(*
ID: $Id$
Author: Jeremy Dawson, NICTA
contains basic definition to do with integers
expressed using Pls, Min, BIT and important resulting theorems,
in particular, bin_rec and related work
*)
header {* Basic Definitions for Binary Integers *}
theory BinGeneral
imports Num_Lemmas
begin
subsection {* Further properties of numerals *}
datatype bit = B0 | B1
definition
Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
"k BIT b = (case b of B0 \<Rightarrow> 0 | B1 \<Rightarrow> 1) + k + k"
lemma BIT_B0_eq_Bit0 [simp]: "w BIT B0 = Int.Bit0 w"
unfolding Bit_def Bit0_def by simp
lemma BIT_B1_eq_Bit1 [simp]: "w BIT B1 = Int.Bit1 w"
unfolding Bit_def Bit1_def by simp
lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1
hide (open) const B0 B1
lemma Min_ne_Pls [iff]:
"Int.Min ~= Int.Pls"
unfolding Min_def Pls_def by auto
lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric]
lemmas PlsMin_defs [intro!] =
Pls_def Min_def Pls_def [symmetric] Min_def [symmetric]
lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI]
lemma number_of_False_cong:
"False \<Longrightarrow> number_of x = number_of y"
by (rule FalseE)
(** ways in which type Bin resembles a datatype **)
lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c"
apply (unfold Bit_def)
apply (simp (no_asm_use) split: bit.split_asm)
apply simp_all
apply (drule_tac f=even in arg_cong, clarsimp)+
done
lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard]
lemma BIT_eq_iff [simp]:
"(u BIT b = v BIT c) = (u = v \<and> b = c)"
by (rule iffI) auto
lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]]
lemma less_Bits:
"(v BIT b < w BIT c) = (v < w | v <= w & b = bit.B0 & c = bit.B1)"
unfolding Bit_def by (auto split: bit.split)
lemma le_Bits:
"(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= bit.B1 | c ~= bit.B0))"
unfolding Bit_def by (auto split: bit.split)
lemma no_no [simp]: "number_of (number_of i) = i"
unfolding number_of_eq by simp
lemma Bit_B0:
"k BIT bit.B0 = k + k"
by (unfold Bit_def) simp
lemma Bit_B1:
"k BIT bit.B1 = k + k + 1"
by (unfold Bit_def) simp
lemma Bit_B0_2t: "k BIT bit.B0 = 2 * k"
by (rule trans, rule Bit_B0) simp
lemma Bit_B1_2t: "k BIT bit.B1 = 2 * k + 1"
by (rule trans, rule Bit_B1) simp
lemma B_mod_2':
"X = 2 ==> (w BIT bit.B1) mod X = 1 & (w BIT bit.B0) mod X = 0"
apply (simp (no_asm) only: Bit_B0 Bit_B1)
apply (simp add: z1pmod2)
done
lemma B1_mod_2 [simp]: "(Int.Bit1 w) mod 2 = 1"
unfolding numeral_simps number_of_is_id by (simp add: z1pmod2)
lemma B0_mod_2 [simp]: "(Int.Bit0 w) mod 2 = 0"
unfolding numeral_simps number_of_is_id by simp
lemma neB1E [elim!]:
assumes ne: "y \<noteq> bit.B1"
assumes y: "y = bit.B0 \<Longrightarrow> P"
shows "P"
apply (rule y)
apply (cases y rule: bit.exhaust, simp)
apply (simp add: ne)
done
lemma bin_ex_rl: "EX w b. w BIT b = bin"
apply (unfold Bit_def)
apply (cases "even bin")
apply (clarsimp simp: even_equiv_def)
apply (auto simp: odd_equiv_def split: bit.split)
done
lemma bin_exhaust:
assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
shows "Q"
apply (insert bin_ex_rl [of bin])
apply (erule exE)+
apply (rule Q)
apply force
done
subsection {* Destructors for binary integers *}
definition bin_rl :: "int \<Rightarrow> int \<times> bit" where
[code del]: "bin_rl w = (THE (r, l). w = r BIT l)"
lemma bin_rl_char: "(bin_rl w = (r, l)) = (r BIT l = w)"
apply (unfold bin_rl_def)
apply safe
apply (cases w rule: bin_exhaust)
apply auto
done
definition
bin_rest_def [code del]: "bin_rest w = fst (bin_rl w)"
definition
bin_last_def [code del] : "bin_last w = snd (bin_rl w)"
primrec bin_nth where
Z: "bin_nth w 0 = (bin_last w = bit.B1)"
| Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"
lemma bin_rl: "bin_rl w = (bin_rest w, bin_last w)"
unfolding bin_rest_def bin_last_def by auto
lemma bin_rl_simps [simp]:
"bin_rl Int.Pls = (Int.Pls, bit.B0)"
"bin_rl Int.Min = (Int.Min, bit.B1)"
"bin_rl (Int.Bit0 r) = (r, bit.B0)"
"bin_rl (Int.Bit1 r) = (r, bit.B1)"
"bin_rl (r BIT b) = (r, b)"
unfolding bin_rl_char by simp_all
declare bin_rl_simps(1-4) [code]
lemmas bin_rl_simp [simp] = iffD1 [OF bin_rl_char bin_rl]
lemma bin_abs_lem:
"bin = (w BIT b) ==> ~ bin = Int.Min --> ~ bin = Int.Pls -->
nat (abs w) < nat (abs bin)"
apply (clarsimp simp add: bin_rl_char)
apply (unfold Pls_def Min_def Bit_def)
apply (cases b)
apply (clarsimp, arith)
apply (clarsimp, arith)
done
lemma bin_induct:
assumes PPls: "P Int.Pls"
and PMin: "P Int.Min"
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 (simp add: measure_def inv_image_def)
apply (case_tac x rule: bin_exhaust)
apply (frule bin_abs_lem)
apply (auto simp add : PPls PMin PBit)
done
lemma numeral_induct:
assumes Pls: "P Int.Pls"
assumes Min: "P Int.Min"
assumes Bit0: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Pls\<rbrakk> \<Longrightarrow> P (Int.Bit0 w)"
assumes Bit1: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Min\<rbrakk> \<Longrightarrow> P (Int.Bit1 w)"
shows "P x"
apply (induct x rule: bin_induct)
apply (rule Pls)
apply (rule Min)
apply (case_tac bit)
apply (case_tac "bin = Int.Pls")
apply simp
apply (simp add: Bit0)
apply (case_tac "bin = Int.Min")
apply simp
apply (simp add: Bit1)
done
lemma bin_rest_simps [simp]:
"bin_rest Int.Pls = Int.Pls"
"bin_rest Int.Min = Int.Min"
"bin_rest (Int.Bit0 w) = w"
"bin_rest (Int.Bit1 w) = w"
"bin_rest (w BIT b) = w"
unfolding bin_rest_def by auto
declare bin_rest_simps(1-4) [code]
lemma bin_last_simps [simp]:
"bin_last Int.Pls = bit.B0"
"bin_last Int.Min = bit.B1"
"bin_last (Int.Bit0 w) = bit.B0"
"bin_last (Int.Bit1 w) = bit.B1"
"bin_last (w BIT b) = b"
unfolding bin_last_def by auto
declare bin_last_simps(1-4) [code]
lemma bin_r_l_extras [simp]:
"bin_last 0 = bit.B0"
"bin_last (- 1) = bit.B1"
"bin_last -1 = bit.B1"
"bin_last 1 = bit.B1"
"bin_rest 1 = 0"
"bin_rest 0 = 0"
"bin_rest (- 1) = - 1"
"bin_rest -1 = -1"
apply (unfold number_of_Min)
apply (unfold Pls_def [symmetric] Min_def [symmetric])
apply (unfold numeral_1_eq_1 [symmetric])
apply (auto simp: number_of_eq)
done
lemma bin_last_mod:
"bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)"
apply (case_tac w rule: bin_exhaust)
apply (case_tac b)
apply auto
done
lemma bin_rest_div:
"bin_rest w = w div 2"
apply (case_tac w rule: bin_exhaust)
apply (rule trans)
apply clarsimp
apply (rule refl)
apply (drule trans)
apply (rule Bit_def)
apply (simp add: z1pdiv2 split: bit.split)
done
lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
unfolding bin_rest_div [symmetric] by auto
lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w"
using Bit_div2 [where b=bit.B0] by simp
lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w"
using Bit_div2 [where b=bit.B1] by simp
lemma 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 del: subset_antisym)
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 del: subset_antisym)
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 (case_tac y rule: bin_exhaust)
apply clarify
apply (erule allE)
apply (erule impE)
prefer 2
apply (erule BIT_eqI)
apply (drule_tac x=0 in fun_cong, force)
apply (rule ext)
apply (drule_tac x="Suc ?x" in fun_cong, force)
done
lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)"
by (auto elim: bin_nth_lem)
lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1], standard]
lemma bin_nth_Pls [simp]: "~ bin_nth Int.Pls n"
by (induct n) auto
lemma bin_nth_Min [simp]: "bin_nth Int.Min n"
by (induct n) auto
lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = bit.B1)"
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_minus_Bit0 [simp]:
"0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)"
using bin_nth_minus [where b=bit.B0] by simp
