src/HOL/Word/Bit_Representation.thy
author huffman
Thu, 17 Nov 2011 12:29:48 +0100
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permissions -rw-r--r--
HOL-Word: add simp rules for bin_rest, bin_last; shorten several proofs
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(* 
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  Author: Jeremy Dawson, NICTA
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  Basic definitions to do with integers, expressed using Pls, Min, BIT.
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*) 
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header {* Basic Definitions for Binary Integers *}
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theory Bit_Representation
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imports Misc_Numeric "~~/src/HOL/Library/Bit"
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begin
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subsection {* Further properties of numerals *}
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definition bitval :: "bit \<Rightarrow> 'a\<Colon>zero_neq_one" where
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  "bitval = bit_case 0 1"
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lemma bitval_simps [simp]:
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  "bitval 0 = 0"
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  "bitval 1 = 1"
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  by (simp_all add: bitval_def)
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definition Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
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  "k BIT b = bitval b + k + k"
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lemma BIT_B0_eq_Bit0 [simp]: "w BIT 0 = Int.Bit0 w"
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  unfolding Bit_def Bit0_def by simp
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lemma BIT_B1_eq_Bit1 [simp]: "w BIT 1 = Int.Bit1 w"
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  unfolding Bit_def Bit1_def by simp
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lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1
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lemma Min_ne_Pls [iff]:  
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  "Int.Min ~= Int.Pls"
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  unfolding Min_def Pls_def by auto
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lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric]
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lemmas PlsMin_defs [intro!] = 
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  Pls_def Min_def Pls_def [symmetric] Min_def [symmetric]
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lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI]
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lemma number_of_False_cong: 
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  "False \<Longrightarrow> number_of x = number_of y"
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  by (rule FalseE)
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(** ways in which type Bin resembles a datatype **)
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lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c"
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  apply (cases b) apply (simp_all)
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  apply (cases c) apply (simp_all)
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  apply (cases c) apply (simp_all)
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  done
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lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard]
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lemma BIT_eq_iff [simp]: 
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  "(u BIT b = v BIT c) = (u = v \<and> b = c)"
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  by (rule iffI) auto
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lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]]
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lemma less_Bits: 
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  "(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))"
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  unfolding Bit_def by (auto simp add: bitval_def split: bit.split)
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lemma le_Bits: 
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  "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))" 
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  unfolding Bit_def by (auto simp add: bitval_def split: bit.split)
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lemma no_no [simp]: "number_of (number_of i) = i"
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  unfolding number_of_eq by simp
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lemma Bit_B0:
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  "k BIT (0::bit) = k + k"
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   by (unfold Bit_def) simp
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lemma Bit_B1:
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  "k BIT (1::bit) = k + k + 1"
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   by (unfold Bit_def) simp
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lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k"
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  by (rule trans, rule Bit_B0) simp
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lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1"
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  by (rule trans, rule Bit_B1) simp
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lemma B_mod_2': 
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  "X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0"
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  apply (simp (no_asm) only: Bit_B0 Bit_B1)
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  apply (simp add: z1pmod2)
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  done
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lemma B1_mod_2 [simp]: "(Int.Bit1 w) mod 2 = 1"
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  unfolding numeral_simps number_of_is_id by (simp add: z1pmod2)
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lemma B0_mod_2 [simp]: "(Int.Bit0 w) mod 2 = 0"
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  unfolding numeral_simps number_of_is_id by simp
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lemma neB1E [elim!]:
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  assumes ne: "y \<noteq> (1::bit)"
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  assumes y: "y = (0::bit) \<Longrightarrow> P"
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  shows "P"
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  apply (rule y)
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  apply (cases y rule: bit.exhaust, simp)
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  apply (simp add: ne)
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  done
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lemma bin_ex_rl: "EX w b. w BIT b = bin"
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  apply (unfold Bit_def)
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  apply (cases "even bin")
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   apply (clarsimp simp: even_equiv_def)
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   apply (auto simp: odd_equiv_def bitval_def split: bit.split)
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  done
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lemma bin_exhaust:
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  assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
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  shows "Q"
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  apply (insert bin_ex_rl [of bin])  
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  apply (erule exE)+
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  apply (rule Q)
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  apply force
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  done
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subsection {* Destructors for binary integers *}
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definition bin_last :: "int \<Rightarrow> bit" where
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  "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"
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definition bin_rest :: "int \<Rightarrow> int" where
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  "bin_rest w = w div 2"
