# Theory Bit_Representation

theory Bit_Representation
imports Misc_Numeric
```(*
Author: Jeremy Dawson, NICTA
*)

section ‹Integers as implict bit strings›

theory Bit_Representation
imports Misc_Numeric
begin

subsection ‹Constructors and destructors for binary integers›

definition Bit :: "int ⇒ bool ⇒ int" (infixl "BIT" 90)
where
"k BIT b = (if b then 1 else 0) + k + k"

lemma Bit_B0:
"k BIT False = k + k"
by (unfold Bit_def) simp

lemma Bit_B1:
"k BIT True = k + k + 1"
by (unfold Bit_def) simp

lemma Bit_B0_2t: "k BIT False = 2 * k"
by (rule trans, rule Bit_B0) simp

lemma Bit_B1_2t: "k BIT True = 2 * k + 1"
by (rule trans, rule Bit_B1) simp

definition bin_last :: "int ⇒ bool"
where
"bin_last w ⟷ w mod 2 = 1"

lemma bin_last_odd:
"bin_last = odd"
by (rule ext) (simp add: bin_last_def even_iff_mod_2_eq_zero)

definition bin_rest :: "int ⇒ int"
where
"bin_rest w = w div 2"

lemma bin_rl_simp [simp]:
"bin_rest w BIT bin_last w = w"
unfolding bin_rest_def bin_last_def Bit_def
using div_mult_mod_eq [of w 2]
by (cases "w mod 2 = 0", simp_all)

lemma bin_rest_BIT [simp]: "bin_rest (x BIT b) = x"
unfolding bin_rest_def Bit_def
by (cases b, simp_all)

lemma bin_last_BIT [simp]: "bin_last (x BIT b) = b"
unfolding bin_last_def Bit_def
by (cases b) simp_all

lemma BIT_eq_iff [iff]: "u BIT b = v BIT c ⟷ u = v ∧ b = c"
apply arith
apply arith
done

lemma BIT_bin_simps [simp]:
"numeral k BIT False = numeral (Num.Bit0 k)"
"numeral k BIT True = numeral (Num.Bit1 k)"
"(- numeral k) BIT False = - numeral (Num.Bit0 k)"
"(- numeral k) BIT True = - numeral (Num.BitM k)"
unfolding numeral.simps numeral_BitM
unfolding Bit_def
by (simp_all del: arith_simps add_numeral_special diff_numeral_special)

lemma BIT_special_simps [simp]:
shows "0 BIT False = 0" and "0 BIT True = 1"
and "1 BIT False = 2" and "1 BIT True = 3"
and "(- 1) BIT False = - 2" and "(- 1) BIT True = - 1"
unfolding Bit_def by simp_all

lemma Bit_eq_0_iff: "w BIT b = 0 ⟷ w = 0 ∧ ¬ b"
apply arith
done

lemma Bit_eq_m1_iff: "w BIT b = -1 ⟷ w = -1 ∧ b"
apply arith
done

lemma BitM_inc: "Num.BitM (Num.inc w) = Num.Bit1 w"
by (induct w, simp_all)

lemma expand_BIT:
"numeral (Num.Bit0 w) = numeral w BIT False"
"numeral (Num.Bit1 w) = numeral w BIT True"
"- numeral (Num.Bit0 w) = (- numeral w) BIT False"
"- numeral (Num.Bit1 w) = (- numeral (w + Num.One)) BIT True"

lemma bin_last_numeral_simps [simp]:
"¬ bin_last 0"
"bin_last 1"
"bin_last (- 1)"
"bin_last Numeral1"
"¬ bin_last (numeral (Num.Bit0 w))"
"bin_last (numeral (Num.Bit1 w))"
"¬ bin_last (- numeral (Num.Bit0 w))"
"bin_last (- numeral (Num.Bit1 w))"

lemma bin_rest_numeral_simps [simp]:
"bin_rest 0 = 0"
"bin_rest 1 = 0"
"bin_rest (- 1) = - 1"
"bin_rest Numeral1 = 0"
"bin_rest (numeral (Num.Bit0 w)) = numeral w"
"bin_rest (numeral (Num.Bit1 w)) = numeral w"
"bin_rest (- numeral (Num.Bit0 w)) = - numeral w"
"bin_rest (- numeral (Num.Bit1 w)) = - numeral (w + Num.One)"

lemma less_Bits:
"v BIT b < w BIT c ⟷ v < w ∨ v ≤ w ∧ ¬ b ∧ c"
unfolding Bit_def by auto

lemma le_Bits:
"v BIT b ≤ w BIT c ⟷ v < w ∨ v ≤ w ∧ (¬ b ∨ c)"
unfolding Bit_def by auto

lemma pred_BIT_simps [simp]:
"x BIT False - 1 = (x - 1) BIT True"
"x BIT True - 1 = x BIT False"

lemma succ_BIT_simps [simp]:
"x BIT False + 1 = x BIT True"
"x BIT True + 1 = (x + 1) BIT False"

"x BIT False + y BIT False = (x + y) BIT False"
"x BIT False + y BIT True = (x + y) BIT True"
"x BIT True + y BIT False = (x + y) BIT True"
"x BIT True + y BIT True = (x + y + 1) BIT False"

lemma mult_BIT_simps [simp]:
"x BIT False * y = (x * y) BIT False"
"x * y BIT False = (x * y) BIT False"
"x BIT True * y = (x * y) BIT False + y"
by (simp_all add: Bit_B0_2t Bit_B1_2t algebra_simps)

lemma B_mod_2':
"X = 2 ==> (w BIT True) mod X = 1 & (w BIT False) mod X = 0"
apply (simp (no_asm) only: Bit_B0 Bit_B1)
apply simp
done

lemma bin_ex_rl: "EX w b. w BIT b = bin"
by (metis bin_rl_simp)

lemma bin_exhaust:
assumes Q: "⋀x b. bin = x BIT b ⟹ Q"
shows "Q"
apply (insert bin_ex_rl [of bin])
apply (erule exE)+
apply (rule Q)
apply force
done

primrec bin_nth where
Z: "bin_nth w 0 ⟷ bin_last w"
| Suc: "bin_nth w (Suc n) ⟷ bin_nth (bin_rest w) n"

lemma bin_abs_lem:
"bin = (w BIT b) ==> bin ~= -1 --> bin ~= 0 -->
nat ¦w¦ < nat ¦bin¦"
apply clarsimp
apply (unfold Bit_def)
apply (cases b)
apply (clarsimp, arith)
apply (clarsimp, arith)
done

lemma bin_induct:
assumes PPls: "P 0"
and PMin: "P (- 1)"
and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"
shows "P bin"
apply (rule_tac P=P and a=bin and f1="nat o abs"
in wf_measure [THEN wf_induct])
apply (case_tac x rule: bin_exhaust)
apply (frule bin_abs_lem)
apply (auto simp add : PPls PMin PBit)
done

lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
unfolding bin_rest_def [symmetric] by (rule bin_rest_BIT)

lemma bin_nth_eq_iff:
"bin_nth x = bin_nth y ⟷ x = y"
proof -
have bin_nth_lem [rule_format]: "ALL y. bin_nth x = bin_nth y --> x = y"
apply (induct x rule: bin_induct)
apply safe
apply (erule rev_mp)
apply (induct_tac y rule: bin_induct)
apply safe
apply (drule_tac x=0 in fun_cong, force)
apply (erule notE, rule ext,
drule_tac x="Suc x" in fun_cong, force)
apply (drule_tac x=0 in fun_cong, force)
apply (erule rev_mp)
apply (induct_tac y rule: bin_induct)
apply safe
apply (drule_tac x=0 in fun_cong, force)
apply (erule notE, rule ext,
drule_tac x="Suc x" in fun_cong, force)
apply (metis Bit_eq_m1_iff Z bin_last_BIT)
apply (case_tac y rule: bin_exhaust)
apply clarify
apply (erule allE)
apply (erule impE)
prefer 2
apply (erule conjI)
apply (drule_tac x=0 in fun_cong, force)
apply (rule ext)
apply (drule_tac x="Suc x" for x in fun_cong, force)
done
show ?thesis
by (auto elim: bin_nth_lem)
qed

lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1]]

lemma bin_eq_iff:
"x = y ⟷ (∀n. bin_nth x n = bin_nth y n)"
using bin_nth_eq_iff by auto

lemma bin_nth_zero [simp]: "¬ bin_nth 0 n"
by (induct n) auto

lemma bin_nth_1 [simp]: "bin_nth 1 n ⟷ n = 0"
by (cases n) simp_all

lemma bin_nth_minus1 [simp]: "bin_nth (- 1) n"
by (induct n) auto

lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 ⟷ b"
by auto

lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
by auto

lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)"
by (cases n) auto

lemma bin_nth_numeral:
"bin_rest x = y ⟹ bin_nth x (numeral n) = bin_nth y (pred_numeral n)"

lemmas bin_nth_numeral_simps [simp] =
bin_nth_numeral [OF bin_rest_numeral_simps(2)]
bin_nth_numeral [OF bin_rest_numeral_simps(5)]
bin_nth_numeral [OF bin_rest_numeral_simps(6)]
bin_nth_numeral [OF bin_rest_numeral_simps(7)]
bin_nth_numeral [OF bin_rest_numeral_simps(8)]

lemmas bin_nth_simps =
bin_nth.Z bin_nth.Suc bin_nth_zero bin_nth_minus1
bin_nth_numeral_simps

subsection ‹Truncating binary integers›

definition bin_sign :: "int ⇒ int"
where
bin_sign_def: "bin_sign k = (if k ≥ 0 then 0 else - 1)"

lemma bin_sign_simps [simp]:
"bin_sign 0 = 0"
"bin_sign 1 = 0"
"bin_sign (- 1) = - 1"
"bin_sign (numeral k) = 0"
"bin_sign (- numeral k) = -1"
"bin_sign (w BIT b) = bin_sign w"
unfolding bin_sign_def Bit_def
