemma rco_lem: |
|
30971 | 737 |
"f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f" |
24333 | 738 |
apply (rule ext) |
739 |
apply (induct_tac n) |
|
740 |
apply (simp_all (no_asm)) |
|
741 |
apply (drule fun_cong) |
|
742 |
apply (unfold o_def) |
|
743 |
apply (erule trans) |
|
744 |
apply simp |
|
745 |
done |
|
746 |
||
747 |
lemmas rco_bintr = bintrunc_rest' |
|
748 |
[THEN rco_lem [THEN fun_cong], unfolded o_def] |
|
749 |
lemmas rco_sbintr = sbintrunc_rest' |
|
750 |
[THEN rco_lem [THEN fun_cong], unfolded o_def] |
|
751 |
||
53062
3af1a6020014
some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents:
47219
diff
changeset
|
752 |
|
24364 | 753 |
subsection {* Splitting and concatenation *} |
754 |
||
26557 | 755 |
primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
756 |
Z: "bin_split 0 w = (w, 0)" |
26557 | 757 |
| Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) |
758 |
in (w1, w2 BIT bin_last w))" |
|
24364 | 759 |
|
37667 | 760 |
lemma [code]: |
761 |
"bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))" |
|
762 |
"bin_split 0 w = (w, 0)" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46607
diff
changeset
|
763 |
by simp_all |
37667 | 764 |
|
26557 | 765 |
primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where |
766 |
Z: "bin_cat w 0 v = w" |
|
767 |
| Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v" |
|
24364 | 768 |
|
53062
3af1a6020014
some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents:
47219
diff
changeset
|
769 |
end |
24364 | 770 |