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definition bin_rl :: "int \<Rightarrow> int \<times> bit" where 
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  "bin_rl w = (bin_rest w, bin_last w)"
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lemma bin_rl_char: "bin_rl w = (r, l) \<longleftrightarrow> r BIT l = w"
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  apply (cases l)
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  apply (auto simp add: bin_rl_def bin_last_def bin_rest_def)
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  unfolding Pls_def Min_def Bit0_def Bit1_def number_of_is_id
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  apply arith+
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  done
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primrec bin_nth where
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  Z: "bin_nth w 0 = (bin_last w = (1::bit))"
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  | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"
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lemma bin_rl_simps [simp]:
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  "bin_rl Int.Pls = (Int.Pls, (0::bit))"
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  "bin_rl Int.Min = (Int.Min, (1::bit))"
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  "bin_rl (Int.Bit0 r) = (r, (0::bit))"
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  "bin_rl (Int.Bit1 r) = (r, (1::bit))"
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  "bin_rl (r BIT b) = (r, b)"
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  unfolding bin_rl_char by simp_all
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lemma bin_rl_simp [simp]:
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  "bin_rest w BIT bin_last w = w"
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  by (simp add: iffD1 [OF bin_rl_char bin_rl_def])
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lemma bin_abs_lem:
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  "bin = (w BIT b) ==> ~ bin = Int.Min --> ~ bin = Int.Pls -->
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    nat (abs w) < nat (abs bin)"
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  apply (clarsimp simp add: bin_rl_char)
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  apply (unfold Pls_def Min_def Bit_def)
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  apply (cases b)
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   apply (clarsimp, arith)
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  apply (clarsimp, arith)
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  done
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lemma bin_induct:
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  assumes PPls: "P Int.Pls"
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    and PMin: "P Int.Min"
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    and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"
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  shows "P bin"
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  apply (rule_tac P=P and a=bin and f1="nat o abs" 
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                  in wf_measure [THEN wf_induct])
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  apply (simp add: measure_def inv_image_def)
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  apply (case_tac x rule: bin_exhaust)
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  apply (frule bin_abs_lem)
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  apply (auto simp add : PPls PMin PBit)
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  done
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lemma numeral_induct:
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  assumes Pls: "P Int.Pls"
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  assumes Min: "P Int.Min"
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  assumes Bit0: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Pls\<rbrakk> \<Longrightarrow> P (Int.Bit0 w)"
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  assumes Bit1: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Min\<rbrakk> \<Longrightarrow> P (Int.Bit1 w)"
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  shows "P x"
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  apply (induct x rule: bin_induct)
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    apply (rule Pls)
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   apply (rule Min)
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  apply (case_tac bit)
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   apply (case_tac "bin = Int.Pls")
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    apply simp
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   apply (simp add: Bit0)
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  apply (case_tac "bin = Int.Min")
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   apply simp
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  apply (simp add: Bit1)
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  done
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lemma bin_rest_simps [simp]: 
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  "bin_rest Int.Pls = Int.Pls"
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  "bin_rest Int.Min = Int.Min"
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  "bin_rest (Int.Bit0 w) = w"
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  "bin_rest (Int.Bit1 w) = w"
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  "bin_rest (w BIT b) = w"
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  using bin_rl_simps bin_rl_def by auto
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lemma bin_last_simps [simp]: 
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  "bin_last Int.Pls = (0::bit)"
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  "bin_last Int.Min = (1::bit)"
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  "bin_last (Int.Bit0 w) = (0::bit)"
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  "bin_last (Int.Bit1 w) = (1::bit)"
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  "bin_last (w BIT b) = b"
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  using bin_rl_simps bin_rl_def by auto
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lemma bin_r_l_extras [simp]:
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  "bin_last 0 = (0::bit)"
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  "bin_last (- 1) = (1::bit)"
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  "bin_last -1 = (1::bit)"
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  "bin_last 1 = (1::bit)"
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  "bin_rest 1 = 0"
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  "bin_rest 0 = 0"
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  "bin_rest (- 1) = - 1"
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  "bin_rest -1 = -1"
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  by (simp_all add: bin_last_def bin_rest_def)
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lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
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  unfolding bin_rest_def [symmetric] by auto
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lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w"
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  using Bit_div2 [where b="(0::bit)"] by simp
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lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w"
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  using Bit_div2 [where b="(1::bit)"] by simp
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lemma bin_nth_lem [rule_format]:
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  "ALL y. bin_nth x = bin_nth y --> x = y"
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  apply (induct x rule: bin_induct)
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    apply safe
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    apply (erule rev_mp)
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    apply (induct_tac y rule: bin_induct)
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      apply (safe del: subset_antisym)
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      apply (drule_tac x=0 in fun_cong, force)
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     apply (erule notE, rule ext, 
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            drule_tac x="Suc x" in fun_cong, force)
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    apply (drule_tac x=0 in fun_cong, force)