lemma bin_nth_minus_Bit1 [simp]:
"0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)"
using bin_nth_minus [where b=bit.B1] by simp
lemmas bin_nth_0 = bin_nth.simps(1)
lemmas bin_nth_Suc = bin_nth.simps(2)
lemmas bin_nth_simps =
bin_nth_0 bin_nth_Suc bin_nth_Pls bin_nth_Min bin_nth_minus
bin_nth_minus_Bit0 bin_nth_minus_Bit1
subsection {* Recursion combinator for binary integers *}
lemma brlem: "(bin = Int.Min) = (- bin + Int.pred 0 = 0)"
unfolding Min_def pred_def by arith
function
bin_rec :: "'a \<Rightarrow> 'a \<Rightarrow> (int \<Rightarrow> bit \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> int \<Rightarrow> 'a"
where
"bin_rec f1 f2 f3 bin = (if bin = Int.Pls then f1
else if bin = Int.Min then f2
else case bin_rl bin of (w, b) => f3 w b (bin_rec f1 f2 f3 w))"
by pat_completeness auto
termination
apply (relation "measure (nat o abs o snd o snd o snd)")
apply simp
apply (simp add: Pls_def brlem)
apply (clarsimp simp: bin_rl_char pred_def)
apply (frule thin_rl [THEN refl [THEN bin_abs_lem [rule_format]]])
apply (unfold Pls_def Min_def number_of_eq)
prefer 2
apply (erule asm_rl)
apply auto
done
declare bin_rec.simps [simp del]
lemma bin_rec_PM:
"f = bin_rec f1 f2 f3 ==> f Int.Pls = f1 & f Int.Min = f2"
by (auto simp add: bin_rec.simps)
lemma bin_rec_Pls: "bin_rec f1 f2 f3 Int.Pls = f1"
by (simp add: bin_rec.simps)
lemma bin_rec_Min: "bin_rec f1 f2 f3 Int.Min = f2"
by (simp add: bin_rec.simps)
lemma bin_rec_Bit0:
"f3 Int.Pls bit.B0 f1 = f1 \<Longrightarrow>
bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w bit.B0 (bin_rec f1 f2 f3 w)"
apply (simp add: bin_rec_Pls bin_rec.simps [of _ _ _ "Int.Bit0 w"])
unfolding Pls_def Min_def Bit0_def
apply auto
apply presburger
apply (simp add: bin_rec.simps)
done
lemma bin_rec_Bit1:
"f3 Int.Min bit.B1 f2 = f2 \<Longrightarrow>
bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w bit.B1 (bin_rec f1 f2 f3 w)"
apply (simp add: bin_rec.simps [of _ _ _ "Int.Bit1 w"])
unfolding Pls_def Min_def Bit1_def
apply auto
apply (cases w)
apply auto
apply (simp add: bin_rec.simps)
unfolding Min_def Pls_def number_of_is_id apply auto
unfolding Bit0_def apply presburger
done
lemma bin_rec_Bit:
"f = bin_rec f1 f2 f3 ==> f3 Int.Pls bit.B0 f1 = f1 ==>
f3 Int.Min bit.B1 f2 = f2 ==> f (w BIT b) = f3 w b (f w)"
by (cases b, simp add: bin_rec_Bit0, simp add: bin_rec_Bit1)
lemmas bin_rec_simps = refl [THEN bin_rec_Bit] bin_rec_Pls bin_rec_Min
bin_rec_Bit0 bin_rec_Bit1
subsection {* Truncating binary integers *}
definition
bin_sign_def [code del] : "bin_sign = bin_rec Int.Pls Int.Min (%w b s. s)"
lemma bin_sign_simps [simp]:
"bin_sign Int.Pls = Int.Pls"
"bin_sign Int.Min = Int.Min"
"bin_sign (Int.Bit0 w) = bin_sign w"
"bin_sign (Int.Bit1 w) = bin_sign w"
"bin_sign (w BIT b) = bin_sign w"
unfolding bin_sign_def by (auto simp: bin_rec_simps)
declare bin_sign_simps(1-4) [code]
lemma bin_sign_rest [simp]:
"bin_sign (bin_rest w) = (bin_sign w)"
by (cases w rule: bin_exhaust) auto
consts
bintrunc :: "nat => int => int"
primrec
Z : "bintrunc 0 bin = Int.Pls"
Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
consts
sbintrunc :: "nat => int => int"
primrec
Z : "sbintrunc 0 bin =
(case bin_last bin of bit.B1 => Int.Min | bit.B0 => Int.Pls)"
Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
lemma sign_bintr:
"!!w. bin_sign (bintrunc n w) = Int.Pls"
by (induct n) auto
lemma bintrunc_mod2p:
"!!w. bintrunc n w = (w mod 2 ^ n :: int)"
apply (induct n, clarsimp)
apply (simp add: bin_last_mod bin_rest_div Bit_def zmod_zmult2_eq
cong: number_of_False_cong)
done
lemma sbintrunc_mod2p:
"!!w. sbintrunc n w = ((w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n :: int)"
apply (induct n)
apply clarsimp
apply (subst zmod_zadd_left_eq)
apply (simp add: bin_last_mod)
apply (simp add: number_of_eq)
apply clarsimp
