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(*
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ID: $Id$
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Author: Jeremy Dawson and Gerwin Klein, NICTA
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contains theorems to do with shifting, rotating, splitting words
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*)
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theory WordShift imports WordBitwise begin
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section "Bit shifting"
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lemma shiftl1_number [simp] :
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"shiftl1 (number_of w) = number_of (w BIT bit.B0)"
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apply (unfold shiftl1_def word_number_of_def)
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apply (simp add: word_ubin.norm_eq_iff [symmetric] word_ubin.eq_norm)
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apply (subst refl [THEN bintrunc_BIT_I, symmetric])
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apply (subst bintrunc_bintrunc_min)
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apply simp
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done
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lemma shiftl1_0 [simp] : "shiftl1 0 = 0"
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unfolding word_0_no shiftl1_number by auto
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lemmas shiftl1_def_u = shiftl1_def [folded word_number_of_def]
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lemma shiftl1_def_s: "shiftl1 w = number_of (sint w BIT bit.B0)"
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by (rule trans [OF _ shiftl1_number]) simp
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lemma shiftr1_0 [simp] : "shiftr1 0 = 0"
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unfolding shiftr1_def
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by simp (simp add: word_0_wi)
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lemma sshiftr1_0 [simp] : "sshiftr1 0 = 0"
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apply (unfold sshiftr1_def)
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apply simp
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apply (simp add : word_0_wi)
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done
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lemma sshiftr1_n1 [simp] : "sshiftr1 -1 = -1"
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unfolding sshiftr1_def by auto
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lemma shiftl_0 [simp] : "(0::'a::len0 word) << n = 0"
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unfolding shiftl_def by (induct n) auto
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lemma shiftr_0 [simp] : "(0::'a::len0 word) >> n = 0"
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unfolding shiftr_def by (induct n) auto
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lemma sshiftr_0 [simp] : "0 >>> n = 0"
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unfolding sshiftr_def by (induct n) auto
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lemma sshiftr_n1 [simp] : "-1 >>> n = -1"
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unfolding sshiftr_def by (induct n) auto
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lemma nth_shiftl1: "shiftl1 w !! n = (n < size w & n > 0 & w !! (n - 1))"
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apply (unfold shiftl1_def word_test_bit_def)
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apply (simp add: nth_bintr word_ubin.eq_norm word_size)
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apply (cases n)
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apply auto
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done
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lemma nth_shiftl' [rule_format]:
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"ALL n. ((w::'a::len0 word) << m) !! n = (n < size w & n >= m & w !! (n - m))"
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apply (unfold shiftl_def)
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apply (induct "m")
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apply (force elim!: test_bit_size)
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apply (clarsimp simp add : nth_shiftl1 word_size)
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apply arith
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done
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lemmas nth_shiftl = nth_shiftl' [unfolded word_size]
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lemma nth_shiftr1: "shiftr1 w !! n = w !! Suc n"
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apply (unfold shiftr1_def word_test_bit_def)
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apply (simp add: nth_bintr word_ubin.eq_norm)
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apply safe
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apply (drule bin_nth.Suc [THEN iffD2, THEN bin_nth_uint_imp])
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apply simp
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done
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lemma nth_shiftr:
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"\<And>n. ((w::'a::len0 word) >> m) !! n = w !! (n + m)"
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apply (unfold shiftr_def)
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apply (induct "m")
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apply (auto simp add : nth_shiftr1)
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done
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(* see paper page 10, (1), (2), shiftr1_def is of the form of (1),
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where f (ie bin_rest) takes normal arguments to normal results,
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thus we get (2) from (1) *)
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lemma uint_shiftr1: "uint (shiftr1 w) = bin_rest (uint w)"
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apply (unfold shiftr1_def word_ubin.eq_norm bin_rest_trunc_i)
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apply (subst bintr_uint [symmetric, OF order_refl])
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apply (simp only : bintrunc_bintrunc_l)
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apply simp
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done
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lemma nth_sshiftr1:
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"sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)"
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apply (unfold sshiftr1_def word_test_bit_def)