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   apply (erule rev_mp)
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   apply (induct_tac y rule: bin_induct)
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     apply (safe del: subset_antisym)
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     apply (drule_tac x=0 in fun_cong, force)
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    apply (erule notE, rule ext, 
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           drule_tac x="Suc x" in fun_cong, force)
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   apply (drule_tac x=0 in fun_cong, force)
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  apply (case_tac y rule: bin_exhaust)
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  apply clarify
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  apply (erule allE)
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  apply (erule impE)
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   prefer 2
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   apply (erule BIT_eqI)
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   apply (drule_tac x=0 in fun_cong, force)
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  apply (rule ext)
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  apply (drule_tac x="Suc ?x" in fun_cong, force)
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  done
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lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)"
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  by (auto elim: bin_nth_lem)
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lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1], standard]
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lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"
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  by (auto intro!: bin_nth_lem del: equalityI)
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lemma bin_nth_Pls [simp]: "~ bin_nth Int.Pls n"
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  by (induct n) auto
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lemma bin_nth_Min [simp]: "bin_nth Int.Min n"
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  by (induct n) auto
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lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))"
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  by auto
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lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
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  by auto
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lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)"
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  by (cases n) auto
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lemma bin_nth_minus_Bit0 [simp]:
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  "0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)"
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  using bin_nth_minus [where b="(0::bit)"] by simp
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lemma bin_nth_minus_Bit1 [simp]:
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  "0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)"
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  using bin_nth_minus [where b="(1::bit)"] by simp
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lemmas bin_nth_0 = bin_nth.simps(1)
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lemmas bin_nth_Suc = bin_nth.simps(2)
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lemmas bin_nth_simps = 
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  bin_nth_0 bin_nth_Suc bin_nth_Pls bin_nth_Min bin_nth_minus
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  bin_nth_minus_Bit0 bin_nth_minus_Bit1
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subsection {* Truncating binary integers *}
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definition
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  bin_sign_def: "bin_sign k = (if k \<ge> 0 then 0 else - 1)"
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lemma bin_sign_simps [simp]:
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  "bin_sign Int.Pls = Int.Pls"
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  "bin_sign Int.Min = Int.Min"
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  "bin_sign (Int.Bit0 w) = bin_sign w"
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  "bin_sign (Int.Bit1 w) = bin_sign w"
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  "bin_sign (w BIT b) = bin_sign w"
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   318
  by (unfold bin_sign_def numeral_simps Bit_def bitval_def) (simp_all split: bit.split)
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9e7f95903b24 more new primrec
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parents: 26514
diff changeset
   319
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31e359126ab6 reorganize into subsections
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parents: 24350
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   320
lemma bin_sign_rest [simp]: 
37667
41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   321
  "bin_sign (bin_rest w) = bin_sign w"
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9e7f95903b24 more new primrec
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parents: 26514
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   322
  by (cases w rule: bin_exhaust) auto
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31e359126ab6 reorganize into subsections
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parents: 24350
diff changeset
   323
37667
41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   324
primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int" where
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   325
  Z : "bintrunc 0 bin = Int.Pls"
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41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   326
| Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
24364
31e359126ab6 reorganize into subsections
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parents: 24350
diff changeset
   327
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41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   328
primrec sbintrunc :: "nat => int => int" where
24364
31e359126ab6 reorganize into subsections
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parents: 24350
diff changeset
   329
  Z : "sbintrunc 0 bin = 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   330
    (case bin_last bin of (1::bit) => Int.Min | (0::bit) => Int.Pls)"
37667
41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   331
| Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   332
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   333
lemma [code]:
41acc0fa6b6c avoid bitstrings in generated code
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parents: 37658
diff changeset
   334
  "sbintrunc 0 bin = 
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   335
    (case bin_last bin of (1::bit) => - 1 | (0::bit) => 0)"
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   336
  "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   337
  apply simp_all apply (cases "bin_last bin") apply simp apply (unfold Min_def number_of_is_id) apply simp done
24364
31e359126ab6 reorganize into subsections
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parents: 24350
diff changeset
   338
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e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   339
lemma sign_bintr:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   340
  "!!w. bin_sign (bintrunc n w) = Int.Pls"
24333
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   341
  by (induct n) auto
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   342
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   343
lemma bintrunc_mod2p:
e77ea0ea7f2c * HOL-Word:
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   344
  "!!w. bintrunc n w = (w mod 2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   345
  apply (induct n, clarsimp)
45529
0e1037d4e049 remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents: 44939
diff changeset
   346
  apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq
24333
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   347
              cong: number_of_False_cong)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   348
  done
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   349
e77ea0ea7f2c * HOL-Word:
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diff changeset
   350
lemma sbintrunc_mod2p:
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   351