by simp_all

lemma bin_sign_rest [simp]:
"bin_sign (bin_rest w) = bin_sign w"
by (cases w rule: bin_exhaust) auto

primrec bintrunc :: "nat ⇒ int ⇒ int" where
Z : "bintrunc 0 bin = 0"
| Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"

primrec sbintrunc :: "nat => int => int" where
Z : "sbintrunc 0 bin = (if bin_last bin then -1 else 0)"
| Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"

lemma sign_bintr: "bin_sign (bintrunc n w) = 0"
by (induct n arbitrary: w) auto

lemma bintrunc_mod2p: "bintrunc n w = (w mod 2 ^ n)"
apply (induct n arbitrary: w, clarsimp)
apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)
done

lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n"
apply (induct n arbitrary: w)
apply simp
apply arith
apply (simp add: bin_last_def bin_rest_def Bit_def)
apply (clarsimp simp: mod_mult_mult1 [symmetric]
mult_div_mod_eq [symmetric, THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
apply (rule trans [symmetric, OF _ emep1])
apply auto
done

subsection "Simplifications for (s)bintrunc"

lemma bintrunc_n_0 [simp]: "bintrunc n 0 = 0"
by (induct n) auto

lemma sbintrunc_n_0 [simp]: "sbintrunc n 0 = 0"
by (induct n) auto

lemma sbintrunc_n_minus1 [simp]: "sbintrunc n (- 1) = -1"
by (induct n) auto

lemma bintrunc_Suc_numeral:
"bintrunc (Suc n) 1 = 1"
"bintrunc (Suc n) (- 1) = bintrunc n (- 1) BIT True"
"bintrunc (Suc n) (numeral (Num.Bit0 w)) = bintrunc n (numeral w) BIT False"
"bintrunc (Suc n) (numeral (Num.Bit1 w)) = bintrunc n (numeral w) BIT True"
"bintrunc (Suc n) (- numeral (Num.Bit0 w)) =
bintrunc n (- numeral w) BIT False"
"bintrunc (Suc n) (- numeral (Num.Bit1 w)) =
bintrunc n (- numeral (w + Num.One)) BIT True"
by simp_all

lemma sbintrunc_0_numeral [simp]:
"sbintrunc 0 1 = -1"
"sbintrunc 0 (numeral (Num.Bit0 w)) = 0"
"sbintrunc 0 (numeral (Num.Bit1 w)) = -1"
"sbintrunc 0 (- numeral (Num.Bit0 w)) = 0"
"sbintrunc 0 (- numeral (Num.Bit1 w)) = -1"
by simp_all

lemma sbintrunc_Suc_numeral:
"sbintrunc (Suc n) 1 = 1"
"sbintrunc (Suc n) (numeral (Num.Bit0 w)) =
sbintrunc n (numeral w) BIT False"
"sbintrunc (Suc n) (numeral (Num.Bit1 w)) =
sbintrunc n (numeral w) BIT True"
"sbintrunc (Suc n) (- numeral (Num.Bit0 w)) =
sbintrunc n (- numeral w) BIT False"
"sbintrunc (Suc n) (- numeral (Num.Bit1 w)) =
sbintrunc n (- numeral (w + Num.One)) BIT True"
by simp_all

lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n"
apply (induct n arbitrary: bin)
apply (case_tac bin rule: bin_exhaust, case_tac b, auto)
done

lemma nth_bintr: "bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
apply (induct n arbitrary: w m)
apply (case_tac m, auto)[1]
apply (case_tac m, auto)[1]
done

lemma nth_sbintr:
"bin_nth (sbintrunc m w) n =
(if n < m then bin_nth w n else bin_nth w m)"
apply (induct n arbitrary: w m)
apply (case_tac m)
apply simp_all
apply (case_tac m)
apply simp_all
done

lemma bin_nth_Bit:
"bin_nth (w BIT b) n = (n = 0 & b | (EX m. n = Suc m & bin_nth w m))"
by (cases n) auto

lemma bin_nth_Bit0:
"bin_nth (numeral (Num.Bit0 w)) n ⟷
(∃m. n = Suc m ∧ bin_nth (numeral w) m)"
using bin_nth_Bit [where w="numeral w" and b="False"] by simp

lemma bin_nth_Bit1:
"bin_nth (numeral (Num.Bit1 w)) n ⟷
n = 0 ∨ (∃m. n = Suc m ∧ bin_nth (numeral w) m)"
using bin_nth_Bit [where w="numeral w" and b="True"] by simp

lemma bintrunc_bintrunc_l:
"n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
by (rule bin_eqI) (auto simp add : nth_bintr)

lemma sbintrunc_sbintrunc_l:
"n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)"