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apply (simp add: nth_bintr word_ubin.eq_norm
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bin_nth.Suc [symmetric] word_size
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del: bin_nth.simps)
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apply (simp add: nth_bintr uint_sint del : bin_nth.simps)
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apply (auto simp add: bin_nth_sint)
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done
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lemma nth_sshiftr [rule_format] :
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"ALL n. sshiftr w m !! n = (n < size w &
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(if n + m >= size w then w !! (size w - 1) else w !! (n + m)))"
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apply (unfold sshiftr_def)
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apply (induct_tac "m")
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apply (simp add: test_bit_bl)
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apply (clarsimp simp add: nth_sshiftr1 word_size)
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apply safe
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apply arith
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apply arith
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apply (erule thin_rl)
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apply (case_tac n)
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apply safe
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apply simp
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apply simp
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apply (erule thin_rl)
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apply (case_tac n)
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apply safe
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apply simp
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apply simp
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apply arith+
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done
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lemma shiftr1_div_2: "uint (shiftr1 w) = uint w div 2"
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apply (unfold shiftr1_def bin_rest_div)
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apply (rule word_uint.Abs_inverse)
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apply (simp add: uints_num pos_imp_zdiv_nonneg_iff)
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apply (rule xtr7)
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prefer 2
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apply (rule zdiv_le_dividend)
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apply auto
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done
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lemma sshiftr1_div_2: "sint (sshiftr1 w) = sint w div 2"
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apply (unfold sshiftr1_def bin_rest_div [symmetric])
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apply (simp add: word_sbin.eq_norm)
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apply (rule trans)
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defer
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apply (subst word_sbin.norm_Rep [symmetric])
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apply (rule refl)
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apply (subst word_sbin.norm_Rep [symmetric])
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apply (unfold One_nat_def)
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apply (rule sbintrunc_rest)
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done
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lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n"
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apply (unfold shiftr_def)
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apply (induct "n")
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apply simp
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apply (simp add: shiftr1_div_2 mult_commute
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zdiv_zmult2_eq [symmetric])
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done
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lemma sshiftr_div_2n: "sint (sshiftr w n) = sint w div 2 ^ n"
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apply (unfold sshiftr_def)
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apply (induct "n")
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apply simp
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apply (simp add: sshiftr1_div_2 mult_commute
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zdiv_zmult2_eq [symmetric])
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done
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subsection "shift functions in terms of lists of bools"
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lemmas bshiftr1_no_bin [simp] =
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bshiftr1_def [where w="number_of ?w", unfolded to_bl_no_bin]
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lemma bshiftr1_bl: "to_bl (bshiftr1 b w) = b # butlast (to_bl w)"
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unfolding bshiftr1_def by (rule word_bl.Abs_inverse) simp
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lemma shiftl1_of_bl: "shiftl1 (of_bl bl) = of_bl (bl @ [False])"
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unfolding uint_bl of_bl_no
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by (simp add: bl_to_bin_aux_append bl_to_bin_def)
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lemmas shiftl1_bl = shiftl1_of_bl
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[where bl = "to_bl (?w :: ?'a :: len0 word)", simplified]
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lemma bl_shiftl1:
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"to_bl (shiftl1 (w :: 'a :: len word)) = tl (to_bl w) @ [False]"
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apply (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons')
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apply (fast intro!: Suc_leI)
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done
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lemma shiftr1_bl: "shiftr1 w = of_bl (butlast (to_bl w))"
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apply (unfold shiftr1_def uint_bl of_bl_def)
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apply (simp add: butlast_rest_bin word_size)
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apply (simp add: bin_rest_trunc [symmetric, unfolded One_nat_def])