apply (simp add: bin_last_mod bin_rest_div Bit_def
cong: number_of_False_cong)
apply (clarsimp simp: zmod_zmult_zmult1 [symmetric]
zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
apply (rule trans [symmetric, OF _ emep1])
apply auto
apply (auto simp: even_def)
done
subsection "Simplifications for (s)bintrunc"
lemma bit_bool:
"(b = (b' = bit.B1)) = (b' = (if b then bit.B1 else bit.B0))"
by (cases b') auto
lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]
lemma bin_sign_lem:
"!!bin. (bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n"
apply (induct n)
apply (case_tac bin rule: bin_exhaust, case_tac b, auto)+
done
lemma nth_bintr:
"!!w m. bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
apply (induct n)
apply (case_tac m, auto)[1]
apply (case_tac m, auto)[1]
done
lemma nth_sbintr:
"!!w m. bin_nth (sbintrunc m w) n =
(if n < m then bin_nth w n else bin_nth w m)"
apply (induct n)
apply (case_tac m, simp_all split: bit.splits)[1]
apply (case_tac m, simp_all split: bit.splits)[1]
done
lemma bin_nth_Bit:
"bin_nth (w BIT b) n = (n = 0 & b = bit.B1 | (EX m. n = Suc m & bin_nth w m))"
by (cases n) auto
lemma bin_nth_Bit0:
"bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)"
using bin_nth_Bit [where b=bit.B0] by simp
lemma bin_nth_Bit1:
"bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))"
using bin_nth_Bit [where b=bit.B1] by simp
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 min_def)
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 (unfold min_def)
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 (unfold min_def)
apply (rule bin_eqI)
apply (auto simp: nth_sbintr)
done
lemmas bintrunc_Pls =
bintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
lemmas bintrunc_Min [simp] =
bintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
lemmas bintrunc_BIT [simp] =
bintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
lemma bintrunc_Bit0 [simp]:
"bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)"
using bintrunc_BIT [where b=bit.B0] by simp
lemma bintrunc_Bit1 [simp]:
"bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)"
using bintrunc_BIT [where b=bit.B1] by simp
lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
bintrunc_Bit0 bintrunc_Bit1
lemmas sbintrunc_Suc_Pls =
sbintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
lemmas sbintrunc_Suc_Min =
sbintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
lemmas sbintrunc_Suc_BIT [simp] =
sbintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
lemma sbintrunc_Suc_Bit0 [simp]:
"sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)"
using sbintrunc_Suc_BIT [where b=bit.B0] by simp
lemma sbintrunc_Suc_Bit1 [simp]:
"sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)"
using sbintrunc_Suc_BIT [where b=bit.B1] by simp
lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1
lemmas sbintrunc_Pls =
sbintrunc.Z [where bin="Int.Pls",
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemmas sbintrunc_Min =
sbintrunc.Z [where bin="Int.Min",
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemmas sbintrunc_0_BIT_B0 [simp] =
sbintrunc.Z [where bin="w BIT bit.B0",
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemmas sbintrunc_0_BIT_B1 [simp] =
sbintrunc.Z [where bin="w BIT bit.B1",
simplified bin_last_simps bin_rest_simps bit.simps, standard]
lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = Int.Pls"
using sbintrunc_0_BIT_B0 by simp
lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = Int.Min"
using sbintrunc_0_BIT_B1 by simp
lemmas sbintrunc_0_simps =
sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
sbintrunc_0_Bit0 sbintrunc_0_Bit1
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], standard]
lemmas sbintrunc_minus_simps =
sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans], standard]