by (rule bin_eqI) (auto simp: nth_sbintr)

lemma bintrunc_bintrunc_ge:
"n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)"
by (rule bin_eqI) (auto simp: nth_bintr)

lemma bintrunc_bintrunc_min [simp]:
"bintrunc m (bintrunc n w) = bintrunc (min m n) w"
apply (rule bin_eqI)
apply (auto simp: nth_bintr)
done

lemma sbintrunc_sbintrunc_min [simp]:
"sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
apply (rule bin_eqI)
apply (auto simp: nth_sbintr min.absorb1 min.absorb2)
done

lemmas bintrunc_Pls =
bintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]

lemmas bintrunc_Min [simp] =
bintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]

lemmas bintrunc_BIT  [simp] =
bintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b

lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
bintrunc_Suc_numeral

lemmas sbintrunc_Suc_Pls =
sbintrunc.Suc [where bin="0", simplified bin_last_numeral_simps bin_rest_numeral_simps]

lemmas sbintrunc_Suc_Min =
sbintrunc.Suc [where bin="-1", simplified bin_last_numeral_simps bin_rest_numeral_simps]

lemmas sbintrunc_Suc_BIT [simp] =
sbintrunc.Suc [where bin="w BIT b", simplified bin_last_BIT bin_rest_BIT] for w b

lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
sbintrunc_Suc_numeral

lemmas sbintrunc_Pls =
sbintrunc.Z [where bin="0",
simplified bin_last_numeral_simps bin_rest_numeral_simps]

lemmas sbintrunc_Min =
sbintrunc.Z [where bin="-1",
simplified bin_last_numeral_simps bin_rest_numeral_simps]

lemmas sbintrunc_0_BIT_B0 [simp] =
sbintrunc.Z [where bin="w BIT False",
simplified bin_last_numeral_simps bin_rest_numeral_simps] for w

lemmas sbintrunc_0_BIT_B1 [simp] =
sbintrunc.Z [where bin="w BIT True",
simplified bin_last_BIT bin_rest_numeral_simps] for w

lemmas sbintrunc_0_simps =
sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1

lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs
lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs

lemma bintrunc_minus:
"0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"
by auto

lemma sbintrunc_minus:
"0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
by auto

lemmas bintrunc_minus_simps =
bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans]]
lemmas sbintrunc_minus_simps =
sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]

lemmas thobini1 = arg_cong [where f = "%w. w BIT b"] for b

lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]
lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]

lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]
lemmas bintrunc_Pls_minus_I = bmsts(1)
lemmas bintrunc_Min_minus_I = bmsts(2)
lemmas bintrunc_BIT_minus_I = bmsts(3)

lemma bintrunc_Suc_lem:
"bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"
by auto

lemmas bintrunc_Suc_Ialts =
bintrunc_Min_I [THEN bintrunc_Suc_lem]
bintrunc_BIT_I [THEN bintrunc_Suc_lem]

lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]

lemmas sbintrunc_Suc_Is =
sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans]]

lemmas sbintrunc_Suc_minus_Is =
sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]

lemma sbintrunc_Suc_lem:
"sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"
by auto

lemmas sbintrunc_Suc_Ialts =
sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]

lemma sbintrunc_bintrunc_lt:
"m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w"
by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)

lemma bintrunc_sbintrunc_le:
"m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w"
apply (rule bin_eqI)
apply (auto simp: nth_sbintr nth_bintr)