  "!!w. sbintrunc n w = ((w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   352
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   353
   apply clarsimp
30034
60f64f112174 removed redundant thms
nipkow
parents: 29631
diff changeset
   354
   apply (subst mod_add_left_eq)
45529
0e1037d4e049 remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents: 44939
diff changeset
   355
   apply (simp add: bin_last_def)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   356
   apply (simp add: number_of_eq)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   357
  apply clarsimp
45529
0e1037d4e049 remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents: 44939
diff changeset
   358
  apply (simp add: bin_last_def bin_rest_def Bit_def 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   359
              cong: number_of_False_cong)
30940
663af91c0720 zmod_zmult_zmult1 now subsumed by mod_mult_mult1
haftmann
parents: 30034
diff changeset
   360
  apply (clarsimp simp: mod_mult_mult1 [symmetric] 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   361
         zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   362
  apply (rule trans [symmetric, OF _ emep1])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   363
     apply auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   364
  apply (auto simp: even_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   365
  done
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   366
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24419
diff changeset
   367
subsection "Simplifications for (s)bintrunc"
70f0214b3ecc revert to Word library version from 2007/08/20
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parents: 24419
diff changeset
   368
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24419
diff changeset
   369
lemma bit_bool:
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   370
  "(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))"
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24419
diff changeset
   371
  by (cases b') auto
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24419
diff changeset
   372
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24419
diff changeset
   373
lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   374
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   375
lemma bin_sign_lem:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   376
  "!!bin. (bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   377
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   378
   apply (case_tac bin rule: bin_exhaust, case_tac b, auto)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   379
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   380
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   381
lemma nth_bintr:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   382
  "!!w m. bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   383
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   384
   apply (case_tac m, auto)[1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   385
  apply (case_tac m, auto)[1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   386
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   387
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   388
lemma nth_sbintr:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   389
  "!!w m. bin_nth (sbintrunc m w) n = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   390
          (if n < m then bin_nth w n else bin_nth w m)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   391
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   392
   apply (case_tac m, simp_all split: bit.splits)[1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   393
  apply (case_tac m, simp_all split: bit.splits)[1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   394
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   395
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   396
lemma bin_nth_Bit:
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   397
  "bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   398
  by (cases n) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   399
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   400
lemma bin_nth_Bit0:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   401
  "bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)"
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   402
  using bin_nth_Bit [where b="(0::bit)"] by simp
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   403
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   404
lemma bin_nth_Bit1:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   405
  "bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))"
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   406
  using bin_nth_Bit [where b="(1::bit)"] by simp
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   407
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   408
lemma bintrunc_bintrunc_l:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   409
  "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   410
  by (rule bin_eqI) (auto simp add : nth_bintr)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   411
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   412
lemma sbintrunc_sbintrunc_l:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   413
  "n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)"
32439
7a91c7bcfe7e tuned proofs
nipkow
parents: 30971
diff changeset
   414
  by (rule bin_eqI) (auto simp: nth_sbintr)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   415
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   416
lemma bintrunc_bintrunc_ge:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   417
  "n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   418
  by (rule bin_eqI) (auto simp: nth_bintr)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   419
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   420
lemma bintrunc_bintrunc_min [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   421
  "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   422
  apply (rule bin_eqI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   423
  apply (auto simp: nth_bintr)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   424
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   425
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   426
lemma sbintrunc_sbintrunc_min [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   427
  "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   428
  apply (rule bin_eqI)
32642
026e7c6a6d08 be more cautious wrt. simp rules: inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer
haftmann
parents: 32439
diff changeset
   429
  apply (auto simp: nth_sbintr min_max.inf_absorb1 min_max.inf_absorb2)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   430
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   431
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   432
lemmas bintrunc_Pls = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   433
  bintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   434
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   435
lemmas bintrunc_Min [simp] = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   436
  bintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   437
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   438
lemmas bintrunc_BIT  [simp] = 
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   439
  bintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   440
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   441
lemma bintrunc_Bit0 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   442
  "bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)"
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   443
  using bintrunc_BIT [where b="(0::bit)"] by simp
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   444
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   445
lemma bintrunc_Bit1 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   446
  "bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)"
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   447
  using bintrunc_BIT [where b="(1::bit)"] by simp
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   448
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   449
lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   450
  bintrunc_Bit0 bintrunc_Bit1
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   451
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   452
lemmas sbintrunc_Suc_Pls = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   453
  sbintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   454
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   455