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done
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lemma bl_shiftr1:
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"to_bl (shiftr1 (w :: 'a :: len word)) = False # butlast (to_bl w)"
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unfolding shiftr1_bl
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by (simp add : word_rep_drop len_gt_0 [THEN Suc_leI])
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(* relate the two above : TODO - remove the :: len restriction on
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this theorem and others depending on it *)
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lemma shiftl1_rev:
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"shiftl1 (w :: 'a :: len word) = word_reverse (shiftr1 (word_reverse w))"
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apply (unfold word_reverse_def)
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apply (rule word_bl.Rep_inverse' [symmetric])
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apply (simp add: bl_shiftl1 bl_shiftr1 word_bl.Abs_inverse)
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apply (cases "to_bl w")
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apply auto
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done
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lemma shiftl_rev:
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"shiftl (w :: 'a :: len word) n = word_reverse (shiftr (word_reverse w) n)"
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apply (unfold shiftl_def shiftr_def)
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apply (induct "n")
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apply (auto simp add : shiftl1_rev)
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done
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lemmas rev_shiftl =
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shiftl_rev [where w = "word_reverse ?w1", simplified, standard]
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lemmas shiftr_rev = rev_shiftl [THEN word_rev_gal', standard]
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lemmas rev_shiftr = shiftl_rev [THEN word_rev_gal', standard]
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lemma bl_sshiftr1:
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"to_bl (sshiftr1 (w :: 'a :: len word)) = hd (to_bl w) # butlast (to_bl w)"
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apply (unfold sshiftr1_def uint_bl word_size)
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apply (simp add: butlast_rest_bin word_ubin.eq_norm)
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apply (simp add: sint_uint)
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apply (rule nth_equalityI)
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apply clarsimp
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apply clarsimp
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apply (case_tac i)
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apply (simp_all add: hd_conv_nth length_0_conv [symmetric]
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nth_bin_to_bl bin_nth.Suc [symmetric]
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nth_sbintr
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del: bin_nth.Suc)
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apply force
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apply (rule impI)
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apply (rule_tac f = "bin_nth (uint w)" in arg_cong)
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apply simp
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done
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lemma drop_shiftr:
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"drop n (to_bl ((w :: 'a :: len word) >> n)) = take (size w - n) (to_bl w)"
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apply (unfold shiftr_def)
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apply (induct n)
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prefer 2
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apply (simp add: drop_Suc bl_shiftr1 butlast_drop [symmetric])
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apply (rule butlast_take [THEN trans])
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apply (auto simp: word_size)
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done
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lemma drop_sshiftr:
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"drop n (to_bl ((w :: 'a :: len word) >>> n)) = take (size w - n) (to_bl w)"
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apply (unfold sshiftr_def)
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apply (induct n)
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prefer 2
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apply (simp add: drop_Suc bl_sshiftr1 butlast_drop [symmetric])
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apply (rule butlast_take [THEN trans])
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apply (auto simp: word_size)
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done
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lemma take_shiftr [rule_format] :
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"n <= size (w :: 'a :: len word) --> take n (to_bl (w >> n)) =
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replicate n False"
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apply (unfold shiftr_def)
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apply (induct n)
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prefer 2
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apply (simp add: bl_shiftr1)
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apply (rule impI)
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apply (rule take_butlast [THEN trans])
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apply (auto simp: word_size)
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done
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lemma take_sshiftr' [rule_format] :
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"n <= size (w :: 'a :: len word) --> hd (to_bl (w >>> n)) = hd (to_bl w) &
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take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))"
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apply (unfold sshiftr_def)
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apply (induct n)
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prefer 2
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apply (simp add: bl_sshiftr1)
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apply (rule impI)
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apply (rule take_butlast [THEN trans])