apply (subgoal_tac "x=n", safe, arith+)[1]
apply (subgoal_tac "x=n", safe, arith+)[1]
done

lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]
lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]
lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]
lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]

lemma bintrunc_sbintrunc' [simp]:
"0 < n ⟹ bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
by (cases n) (auto simp del: bintrunc.Suc)

lemma sbintrunc_bintrunc' [simp]:
"0 < n ⟹ sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
by (cases n) (auto simp del: bintrunc.Suc)

lemma bin_sbin_eq_iff:
"bintrunc (Suc n) x = bintrunc (Suc n) y ⟷
sbintrunc n x = sbintrunc n y"
apply (rule iffI)
apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])
apply simp
apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc])
apply simp
done

lemma bin_sbin_eq_iff':
"0 < n ⟹ bintrunc n x = bintrunc n y ⟷
sbintrunc (n - 1) x = sbintrunc (n - 1) y"
by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc)

lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]
lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def]

lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]
lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]

(* although bintrunc_minus_simps, if added to default simpset,
tends to get applied where it's not wanted in developing the theories,
we get a version for when the word length is given literally *)

lemmas nat_non0_gr =
trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl]

lemma bintrunc_numeral:
"bintrunc (numeral k) x =
bintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"

lemma sbintrunc_numeral:
"sbintrunc (numeral k) x =
sbintrunc (pred_numeral k) (bin_rest x) BIT bin_last x"

lemma bintrunc_numeral_simps [simp]:
"bintrunc (numeral k) (numeral (Num.Bit0 w)) =
bintrunc (pred_numeral k) (numeral w) BIT False"
"bintrunc (numeral k) (numeral (Num.Bit1 w)) =
bintrunc (pred_numeral k) (numeral w) BIT True"
"bintrunc (numeral k) (- numeral (Num.Bit0 w)) =
bintrunc (pred_numeral k) (- numeral w) BIT False"
"bintrunc (numeral k) (- numeral (Num.Bit1 w)) =
bintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
"bintrunc (numeral k) 1 = 1"

lemma sbintrunc_numeral_simps [simp]:
"sbintrunc (numeral k) (numeral (Num.Bit0 w)) =
sbintrunc (pred_numeral k) (numeral w) BIT False"
"sbintrunc (numeral k) (numeral (Num.Bit1 w)) =
sbintrunc (pred_numeral k) (numeral w) BIT True"
"sbintrunc (numeral k) (- numeral (Num.Bit0 w)) =
sbintrunc (pred_numeral k) (- numeral w) BIT False"
"sbintrunc (numeral k) (- numeral (Num.Bit1 w)) =
sbintrunc (pred_numeral k) (- numeral (w + Num.One)) BIT True"
"sbintrunc (numeral k) 1 = 1"

lemma no_bintr_alt1: "bintrunc n = (λw. w mod 2 ^ n :: int)"
by (rule ext) (rule bintrunc_mod2p)

lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}"
apply (unfold no_bintr_alt1)
apply (rule exI)
apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
done

lemma no_sbintr_alt2:
"sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
by (rule ext) (simp add : sbintrunc_mod2p)

lemma range_sbintrunc:
"range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}"
apply (unfold no_sbintr_alt2)
apply (auto simp add: image_iff eq_diff_eq)
apply (rule exI)
apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
done

lemma sb_inc_lem:
"(a::int) + 2^k < 0 ⟹ a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p])
apply (rule TrueI)
done

lemma sb_inc_lem':
"(a::int) < - (2^k) ⟹ a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