lemmas sbintrunc_Suc_Min = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   456
  sbintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   457
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   458
lemmas sbintrunc_Suc_BIT [simp] = 
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   459
  sbintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   460
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   461
lemma sbintrunc_Suc_Bit0 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   462
  "sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)"
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   463
  using sbintrunc_Suc_BIT [where b="(0::bit)"] by simp
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   464
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   465
lemma sbintrunc_Suc_Bit1 [simp]:
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   466
  "sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)"
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   467
  using sbintrunc_Suc_BIT [where b="(1::bit)"] by simp
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   468
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   469
lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   470
  sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   471
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   472
lemmas sbintrunc_Pls = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   473
  sbintrunc.Z [where bin="Int.Pls", 
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   474
               simplified bin_last_simps bin_rest_simps bit.simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   475
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   476
lemmas sbintrunc_Min = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   477
  sbintrunc.Z [where bin="Int.Min", 
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   478
               simplified bin_last_simps bin_rest_simps bit.simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   479
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   480
lemmas sbintrunc_0_BIT_B0 [simp] = 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   481
  sbintrunc.Z [where bin="w BIT (0::bit)", 
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   482
               simplified bin_last_simps bin_rest_simps bit.simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   483
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   484
lemmas sbintrunc_0_BIT_B1 [simp] = 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   485
  sbintrunc.Z [where bin="w BIT (1::bit)", 
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   486
               simplified bin_last_simps bin_rest_simps bit.simps, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   487
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   488
lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = Int.Pls"
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   489
  using sbintrunc_0_BIT_B0 by simp
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   490
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   491
lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = Int.Min"
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   492
  using sbintrunc_0_BIT_B1 by simp
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   493
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   494
lemmas sbintrunc_0_simps =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   495
  sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   496
  sbintrunc_0_Bit0 sbintrunc_0_Bit1
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   497
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   498
lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   499
lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   500
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   501
lemma bintrunc_minus:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   502
  "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   503
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   504
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   505
lemma sbintrunc_minus:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   506
  "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   507
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   508
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   509
lemmas bintrunc_minus_simps = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   510
  bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans], standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   511
lemmas sbintrunc_minus_simps = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   512
  sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans], standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   513
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   514
lemma bintrunc_n_Pls [simp]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   515
  "bintrunc n Int.Pls = Int.Pls"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   516
  by (induct n) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   517
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   518
lemma sbintrunc_n_PM [simp]:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   519
  "sbintrunc n Int.Pls = Int.Pls"
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   520
  "sbintrunc n Int.Min = Int.Min"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   521
  by (induct n) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   522
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   523
lemmas thobini1 = arg_cong [where f = "%w. w BIT b", standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   524
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   525
lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   526
lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   527
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   528
lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   529
lemmas bintrunc_Pls_minus_I = bmsts(1)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   530
lemmas bintrunc_Min_minus_I = bmsts(2)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   531
lemmas bintrunc_BIT_minus_I = bmsts(3)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   532
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   533
lemma bintrunc_0_Min: "bintrunc 0 Int.Min = Int.Pls"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   534
  by auto
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   535
lemma bintrunc_0_BIT: "bintrunc 0 (w BIT b) = Int.Pls"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   536
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   537
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   538
lemma bintrunc_Suc_lem:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   539
  "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   540
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   541
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   542
lemmas bintrunc_Suc_Ialts = 
26294
c5fe289de634 fixed broken bintrunc lemma
kleing
parents: 26086
diff changeset
   543
  bintrunc_Min_I [THEN bintrunc_Suc_lem, standard]
c5fe289de634 fixed broken bintrunc lemma
kleing
parents: 26086
diff changeset
   544
  bintrunc_BIT_I [THEN bintrunc_Suc_lem, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   545
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   546
lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   547
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   548
lemmas sbintrunc_Suc_Is = 
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   549
  sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans], standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   550
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   551
lemmas sbintrunc_Suc_minus_Is = 
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   552
  sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   553
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   554
lemma sbintrunc_Suc_lem:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   555
  "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   556
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   557
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   558
lemmas sbintrunc_Suc_Ialts = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   559
  sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem, standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   560
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   561
lemma sbintrunc_bintrunc_lt:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   562
  "m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   563
  by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   564