lemma bintrunc_n_Pls [simp]:
"bintrunc n Int.Pls = Int.Pls"
by (induct n) auto
lemma sbintrunc_n_PM [simp]:
"sbintrunc n Int.Pls = Int.Pls"
"sbintrunc n Int.Min = Int.Min"
by (induct n) auto
lemmas thobini1 = arg_cong [where f = "%w. w BIT b", standard]
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], standard]
lemmas bintrunc_Pls_minus_I = bmsts(1)
lemmas bintrunc_Min_minus_I = bmsts(2)
lemmas bintrunc_BIT_minus_I = bmsts(3)
lemma bintrunc_0_Min: "bintrunc 0 Int.Min = Int.Pls"
by auto
lemma bintrunc_0_BIT: "bintrunc 0 (w BIT b) = Int.Pls"
by auto
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, standard]
bintrunc_BIT_I [THEN bintrunc_Suc_lem, standard]
lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]
lemmas sbintrunc_Suc_Is =
sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans], standard]
lemmas sbintrunc_Suc_minus_Is =
sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard]
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, standard]
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 \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
by (cases n) (auto simp del: bintrunc.Suc)
lemma sbintrunc_bintrunc' [simp]:
"0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
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 \<Longrightarrow> 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, standard]
lemmas bintrunc_pred_simps [simp] =
bintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
lemmas sbintrunc_pred_simps [simp] =
sbintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
lemma no_bintr_alt:
"number_of (bintrunc n w) = w mod 2 ^ n"
by (simp add: number_of_eq bintrunc_mod2p)
lemma no_bintr_alt1: "bintrunc n = (%w. 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 (auto simp add: image_iff)
apply (rule exI)
apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
done
lemma no_bintr:
"number_of (bintrunc n w) = (number_of w mod 2 ^ n :: int)"
by (simp add : bintrunc_mod2p number_of_eq)
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 no_sbintr:
"number_of (sbintrunc n w) =
((number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
by (simp add : no_sbintr_alt2 number_of_eq)
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 \<Longrightarrow> 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) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
by (rule iffD1 [OF less_diff_eq, THEN sb_inc_lem, simplified OrderedGroup.diff_0])
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"
by (rule int_mod_le' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k",
simplified zless2p, OF _ TrueI, simplified])
lemma sb_dec_lem':
"(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
by (rule iffD1 [OF diff_le_eq', THEN sb_dec_lem, simplified])
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' = zmod_uminus [where b="c", standard]
lemmas zpower_zmod' = zpower_zmod [where m="c" and y="k", standard]
lemmas brdmod1s' [symmetric] =
zmod_zadd_left_eq zmod_zadd_right_eq
zmod_zsub_left_eq zmod_zsub_right_eq
zmod_zmult1_eq zmod_zmult1_eq_rev
lemmas brdmods' [symmetric] =
zpower_zmod' [symmetric]
trans [OF zmod_zadd_left_eq zmod_zadd_right_eq]
trans [OF zmod_zsub_left_eq zmod_zsub_right_eq]
trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev]
zmod_uminus' [symmetric]
zmod_zadd_left_eq [where b = "1"]
zmod_zsub_left_eq [where b = "1"]
lemmas bintr_arith1s =
brdmod1s' [where c="2^n", folded pred_def succ_def bintrunc_mod2p, standard]
lemmas bintr_ariths =
brdmods' [where c="2^n", folded pred_def succ_def bintrunc_mod2p, standard]
lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p, standard]
lemma bintr_ge0: "(0 :: int) <= number_of (bintrunc n w)"
by (simp add : no_bintr m2pths)
lemma bintr_lt2p: "number_of (bintrunc n w) < (2 ^ n :: int)"
by (simp add : no_bintr m2pths)