by (rule sb_inc_lem) simp

lemma sbintrunc_inc:
"x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"
unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp

lemma sb_dec_lem:
"(0::int) ≤ - (2 ^ k) + a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp

lemma sb_dec_lem':
"(2::int) ^ k ≤ a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
by (rule sb_dec_lem) simp

lemma sbintrunc_dec:
"x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"
unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp

lemmas zmod_uminus' = zminus_zmod [where m=c] for c
lemmas zpower_zmod' = power_mod [where b=c and n=k] for c k

lemmas brdmod1s' [symmetric] =
mod_diff_left_eq mod_diff_right_eq
mod_mult_left_eq mod_mult_right_eq

lemmas brdmods' [symmetric] =
zpower_zmod' [symmetric]
trans [OF mod_diff_left_eq mod_diff_right_eq]
trans [OF mod_mult_right_eq mod_mult_left_eq]
zmod_uminus' [symmetric]
mod_diff_left_eq [where b = "1::int"]

lemmas bintr_arith1s =
brdmod1s' [where c="2^n::int", folded bintrunc_mod2p] for n
lemmas bintr_ariths =
brdmods' [where c="2^n::int", folded bintrunc_mod2p] for n

lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]

lemma bintr_ge0: "0 ≤ bintrunc n w"

lemma bintr_lt2p: "bintrunc n w < 2 ^ n"

lemma bintr_Min: "bintrunc n (- 1) = 2 ^ n - 1"

lemma sbintr_ge: "- (2 ^ n) ≤ sbintrunc n w"

lemma sbintr_lt: "sbintrunc n w < 2 ^ n"

lemma sign_Pls_ge_0:
"(bin_sign bin = 0) = (bin >= (0 :: int))"
unfolding bin_sign_def by simp

lemma sign_Min_lt_0:
"(bin_sign bin = -1) = (bin < (0 :: int))"
unfolding bin_sign_def by simp

lemma bin_rest_trunc:
"(bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
by (induct n arbitrary: bin) auto

lemma bin_rest_power_trunc:
"(bin_rest ^^ k) (bintrunc n bin) =
bintrunc (n - k) ((bin_rest ^^ k) bin)"
by (induct k) (auto simp: bin_rest_trunc)

lemma bin_rest_trunc_i:
"bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
by auto

lemma bin_rest_strunc:
"bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
by (induct n arbitrary: bin) auto

lemma bintrunc_rest [simp]:
"bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
apply (induct n arbitrary: bin, simp)
apply (case_tac bin rule: bin_exhaust)
apply (auto simp: bintrunc_bintrunc_l)
done

lemma sbintrunc_rest [simp]:
"sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
apply (induct n arbitrary: bin, simp)
apply (case_tac bin rule: bin_exhaust)
apply (auto simp: bintrunc_bintrunc_l split: bool.splits)
done

lemma bintrunc_rest':
"bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"
by (rule ext) auto

lemma sbintrunc_rest' :
"sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"
by (rule ext) auto

lemma rco_lem:
"f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f"
apply (rule ext)
apply (induct_tac n)
apply (simp_all (no_asm))
apply (drule fun_cong)
apply (unfold o_def)
apply (erule trans)
apply simp
done

lemmas rco_bintr = bintrunc_rest'
[THEN rco_lem [THEN fun_cong], unfolded o_def]
lemmas rco_sbintr = sbintrunc_rest'
[THEN rco_lem [THEN fun_cong], unfolded o_def]

subsection ‹Splitting and concatenation›

primrec bin_split :: "nat ⇒ int ⇒ int × int" where
Z: "bin_split 0 w = (w, 0)"
| Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
in (w1, w2 BIT bin_last w))"

lemma [code]:
"bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))"
"bin_split 0 w = (w, 0)"
by simp_all

primrec bin_cat :: "int ⇒ nat ⇒ int ⇒ int" where
Z: "bin_cat w 0 v = w"
| Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"

end

```