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   565
lemma bintrunc_sbintrunc_le:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   566
  "m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   567
  apply (rule bin_eqI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   568
  apply (auto simp: nth_sbintr nth_bintr)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   569
   apply (subgoal_tac "x=n", safe, arith+)[1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   570
  apply (subgoal_tac "x=n", safe, arith+)[1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   571
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   572
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   573
lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   574
lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   575
lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   576
lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l] 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   577
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   578
lemma bintrunc_sbintrunc' [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   579
  "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   580
  by (cases n) (auto simp del: bintrunc.Suc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   581
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   582
lemma sbintrunc_bintrunc' [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   583
  "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   584
  by (cases n) (auto simp del: bintrunc.Suc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   585
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   586
lemma bin_sbin_eq_iff: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   587
  "bintrunc (Suc n) x = bintrunc (Suc n) y <-> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   588
   sbintrunc n x = sbintrunc n y"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   589
  apply (rule iffI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   590
   apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   591
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   592
  apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   593
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   594
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   595
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   596
lemma bin_sbin_eq_iff':
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   597
  "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y <-> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   598
            sbintrunc (n - 1) x = sbintrunc (n - 1) y"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   599
  by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   600
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   601
lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   602
lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   603
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   604
lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   605
lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   606
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   607
(* although bintrunc_minus_simps, if added to default simpset,
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   608
  tends to get applied where it's not wanted in developing the theories,
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   609
  we get a version for when the word length is given literally *)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   610
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   611
lemmas nat_non0_gr = 
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 24465
diff changeset
   612
  trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   613
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   614
lemmas bintrunc_pred_simps [simp] = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   615
  bintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   616
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   617
lemmas sbintrunc_pred_simps [simp] = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   618
  sbintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   619
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   620
lemma no_bintr_alt:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   621
  "number_of (bintrunc n w) = w mod 2 ^ n"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   622
  by (simp add: number_of_eq bintrunc_mod2p)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   623
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   624
lemma no_bintr_alt1: "bintrunc n = (%w. w mod 2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   625
  by (rule ext) (rule bintrunc_mod2p)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   626
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   627
lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   628
  apply (unfold no_bintr_alt1)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   629
  apply (auto simp add: image_iff)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   630
  apply (rule exI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   631
  apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   632
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   633
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   634
lemma no_bintr: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   635
  "number_of (bintrunc n w) = (number_of w mod 2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   636
  by (simp add : bintrunc_mod2p number_of_eq)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   637
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   638
lemma no_sbintr_alt2: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   639
  "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   640
  by (rule ext) (simp add : sbintrunc_mod2p)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   641
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   642
lemma no_sbintr: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   643
  "number_of (sbintrunc n w) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   644
   ((number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   645
  by (simp add : no_sbintr_alt2 number_of_eq)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   646
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   647
lemma range_sbintrunc: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   648
  "range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   649
  apply (unfold no_sbintr_alt2)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   650
  apply (auto simp add: image_iff eq_diff_eq)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   651
  apply (rule exI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   652
  apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   653
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   654
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   655
lemma sb_inc_lem:
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   656
  "(a::int) + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   657
  apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p])
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   658
  apply (rule TrueI)
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   659
  done
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   660
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   661
lemma sb_inc_lem':
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   662
  "(a::int) < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
35048
82ab78fff970 tuned proofs
haftmann
parents: 32642
diff changeset
   663
  by (rule sb_inc_lem) simp
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   664
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   665
lemma sbintrunc_inc:
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   666
  "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   667
  unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   668
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   669
lemma sb_dec_lem:
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   670
  "(0::int) <= - (2^k) + a ==> (a + 2^k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   671
  by (rule int_mod_le' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k",
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   672
    simplified zless2p, OF _ TrueI, simplified])
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   673
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   674
lemma sb_dec_lem':
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   675
  "(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   676