lemma bintr_Min:
"number_of (bintrunc n Int.Min) = (2 ^ n :: int) - 1"
by (simp add : no_bintr m1mod2k)
lemma sbintr_ge: "(- (2 ^ n) :: int) <= number_of (sbintrunc n w)"
by (simp add : no_sbintr m2pths)
lemma sbintr_lt: "number_of (sbintrunc n w) < (2 ^ n :: int)"
by (simp add : no_sbintr m2pths)
lemma bintrunc_Suc:
"bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin"
by (case_tac bin rule: bin_exhaust) auto
lemma sign_Pls_ge_0:
"(bin_sign bin = Int.Pls) = (number_of bin >= (0 :: int))"
by (induct bin rule: numeral_induct) auto
lemma sign_Min_lt_0:
"(bin_sign bin = Int.Min) = (number_of bin < (0 :: int))"
by (induct bin rule: numeral_induct) auto
lemmas sign_Min_neg = trans [OF sign_Min_lt_0 neg_def [symmetric]]
lemma bin_rest_trunc:
"!!bin. (bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
by (induct n) auto
lemma bin_rest_power_trunc [rule_format] :
"(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. bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
by (induct n) auto
lemma bintrunc_rest [simp]:
"!!bin. bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
apply (induct n, simp)
apply (case_tac bin rule: bin_exhaust)
apply (auto simp: bintrunc_bintrunc_l)
done
lemma sbintrunc_rest [simp]:
"!!bin. sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
apply (induct n, simp)
apply (case_tac bin rule: bin_exhaust)
apply (auto simp: bintrunc_bintrunc_l split: bit.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
lemma rco_alt: "(f o g) ^ n o f = f o (g o f) ^ n"
apply (rule ext)
apply (induct n)
apply (simp_all add: o_def)
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 \<Rightarrow> int \<Rightarrow> int \<times> int" where
Z: "bin_split 0 w = (w, Int.Pls)"
| Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
in (w1, w2 BIT bin_last w))"
primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where
Z: "bin_cat w 0 v = w"
| Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"
subsection {* Miscellaneous lemmas *}
lemmas funpow_minus_simp =
trans [OF gen_minus [where f = "power f"] funpow_Suc, standard]
lemmas funpow_pred_simp [simp] =
funpow_minus_simp [of "number_of bin", simplified nobm1, standard]
lemmas replicate_minus_simp =
trans [OF gen_minus [where f = "%n. replicate n x"] replicate.replicate_Suc,
standard]
lemmas replicate_pred_simp [simp] =
replicate_minus_simp [of "number_of bin", simplified nobm1, standard]
lemmas power_Suc_no [simp] = power_Suc [of "number_of a", standard]
lemmas power_minus_simp =
trans [OF gen_minus [where f = "power f"] power_Suc, standard]
lemmas power_pred_simp =
power_minus_simp [of "number_of bin", simplified nobm1, standard]
lemmas power_pred_simp_no [simp] = power_pred_simp [where f= "number_of f", standard]
lemma list_exhaust_size_gt0:
assumes y: "\<And>a list. y = a # list \<Longrightarrow> P"
shows "0 < length y \<Longrightarrow> P"
apply (cases y, simp)
apply (rule y)
apply fastsimp
done
lemma list_exhaust_size_eq0:
assumes y: "y = [] \<Longrightarrow> P"
shows "length y = 0 \<Longrightarrow> P"
apply (cases y)
apply (rule y, simp)
apply simp
done
lemma size_Cons_lem_eq:
"y = xa # list ==> size y = Suc k ==> size list = k"
by auto
lemma size_Cons_lem_eq_bin:
"y = xa # list ==> size y = number_of (Int.succ k) ==>
size list = number_of k"
by (auto simp: pred_def succ_def split add : split_if_asm)
lemmas ls_splits =
prod.split split_split prod.split_asm split_split_asm split_if_asm
lemma not_B1_is_B0: "y \<noteq> bit.B1 \<Longrightarrow> y = bit.B0"
by (cases y) auto
lemma B1_ass_B0:
assumes y: "y = bit.B0 \<Longrightarrow> y = bit.B1"
shows "y = bit.B1"
apply (rule classical)
apply (drule not_B1_is_B0)
apply (erule y)
done
-- "simplifications for specific word lengths"
lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'
lemmas s2n_ths = n2s_ths [symmetric]
end