Theory BinGeneral

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theory BinGeneral
imports Num_Lemmas
begin

(* 
  ID:     $Id: BinGeneral.thy,v 1.19 2008/05/07 08:59:42 berghofe Exp $
  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 => bit => int" (infixl "BIT" 90) where
  "k BIT b = (case b of B0 => 0 | B1 => 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 ==> 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 ∧ 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 ≠ bit.B1"
  assumes y: "y = bit.B0 ==> 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: "!!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


subsection {* Destructors for binary integers *}

definition bin_rl :: "int => int × bit" where 
  [code func 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 func del]: "bin_rest w = fst (bin_rl w)"

definition
  bin_last_def [code func 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 func]

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: "!!w. [|P w; w ≠ Int.Pls|] ==> P (Int.Bit0 w)"
  assumes Bit1: "!!w. [|P w; w ≠ Int.Min|] ==> 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 func]

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 func]

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 => 'a => (int => bit => 'a => 'a) => int => '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 ==>
    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 ==>
    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 func 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 func]

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 ==> 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, 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 ==> 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 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 => int => int × 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 => 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"

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: "!!a list. y = a # list ==> P"
  shows "0 < length y ==> P"
  apply (cases y, simp)
  apply (rule y)
  apply fastsimp
  done

lemma list_exhaust_size_eq0:
  assumes y: "y = [] ==> P"
  shows "length y = 0 ==> 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 ≠ bit.B1 ==> y = bit.B0"
  by (cases y) auto

lemma B1_ass_B0: 
  assumes y: "y = bit.B0 ==> 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

Further properties of numerals

lemma BIT_B0_eq_Bit0:

  w BIT B0 = Int.Bit0 w

lemma BIT_B1_eq_Bit1:

  w BIT B1 = Int.Bit1 w

lemma BIT_simps:

  w BIT B0 = Int.Bit0 w
  w BIT B1 = Int.Bit1 w

lemma Min_ne_Pls:

  Int.Min  Int.Pls

lemma Pls_ne_Min:

  Int.Pls  Int.Min

lemma PlsMin_defs:

  Int.Pls = 0
  Int.Min = - 1
  0 = Int.Pls
  - 1 = Int.Min

lemma PlsMin_simps:

  Int.Pls = 0 == True
  Int.Min = - 1 == True
  0 = Int.Pls == True
  - 1 = Int.Min == True

lemma number_of_False_cong:

  False ==> number_of x = number_of y

lemma BIT_eq:

  u BIT b = v BIT c ==> u = vb = c

lemma BIT_eqE:

  [| u BIT b = v BIT c; [| u = v; b = c |] ==> R |] ==> R

lemma BIT_eq_iff:

  (u BIT b = v BIT c) = (u = vb = c)

lemma BIT_eqI:

  [| u1 = v1; b1 = c1 |] ==> u1 BIT b1 = v1 BIT c1

lemma less_Bits:

  (v BIT b < w BIT c) = (v < wv  wb = bit.B0 ∧ c = bit.B1)

lemma le_Bits:

  (v BIT b  w BIT c) = (v < wv  w ∧ (b  bit.B1 ∨ c  bit.B0))

lemma no_no:

  number_of (number_of i) = i

lemma Bit_B0:

  k BIT bit.B0 = k + k

lemma Bit_B1:

  k BIT bit.B1 = k + k + 1

lemma Bit_B0_2t:

  k BIT bit.B0 = 2 * k

lemma Bit_B1_2t:

  k BIT bit.B1 = 2 * k + 1

lemma B_mod_2':

  X = 2 ==> w BIT bit.B1 mod X = 1w BIT bit.B0 mod X = 0

lemma B1_mod_2:

  Int.Bit1 w mod 2 = 1

lemma B0_mod_2:

  Int.Bit0 w mod 2 = 0

lemma neB1E:

  [| y  bit.B1; y = bit.B0 ==> P |] ==> P

lemma bin_ex_rl:

  w b. w BIT b = bin

lemma bin_exhaust:

  (!!x b. bin = x BIT b ==> Q) ==> Q

Destructors for binary integers

lemma bin_rl_char:

  (bin_rl w = (r, l)) = (r BIT l = w)

lemma bin_rl:

  bin_rl w = (bin_rest w, bin_last w)

lemma bin_rl_simps:

  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)

lemma bin_rl_simp:

  bin_rest w1 BIT bin_last w1 = w1

lemma bin_abs_lem:

  bin = w BIT b ==> bin  Int.Min --> bin  Int.Pls --> nat ¦w¦ < nat ¦bin¦

lemma bin_induct:

  [| P Int.Pls; P Int.Min; !!bin bit. P bin ==> P (bin BIT bit) |] ==> P bin

lemma numeral_induct:

  [| P Int.Pls; P Int.Min; !!w. [| P w; w  Int.Pls |] ==> P (Int.Bit0 w);
     !!w. [| P w; w  Int.Min |] ==> P (Int.Bit1 w) |]
  ==> P x

lemma bin_rest_simps:

  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

lemma bin_last_simps:

  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

lemma bin_r_l_extras:

  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

lemma bin_last_mod:

  bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)

lemma bin_rest_div:

  bin_rest w = w div 2

lemma Bit_div2:

  w BIT b div 2 = w

lemma Bit0_div2:

  Int.Bit0 w div 2 = w

lemma Bit1_div2:

  Int.Bit1 w div 2 = w

lemma bin_nth_lem:

  bin_nth x = bin_nth y ==> x = y

lemma bin_nth_eq_iff:

  (bin_nth x = bin_nth y) = (x = y)

lemma bin_eqI:

  (!!x. bin_nth x x = bin_nth y x) ==> x = y

lemma bin_nth_Pls:

  ¬ bin_nth Int.Pls n

lemma bin_nth_Min:

  bin_nth Int.Min n

lemma bin_nth_0_BIT:

  bin_nth (w BIT b) 0 = (b = bit.B1)

lemma bin_nth_Suc_BIT:

  bin_nth (w BIT b) (Suc n) = bin_nth w n

lemma bin_nth_minus:

  0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)

lemma bin_nth_minus_Bit0:

  0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)

lemma bin_nth_minus_Bit1:

  0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)

lemma bin_nth_0:

  bin_nth w 0 = (bin_last w = bit.B1)

lemma bin_nth_Suc:

  bin_nth w (Suc n) = bin_nth (bin_rest w) n

lemma bin_nth_simps:

  bin_nth w 0 = (bin_last w = bit.B1)
  bin_nth w (Suc n) = bin_nth (bin_rest w) n
  ¬ bin_nth Int.Pls n
  bin_nth Int.Min n
  0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)
  0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)
  0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)

Recursion combinator for binary integers

lemma brlem:

  (bin = Int.Min) = (- bin + Int.pred 0 = 0)

lemma bin_rec_PM:

  f = bin_rec f1.0 f2.0 f3.0 ==> f Int.Pls = f1.0f Int.Min = f2.0

lemma bin_rec_Pls:

  bin_rec f1.0 f2.0 f3.0 Int.Pls = f1.0

lemma bin_rec_Min:

  bin_rec f1.0 f2.0 f3.0 Int.Min = f2.0

lemma bin_rec_Bit0:

  f3.0 Int.Pls bit.B0 f1.0 = f1.0
  ==> bin_rec f1.0 f2.0 f3.0 (Int.Bit0 w) =
      f3.0 w bit.B0 (bin_rec f1.0 f2.0 f3.0 w)

lemma bin_rec_Bit1:

  f3.0 Int.Min bit.B1 f2.0 = f2.0
  ==> bin_rec f1.0 f2.0 f3.0 (Int.Bit1 w) =
      f3.0 w bit.B1 (bin_rec f1.0 f2.0 f3.0 w)

lemma bin_rec_Bit:

  [| f = bin_rec f1.0 f2.0 f3.0; f3.0 Int.Pls bit.B0 f1.0 = f1.0;
     f3.0 Int.Min bit.B1 f2.0 = f2.0 |]
  ==> f (w BIT b) = f3.0 w b (f w)

lemma bin_rec_simps:

  [| f3.0 Int.Pls bit.B0 f1.0 = f1.0; f3.0 Int.Min bit.B1 f2.0 = f2.0 |]
  ==> bin_rec f1.0 f2.0 f3.0 (w BIT b) = f3.0 w b (bin_rec f1.0 f2.0 f3.0 w)
  bin_rec f1.0 f2.0 f3.0 Int.Pls = f1.0
  bin_rec f1.0 f2.0 f3.0 Int.Min = f2.0
  f3.0 Int.Pls bit.B0 f1.0 = f1.0
  ==> bin_rec f1.0 f2.0 f3.0 (Int.Bit0 w) =
      f3.0 w bit.B0 (bin_rec f1.0 f2.0 f3.0 w)
  f3.0 Int.Min bit.B1 f2.0 = f2.0
  ==> bin_rec f1.0 f2.0 f3.0 (Int.Bit1 w) =
      f3.0 w bit.B1 (bin_rec f1.0 f2.0 f3.0 w)

Truncating binary integers

lemma bin_sign_simps:

  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

lemma bin_sign_rest:

  bin_sign (bin_rest w) = bin_sign w

lemma sign_bintr:

  bin_sign (bintrunc n w) = Int.Pls

lemma bintrunc_mod2p:

  bintrunc n w = w mod 2 ^ n

lemma sbintrunc_mod2p:

  sbintrunc n w = (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n

Simplifications for (s)bintrunc

lemma bit_bool:

  (b = (b' = bit.B1)) = (b' = (if b then bit.B1 else bit.B0))

lemma bit_bool1:

  (if s = bit.B1 then bit.B1 else bit.B0) = s

lemma bin_sign_lem:

  (bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n

lemma nth_bintr:

  bin_nth (bintrunc m w) n = (n < mbin_nth w n)

lemma nth_sbintr:

  bin_nth (sbintrunc m w) n = (if n < m then bin_nth w n else bin_nth w m)

lemma bin_nth_Bit:

  bin_nth (w BIT b) n = (n = 0b = bit.B1 ∨ (∃m. n = Suc mbin_nth w m))

lemma bin_nth_Bit0:

  bin_nth (Int.Bit0 w) n = (∃m. n = Suc mbin_nth w m)

lemma bin_nth_Bit1:

  bin_nth (Int.Bit1 w) n = (n = 0 ∨ (∃m. n = Suc mbin_nth w m))

lemma bintrunc_bintrunc_l:

  n  m ==> bintrunc m (bintrunc n w) = bintrunc n w

lemma sbintrunc_sbintrunc_l:

  n  m ==> sbintrunc m (sbintrunc n w) = sbintrunc n w

lemma bintrunc_bintrunc_ge:

  n  m ==> bintrunc n (bintrunc m w) = bintrunc n w

lemma bintrunc_bintrunc_min:

  bintrunc m (bintrunc n w) = bintrunc (min m n) w

lemma sbintrunc_sbintrunc_min:

  sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w

lemma bintrunc_Pls:

  bintrunc (Suc n) Int.Pls = bintrunc n Int.Pls BIT bit.B0

lemma bintrunc_Min:

  bintrunc (Suc n) Int.Min = bintrunc n Int.Min BIT bit.B1

lemma bintrunc_BIT:

  bintrunc (Suc n) (w BIT b) = bintrunc n w BIT b

lemma bintrunc_Bit0:

  bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)

lemma bintrunc_Bit1:

  bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)

lemma bintrunc_Sucs:

  bintrunc (Suc n) Int.Pls = bintrunc n Int.Pls BIT bit.B0
  bintrunc (Suc n) Int.Min = bintrunc n Int.Min BIT bit.B1
  bintrunc (Suc n) (w BIT b) = bintrunc n w BIT b
  bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)
  bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)

lemma sbintrunc_Suc_Pls:

  sbintrunc (Suc n) Int.Pls = sbintrunc n Int.Pls BIT bit.B0

lemma sbintrunc_Suc_Min:

  sbintrunc (Suc n) Int.Min = sbintrunc n Int.Min BIT bit.B1

lemma sbintrunc_Suc_BIT:

  sbintrunc (Suc n) (w BIT b) = sbintrunc n w BIT b

lemma sbintrunc_Suc_Bit0:

  sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)

lemma sbintrunc_Suc_Bit1:

  sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)

lemma sbintrunc_Sucs:

  sbintrunc (Suc n) Int.Pls = sbintrunc n Int.Pls BIT bit.B0
  sbintrunc (Suc n) Int.Min = sbintrunc n Int.Min BIT bit.B1
  sbintrunc (Suc n) (w BIT b) = sbintrunc n w BIT b
  sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)
  sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)

lemma sbintrunc_Pls:

  sbintrunc 0 Int.Pls = Int.Pls

lemma sbintrunc_Min:

  sbintrunc 0 Int.Min = Int.Min

lemma sbintrunc_0_BIT_B0:

  sbintrunc 0 (w BIT bit.B0) = Int.Pls

lemma sbintrunc_0_BIT_B1:

  sbintrunc 0 (w BIT bit.B1) = Int.Min

lemma sbintrunc_0_Bit0:

  sbintrunc 0 (Int.Bit0 w) = Int.Pls

lemma sbintrunc_0_Bit1:

  sbintrunc 0 (Int.Bit1 w) = Int.Min

lemma sbintrunc_0_simps:

  sbintrunc 0 Int.Pls = Int.Pls
  sbintrunc 0 Int.Min = Int.Min
  sbintrunc 0 (w BIT bit.B0) = Int.Pls
  sbintrunc 0 (w BIT bit.B1) = Int.Min
  sbintrunc 0 (Int.Bit0 w) = Int.Pls
  sbintrunc 0 (Int.Bit1 w) = Int.Min

lemma bintrunc_simps:

  bintrunc 0 bin = Int.Pls
  bintrunc (Suc n) Int.Pls = bintrunc n Int.Pls BIT bit.B0
  bintrunc (Suc n) Int.Min = bintrunc n Int.Min BIT bit.B1
  bintrunc (Suc n) (w BIT b) = bintrunc n w BIT b
  bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)
  bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)

lemma sbintrunc_simps:

  sbintrunc 0 Int.Pls = Int.Pls
  sbintrunc 0 Int.Min = Int.Min
  sbintrunc 0 (w BIT bit.B0) = Int.Pls
  sbintrunc 0 (w BIT bit.B1) = Int.Min
  sbintrunc 0 (Int.Bit0 w) = Int.Pls
  sbintrunc 0 (Int.Bit1 w) = Int.Min
  sbintrunc (Suc n) Int.Pls = sbintrunc n Int.Pls BIT bit.B0
  sbintrunc (Suc n) Int.Min = sbintrunc n Int.Min BIT bit.B1
  sbintrunc (Suc n) (w BIT b) = sbintrunc n w BIT b
  sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)
  sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)

lemma bintrunc_minus:

  0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w

lemma sbintrunc_minus:

  0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w

lemma bintrunc_minus_simps:

  0 < n ==> bintrunc n Int.Pls = bintrunc (n - 1) Int.Pls BIT bit.B0
  0 < n ==> bintrunc n Int.Min = bintrunc (n - 1) Int.Min BIT bit.B1
  0 < n ==> bintrunc n (w BIT b) = bintrunc (n - 1) w BIT b
  0 < n ==> bintrunc n (Int.Bit0 w) = Int.Bit0 (bintrunc (n - 1) w)
  0 < n ==> bintrunc n (Int.Bit1 w) = Int.Bit1 (bintrunc (n - 1) w)

lemma sbintrunc_minus_simps:

  0 < n ==> sbintrunc n Int.Pls = sbintrunc (n - 1) Int.Pls BIT bit.B0
  0 < n ==> sbintrunc n Int.Min = sbintrunc (n - 1) Int.Min BIT bit.B1
  0 < n ==> sbintrunc n (w BIT b) = sbintrunc (n - 1) w BIT b
  0 < n ==> sbintrunc n (Int.Bit0 w) = Int.Bit0 (sbintrunc (n - 1) w)
  0 < n ==> sbintrunc n (Int.Bit1 w) = Int.Bit1 (sbintrunc (n - 1) w)

lemma bintrunc_n_Pls:

  bintrunc n Int.Pls = Int.Pls

lemma sbintrunc_n_PM:

  sbintrunc n Int.Pls = Int.Pls
  sbintrunc n Int.Min = Int.Min

lemma thobini1:

  x = y ==> x BIT b = y BIT b

lemma bintrunc_BIT_I:

  bintrunc n2 w2 = y1 ==> bintrunc (Suc n2) (w2 BIT b1) = y1 BIT b1

lemma bintrunc_Min_I:

  bintrunc n2 Int.Min = y1 ==> bintrunc (Suc n2) Int.Min = y1 BIT bit.B1

lemma bmsts:

  [| 0 < n; bintrunc (n - 1) Int.Pls = y |] ==> bintrunc n Int.Pls = y BIT bit.B0
  [| 0 < n; bintrunc (n - 1) Int.Min = y |] ==> bintrunc n Int.Min = y BIT bit.B1
  [| 0 < n; bintrunc (n - 1) w = y |] ==> bintrunc n (w BIT b) = y BIT b

lemma bintrunc_Pls_minus_I:

  [| 0 < n; bintrunc (n - 1) Int.Pls = y |] ==> bintrunc n Int.Pls = y BIT bit.B0

lemma bintrunc_Min_minus_I:

  [| 0 < n; bintrunc (n - 1) Int.Min = y |] ==> bintrunc n Int.Min = y BIT bit.B1

lemma bintrunc_BIT_minus_I:

  [| 0 < n; bintrunc (n - 1) w = y |] ==> bintrunc n (w BIT b) = y BIT b

lemma bintrunc_0_Min:

  bintrunc 0 Int.Min = Int.Pls

lemma bintrunc_0_BIT:

  bintrunc 0 (w BIT b) = Int.Pls

lemma bintrunc_Suc_lem:

  [| bintrunc (Suc n) x = y; m = Suc n |] ==> bintrunc m x = y

lemma bintrunc_Suc_Ialts:

  [| bintrunc n Int.Min = y; m = Suc n |] ==> bintrunc m Int.Min = y BIT bit.B1
  [| bintrunc n w = y; m = Suc n |] ==> bintrunc m (w BIT b) = y BIT b

lemma sbintrunc_BIT_I:

  sbintrunc n2 w2 = y1 ==> sbintrunc (Suc n2) (w2 BIT b1) = y1 BIT b1

lemma sbintrunc_Suc_Is:

  sbintrunc n Int.Pls = y ==> sbintrunc (Suc n) Int.Pls = y BIT bit.B0
  sbintrunc n Int.Min = y ==> sbintrunc (Suc n) Int.Min = y BIT bit.B1
  sbintrunc n w = y ==> sbintrunc (Suc n) (w BIT b) = y BIT b

lemma sbintrunc_Suc_minus_Is:

  [| 0 < n; sbintrunc (n - 1) Int.Pls = y |]
  ==> sbintrunc n Int.Pls = y BIT bit.B0
  [| 0 < n; sbintrunc (n - 1) Int.Min = y |]
  ==> sbintrunc n Int.Min = y BIT bit.B1
  [| 0 < n; sbintrunc (n - 1) w = y |] ==> sbintrunc n (w BIT b) = y BIT b

lemma sbintrunc_Suc_lem:

  [| sbintrunc (Suc n) x = y; m = Suc n |] ==> sbintrunc m x = y

lemma sbintrunc_Suc_Ialts:

  [| sbintrunc n Int.Pls = y; m = Suc n |] ==> sbintrunc m Int.Pls = y BIT bit.B0
  [| sbintrunc n Int.Min = y; m = Suc n |] ==> sbintrunc m Int.Min = y BIT bit.B1
  [| sbintrunc n w = y; m = Suc n |] ==> sbintrunc m (w BIT b) = y BIT b

lemma sbintrunc_bintrunc_lt:

  n < m ==> sbintrunc n (bintrunc m w) = sbintrunc n w

lemma bintrunc_sbintrunc_le:

  m  Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w

lemma bintrunc_sbintrunc:

  bintrunc (Suc n) (sbintrunc n w) = bintrunc (Suc n) w

lemma sbintrunc_bintrunc:

  sbintrunc n (bintrunc (Suc n) w) = sbintrunc n w

lemma bintrunc_bintrunc:

  bintrunc m (bintrunc m w) = bintrunc m w

lemma sbintrunc_sbintrunc:

  sbintrunc m (sbintrunc m w) = sbintrunc m w

lemma bintrunc_sbintrunc':

  0 < n ==> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w

lemma sbintrunc_bintrunc':

  0 < n ==> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w

lemma bin_sbin_eq_iff:

  (bintrunc (Suc n) x = bintrunc (Suc n) y) = (sbintrunc n x = sbintrunc n y)

lemma bin_sbin_eq_iff':

  0 < n
  ==> (bintrunc n x = bintrunc n y) = (sbintrunc (n - 1) x = sbintrunc (n - 1) y)

lemma bintrunc_sbintruncS0:

  0 < n ==> bintrunc n (sbintrunc (n - Suc 0) w) = bintrunc n w

lemma sbintrunc_bintruncS0:

  0 < n ==> sbintrunc (n - Suc 0) (bintrunc n w) = sbintrunc (n - Suc 0) w

lemma bintrunc_bintrunc_l':

  bintrunc (n + m1) (bintrunc n w) = bintrunc n w

lemma sbintrunc_sbintrunc_l':

  sbintrunc (n + m1) (sbintrunc n w) = sbintrunc n w

lemma nat_non0_gr:

  iszero z) = (z  (0::'a))

lemma bintrunc_pred_simps:

  0 < number_of bin
  ==> bintrunc (number_of bin) Int.Pls =
      bintrunc (number_of (Int.pred bin)) Int.Pls BIT bit.B0
  0 < number_of bin
  ==> bintrunc (number_of bin) Int.Min =
      bintrunc (number_of (Int.pred bin)) Int.Min BIT bit.B1
  0 < number_of bin
  ==> bintrunc (number_of bin) (w BIT b) =
      bintrunc (number_of (Int.pred bin)) w BIT b
  0 < number_of bin
  ==> bintrunc (number_of bin) (Int.Bit0 w) =
      Int.Bit0 (bintrunc (number_of (Int.pred bin)) w)
  0 < number_of bin
  ==> bintrunc (number_of bin) (Int.Bit1 w) =
      Int.Bit1 (bintrunc (number_of (Int.pred bin)) w)

lemma sbintrunc_pred_simps:

  0 < number_of bin
  ==> sbintrunc (number_of bin) Int.Pls =
      sbintrunc (number_of (Int.pred bin)) Int.Pls BIT bit.B0
  0 < number_of bin
  ==> sbintrunc (number_of bin) Int.Min =
      sbintrunc (number_of (Int.pred bin)) Int.Min BIT bit.B1
  0 < number_of bin
  ==> sbintrunc (number_of bin) (w BIT b) =
      sbintrunc (number_of (Int.pred bin)) w BIT b
  0 < number_of bin
  ==> sbintrunc (number_of bin) (Int.Bit0 w) =
      Int.Bit0 (sbintrunc (number_of (Int.pred bin)) w)
  0 < number_of bin
  ==> sbintrunc (number_of bin) (Int.Bit1 w) =
      Int.Bit1 (sbintrunc (number_of (Int.pred bin)) w)

lemma no_bintr_alt:

  number_of (bintrunc n w) = w mod 2 ^ n

lemma no_bintr_alt1:

  bintrunc n = (λw. w mod 2 ^ n)

lemma range_bintrunc:

  range (bintrunc n) = {i. 0  ii < 2 ^ n}

lemma no_bintr:

  number_of (bintrunc n w) = number_of w mod 2 ^ n

lemma no_sbintr_alt2:

  sbintrunc n = (λw. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n)

lemma no_sbintr:

  number_of (sbintrunc n w) = (number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n

lemma range_sbintrunc:

  range (sbintrunc n) = {i. - (2 ^ n)  ii < 2 ^ n}

lemma sb_inc_lem:

  a + 2 ^ k < 0 ==> a + 2 ^ k + 2 ^ Suc k  (a + 2 ^ k) mod 2 ^ Suc k

lemma sb_inc_lem':

  a < - (2 ^ k) ==> a + 2 ^ k + 2 ^ Suc k  (a + 2 ^ k) mod 2 ^ Suc k

lemma sbintrunc_inc:

  x < - (2 ^ n) ==> x + 2 ^ Suc n  sbintrunc n x

lemma sb_dec_lem:

  0  - (2 ^ k) + a ==> (a + 2 ^ k) mod (2 * 2 ^ k)  - (2 ^ k) + a

lemma sb_dec_lem':

  2 ^ k  a ==> (a + 2 ^ k) mod (2 * 2 ^ k)  - (2 ^ k) + a

lemma sbintrunc_dec:

  2 ^ n  x ==> sbintrunc n x  x - 2 ^ Suc n

lemma zmod_uminus':

  - (a mod c) mod c = - a mod c

lemma zpower_zmod':

  (x mod c) ^ k mod c = x ^ k mod c

lemma brdmod1s':

  (a mod c + b) mod c = (a + b) mod c
  (a + b mod c) mod c = (a + b) mod c
  (a mod c - b) mod c = (a - b) mod c
  (a - b mod c) mod c = (a - b) mod c
  a * (b mod c) mod c = a * b mod c
  b mod c * a mod c = b * a mod c

lemma brdmods':

  (x mod c) ^ k mod c = x ^ k mod c
  (a mod c + b mod c) mod c = (a + b) mod c
  (a mod c - b mod c) mod c = (a - b) mod c
  b mod c * (ba mod c) mod c = b * ba mod c
  - (a mod c) mod c = - a mod c
  (a mod c + 1) mod c = (a + 1) mod c
  (a mod c - 1) mod c = (a - 1) mod c

lemma bintr_arith1s:

  bintrunc n (bintrunc n a + b) = bintrunc n (a + b)
  bintrunc n (a + bintrunc n b) = bintrunc n (a + b)
  bintrunc n (bintrunc n a - b) = bintrunc n (a - b)
  bintrunc n (a - bintrunc n b) = bintrunc n (a - b)
  bintrunc n (a * bintrunc n b) = bintrunc n (a * b)
  bintrunc n (bintrunc n b * a) = bintrunc n (b * a)

lemma bintr_ariths:

  bintrunc n (bintrunc n x ^ k) = bintrunc n (x ^ k)
  bintrunc n (bintrunc n a + bintrunc n b) = bintrunc n (a + b)
  bintrunc n (bintrunc n a - bintrunc n b) = bintrunc n (a - b)
  bintrunc n (bintrunc n b * bintrunc n ba) = bintrunc n (b * ba)
  bintrunc n (- bintrunc n a) = bintrunc n (- a)
  bintrunc n (Int.succ (bintrunc n a)) = bintrunc n (Int.succ a)
  bintrunc n (Int.pred (bintrunc n a)) = bintrunc n (Int.pred a)

lemma m2pths:

  0 < b ==> 0  a mod b
  a mod 2 ^ n < 2 ^ n

lemma bintr_ge0:

  0  number_of (bintrunc n w)

lemma bintr_lt2p:

  number_of (bintrunc n w) < 2 ^ n

lemma bintr_Min:

  number_of (bintrunc n Int.Min) = 2 ^ n - 1

lemma sbintr_ge:

  - (2 ^ n)  number_of (sbintrunc n w)

lemma sbintr_lt:

  number_of (sbintrunc n w) < 2 ^ n

lemma bintrunc_Suc:

  bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin

lemma sign_Pls_ge_0:

  (bin_sign bin = Int.Pls) = (0  number_of bin)

lemma sign_Min_lt_0:

  (bin_sign bin = Int.Min) = (number_of bin < 0)

lemma sign_Min_neg:

  (bin_sign bin2 = Int.Min) = neg (number_of bin2)

lemma bin_rest_trunc:

  bin_rest (bintrunc n bin) = bintrunc (n - 1) (bin_rest bin)

lemma bin_rest_power_trunc:

  (bin_rest ^ k) (bintrunc n bin) = bintrunc (n - k) ((bin_rest ^ k) bin)

lemma bin_rest_trunc_i:

  bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)

lemma bin_rest_strunc:

  bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)

lemma bintrunc_rest:

  bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)

lemma sbintrunc_rest:

  sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)

lemma bintrunc_rest':

  bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n

lemma sbintrunc_rest':

  sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n

lemma rco_lem:

  f o g o f = g o f ==> f o (g o f) ^ n = g ^ n o f

lemma rco_alt:

  (f o g) ^ n o f = f o (g o f) ^ n

lemma rco_bintr:

  bintrunc n2 (((λx. bin_rest (bintrunc n2 x)) ^ n1) x) =
  (bin_rest ^ n1) (bintrunc n2 x)

lemma rco_sbintr:

  sbintrunc n2 (((λx. bin_rest (sbintrunc n2 x)) ^ n1) x) =
  (bin_rest ^ n1) (sbintrunc n2 x)

Splitting and concatenation

Miscellaneous lemmas

lemma funpow_minus_simp:

  0 < n ==> f ^ n = f o f ^ (n - 1)

lemma funpow_pred_simp:

  0 < number_of bin ==> f ^ number_of bin = f o f ^ number_of (Int.pred bin)

lemma replicate_minus_simp:

  0 < n ==> replicate n x = x # replicate (n - 1) x

lemma replicate_pred_simp:

  0 < number_of bin
  ==> replicate (number_of bin) x = x # replicate (number_of (Int.pred bin)) x

lemma power_Suc_no:

  number_of a ^ Suc n = number_of a * number_of a ^ n

lemma power_minus_simp:

  0 < n ==> f ^ n = f * f ^ (n - 1)

lemma power_pred_simp:

  0 < number_of bin ==> f ^ number_of bin = f * f ^ number_of (Int.pred bin)

lemma power_pred_simp_no:

  0 < number_of bin
  ==> number_of f ^ number_of bin =
      number_of f * number_of f ^ number_of (Int.pred bin)

lemma list_exhaust_size_gt0:

  [| !!a list. y = a # list ==> P; 0 < length y |] ==> P

lemma list_exhaust_size_eq0:

  [| y = [] ==> P; length y = 0 |] ==> P

lemma size_Cons_lem_eq:

  [| y = xa # list; length y = Suc k |] ==> length list = k

lemma size_Cons_lem_eq_bin:

  [| y = xa # list; length y = number_of (Int.succ k) |]
  ==> length list = number_of k

lemma ls_splits:

  P (prod_case f1.0 x) = (∀a b. x = (a, b) --> P (f1.0 a b))
  P (split f1.0 x) = (∀a b. x = (a, b) --> P (f1.0 a b))
  P (prod_case f1.0 x) = (¬ (∃a b. x = (a, b) ∧ ¬ P (f1.0 a b)))
  P (split f1.0 x) = (¬ (∃a b. x = (a, b) ∧ ¬ P (f1.0 a b)))
  P (if Q then x else y) = (¬ (Q ∧ ¬ P x ∨ ¬ Q ∧ ¬ P y))

lemma not_B1_is_B0:

  y  bit.B1 ==> y = bit.B0

lemma B1_ass_B0:

  (y = bit.B0 ==> y = bit.B1) ==> y = bit.B1

lemma n2s_ths:

  2 + n1 == Suc (Suc n1)
  n1 + 2 == Suc (Suc n1)

lemma s2n_ths:

  Suc (Suc n) == 2 + n
  Suc (Suc n) == n + 2