  by (rule iffD1 [OF diff_le_eq', THEN sb_dec_lem, simplified])
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   677
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   678
lemma sbintrunc_dec:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   679
  "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   680
  unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   681
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   682
lemmas zmod_uminus' = zmod_uminus [where b="c", standard]
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   683
lemmas zpower_zmod' = zpower_zmod [where m="c" and y="k", standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   684
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   685
lemmas brdmod1s' [symmetric] = 
30034
60f64f112174 removed redundant thms
nipkow
parents: 29631
diff changeset
   686
  mod_add_left_eq mod_add_right_eq 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   687
  zmod_zsub_left_eq zmod_zsub_right_eq 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   688
  zmod_zmult1_eq zmod_zmult1_eq_rev 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   689
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   690
lemmas brdmods' [symmetric] = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   691
  zpower_zmod' [symmetric]
30034
60f64f112174 removed redundant thms
nipkow
parents: 29631
diff changeset
   692
  trans [OF mod_add_left_eq mod_add_right_eq] 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   693
  trans [OF zmod_zsub_left_eq zmod_zsub_right_eq] 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   694
  trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev] 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   695
  zmod_uminus' [symmetric]
30034
60f64f112174 removed redundant thms
nipkow
parents: 29631
diff changeset
   696
  mod_add_left_eq [where b = "1::int"]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   697
  zmod_zsub_left_eq [where b = "1"]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   698
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   699
lemmas bintr_arith1s =
30034
60f64f112174 removed redundant thms
nipkow
parents: 29631
diff changeset
   700
  brdmod1s' [where c="2^n::int", folded pred_def succ_def bintrunc_mod2p, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   701
lemmas bintr_ariths =
30034
60f64f112174 removed redundant thms
nipkow
parents: 29631
diff changeset
   702
  brdmods' [where c="2^n::int", folded pred_def succ_def bintrunc_mod2p, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   703
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   704
lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p, standard] 
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   705
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   706
lemma bintr_ge0: "(0 :: int) <= number_of (bintrunc n w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   707
  by (simp add : no_bintr m2pths)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   708
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   709
lemma bintr_lt2p: "number_of (bintrunc n w) < (2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   710
  by (simp add : no_bintr m2pths)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   711
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   712
lemma bintr_Min: 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   713
  "number_of (bintrunc n Int.Min) = (2 ^ n :: int) - 1"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   714
  by (simp add : no_bintr m1mod2k)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   715
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   716
lemma sbintr_ge: "(- (2 ^ n) :: int) <= number_of (sbintrunc n w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   717
  by (simp add : no_sbintr m2pths)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   718
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   719
lemma sbintr_lt: "number_of (sbintrunc n w) < (2 ^ n :: int)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   720
  by (simp add : no_sbintr m2pths)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   721
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   722
lemma bintrunc_Suc:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   723
  "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   724
  by (case_tac bin rule: bin_exhaust) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   725
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   726
lemma sign_Pls_ge_0: 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   727
  "(bin_sign bin = Int.Pls) = (number_of bin >= (0 :: int))"
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   728
  by (induct bin rule: numeral_induct) auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   729
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   730
lemma sign_Min_lt_0: 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   731
  "(bin_sign bin = Int.Min) = (number_of bin < (0 :: int))"
26086
3c243098b64a New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents: 25919
diff changeset
   732
  by (induct bin rule: numeral_induct) auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   733
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   734
lemmas sign_Min_neg = trans [OF sign_Min_lt_0 neg_def [symmetric]] 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   735
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   736
lemma bin_rest_trunc:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   737
  "!!bin. (bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   738
  by (induct n) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   739
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   740
lemma bin_rest_power_trunc [rule_format] :
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   741
  "(bin_rest ^^ k) (bintrunc n bin) = 
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   742
    bintrunc (n - k) ((bin_rest ^^ k) bin)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   743
  by (induct k) (auto simp: bin_rest_trunc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   744
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   745
lemma bin_rest_trunc_i:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   746
  "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   747
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   748
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   749
lemma bin_rest_strunc:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   750
  "!!bin. bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   751
  by (induct n) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   752
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   753
lemma bintrunc_rest [simp]: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   754
  "!!bin. bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   755
  apply (induct n, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   756
  apply (case_tac bin rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   757
  apply (auto simp: bintrunc_bintrunc_l)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   758
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   759
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   760
lemma sbintrunc_rest [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   761
  "!!bin. sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   762
  apply (induct n, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   763
  apply (case_tac bin rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   764
  apply (auto simp: bintrunc_bintrunc_l split: bit.splits)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   765
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   766
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   767
lemma bintrunc_rest':
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   768
  "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   769
  by (rule ext) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   770
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   771
lemma sbintrunc_rest' :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   772
  "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   773
  by (rule ext) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   774
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   775
lemma rco_lem:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   776
  "f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   777
  apply (rule ext)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   778
  apply (induct_tac n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   779
   apply (simp_all (no_asm))
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   780
  apply (drule fun_cong)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   781
  apply (unfold o_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   782
  apply (erule trans)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   783
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   784
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   785
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   786
lemma rco_alt: "(f o g) ^^ n o f = f o (g o f) ^^ n"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   787
  apply (rule ext)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   788
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   789
   apply (simp_all add: o_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   790
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   791
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   792
lemmas rco_bintr = bintrunc_rest' 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   793
  [THEN rco_lem [THEN fun_cong], unfolded o_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   794
lemmas rco_sbintr = sbintrunc_rest' 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   795
  [THEN rco_lem [THEN fun_cong], unfolded o_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   796
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   797
subsection {* Splitting and concatenation *}
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   798
26557
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   799
primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   800
  Z: "bin_split 0 w = (w, Int.Pls)"
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   801
  | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   802
        in (w1, w2 BIT bin_last w))"
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   803
37667
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   804
lemma [code]:
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   805
  "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))"
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   806
  "bin_split 0 w = (w, 0)"
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   807
  by (simp_all add: Pls_def)
41acc0fa6b6c avoid bitstrings in generated code
haftmann
parents: 37658
diff changeset
   808
26557
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   809
primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   810
  Z: "bin_cat w 0 v = w"
9e7f95903b24 more new primrec
haftmann
parents: 26514
diff changeset
   811
  | Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   812
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   813
subsection {* Miscellaneous lemmas *}
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   814
30952
7ab2716dd93b power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents: 30940
diff changeset
   815
lemma funpow_minus_simp:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   816
  "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"
30952
7ab2716dd93b power operation on functions with syntax o^; power operation on relations with syntax ^^
haftmann
parents: 30940
diff changeset
   817
  by (cases n) simp_all
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   818
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   819
lemmas funpow_pred_simp [simp] =
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   820
  funpow_minus_simp [of "number_of bin", simplified nobm1, standard]
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   821
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   822
lemmas replicate_minus_simp = 
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   823
  trans [OF gen_minus [where f = "%n. replicate n x"] replicate.replicate_Suc,
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   824
         standard]
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   825
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   826
lemmas replicate_pred_simp [simp] =
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   827
  replicate_minus_simp [of "number_of bin", simplified nobm1, standard]
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   828
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   829
lemmas power_Suc_no [simp] = power_Suc [of "number_of a", standard]
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   830
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   831
lemmas power_minus_simp = 
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   832
  trans [OF gen_minus [where f = "power f"] power_Suc, standard]
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   833
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   834
lemmas power_pred_simp = 
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   835
  power_minus_simp [of "number_of bin", simplified nobm1, standard]
25349
0d46bea01741 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   836
lemmas power_pred_simp_no [simp] = power_pred_simp [where f= "number_of f", standard]
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   837
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   838
lemma list_exhaust_size_gt0:
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   839
  assumes y: "\<And>a list. y = a # list \<Longrightarrow> P"
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   840
  shows "0 < length y \<Longrightarrow> P"
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   841
  apply (cases y, simp)
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   842
  apply (rule y)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 41413
diff changeset
   843
  apply fastforce
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   844
  done
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   845
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   846
lemma list_exhaust_size_eq0:
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   847
  assumes y: "y = [] \<Longrightarrow> P"
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   848
  shows "length y = 0 \<Longrightarrow> P"
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   849
  apply (cases y)
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   850
   apply (rule y, simp)
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   851
  apply simp
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   852
  done
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   853
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   854
lemma size_Cons_lem_eq:
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   855
  "y = xa # list ==> size y = Suc k ==> size list = k"
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   856
  by auto
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   857
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   858
lemma size_Cons_lem_eq_bin:
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25349
diff changeset
   859
  "y = xa # list ==> size y = number_of (Int.succ k) ==> 
24364
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   860
    size list = number_of k"
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   861
  by (auto simp: pred_def succ_def split add : split_if_asm)
31e359126ab6 reorganize into subsections
huffman
parents: 24350
diff changeset
   862
44939
5930d35c976d removed unused legacy lemma names, some comment cleanup.
kleing
parents: 44890
diff changeset
   863
lemmas ls_splits = prod.split prod.split_asm split_if_asm
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   864
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   865
lemma not_B1_is_B0: "y \<noteq> (1::bit) \<Longrightarrow> y = (0::bit)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   866
  by (cases y) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   867
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   868
lemma B1_ass_B0: 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   869
  assumes y: "y = (0::bit) \<Longrightarrow> y = (1::bit)"
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 37546
diff changeset
   870
  shows "y = (1::bit)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   871
  apply (rule classical)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   872
  apply (drule not_B1_is_B0)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   873
  apply (erule y)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   874
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   875
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   876
-- "simplifications for specific word lengths"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   877
lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   878
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   879
lemmas s2n_ths = n2s_ths [symmetric]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   880
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   881
end