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apply (auto simp: word_size)
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done
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lemmas hd_sshiftr = take_sshiftr' [THEN conjunct1, standard]
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lemmas take_sshiftr = take_sshiftr' [THEN conjunct2, standard]
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lemma atd_lem: "take n xs = t ==> drop n xs = d ==> xs = t @ d"
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by (auto intro: append_take_drop_id [symmetric])
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lemmas bl_shiftr = atd_lem [OF take_shiftr drop_shiftr]
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lemmas bl_sshiftr = atd_lem [OF take_sshiftr drop_sshiftr]
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lemma shiftl_of_bl: "of_bl bl << n = of_bl (bl @ replicate n False)"
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unfolding shiftl_def
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by (induct n) (auto simp: shiftl1_of_bl replicate_app_Cons_same)
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lemmas shiftl_bl =
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shiftl_of_bl [where bl = "to_bl (?w :: ?'a :: len0 word)", simplified]
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lemmas shiftl_number [simp] = shiftl_def [where w="number_of ?w"]
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lemma bl_shiftl:
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"to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False"
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by (simp add: shiftl_bl word_rep_drop word_size min_def)
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lemma shiftl_zero_size:
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fixes x :: "'a::len0 word"
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shows "size x <= n ==> x << n = 0"
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apply (unfold word_size)
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apply (rule word_eqI)
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apply (clarsimp simp add: shiftl_bl word_size test_bit_of_bl nth_append)
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done
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(* note - the following results use 'a :: len word < number_ring *)
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lemma shiftl1_2t: "shiftl1 (w :: 'a :: len word) = 2 * w"
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apply (simp add: shiftl1_def_u)
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apply (simp only: double_number_of_BIT [symmetric])
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apply simp
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done
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lemma shiftl1_p: "shiftl1 (w :: 'a :: len word) = w + w"
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apply (simp add: shiftl1_def_u)
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apply (simp only: double_number_of_BIT [symmetric])
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apply simp
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done
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lemma shiftl_t2n: "shiftl (w :: 'a :: len word) n = 2 ^ n * w"
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unfolding shiftl_def
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by (induct n) (auto simp: shiftl1_2t power_Suc)
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lemma shiftr1_bintr [simp]:
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"(shiftr1 (number_of w) :: 'a :: len0 word) =
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number_of (bin_rest (bintrunc (len_of TYPE ('a)) w))"
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unfolding shiftr1_def word_number_of_def
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by (simp add : word_ubin.eq_norm)
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lemma sshiftr1_sbintr [simp] :
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"(sshiftr1 (number_of w) :: 'a :: len word) =
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number_of (bin_rest (sbintrunc (len_of TYPE ('a) - 1) w))"
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unfolding sshiftr1_def word_number_of_def
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by (simp add : word_sbin.eq_norm)
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lemma shiftr_no':
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"w = number_of bin ==>
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(w::'a::len0 word) >> n = number_of ((bin_rest ^ n) (bintrunc (size w) bin))"
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apply clarsimp
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apply (rule word_eqI)
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apply (auto simp: nth_shiftr nth_rest_power_bin nth_bintr word_size)
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done
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lemma sshiftr_no':
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"w = number_of bin ==> w >>> n = number_of ((bin_rest ^ n)
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(sbintrunc (size w - 1) bin))"
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360 |
apply clarsimp
|
|
361 |
apply (rule word_eqI)
|
|
362 |
apply (auto simp: nth_sshiftr nth_rest_power_bin nth_sbintr word_size)
|
|
363 |
apply (subgoal_tac "na + n = len_of TYPE('a) - Suc 0", simp, simp)+
|
|
364 |
done
|
|
365 |
|
|
366 |
lemmas sshiftr_no [simp] =
|
|
367 |
sshiftr_no' [where w = "number_of ?w", OF refl, unfolded word_size]
|
|
368 |
|
|
369 |
lemmas shiftr_no [simp] =
|
|
370 |
shiftr_no' [where w = "number_of ?w", OF refl, unfolded word_size]
|
|
371 |
|
|
372 |
lemma shiftr1_bl_of':
|
|
373 |
"us = shiftr1 (of_bl bl) ==> length bl <= size us ==>
|
|
374 |
us = of_bl (butlast bl)"
|
|
375 |
by (clarsimp simp: shiftr1_def of_bl_def word_size butlast_rest_bl2bin
|
|
376 |
word_ubin.eq_norm trunc_bl2bin)
|
|
377 |
|
|
378 |
lemmas shiftr1_bl_of = refl [THEN shiftr1_bl_of', unfolded word_size]
|
|
379 |
|
|
380 |
lemma shiftr_bl_of' [rule_format]:
|
|
381 |
"us = of_bl bl >> n ==> length bl <= size us -->
|
|
382 |
us = of_bl (take (length bl - n) bl)"
|
|
383 |
apply (unfold shiftr_def)
|
|
384 |
apply hypsubst
|
|
385 |
apply (unfold word_size)
|
|
386 |
apply (induct n)
|
|
387 |
apply clarsimp
|
|
388 |
apply clarsimp
|
|
389 |
apply (subst shiftr1_bl_of)
|
|
390 |
apply simp
|
|
391 |
apply (simp add: butlast_take)
|
|
392 |
done
|
|
393 |
|
|
394 |
lemmas shiftr_bl_of = refl [THEN shiftr_bl_of', unfolded word_size]
|
|
395 |
|
|
396 |
lemmas shiftr_bl = word_bl.Rep' [THEN eq_imp_le, THEN shiftr_bl_of,
|
|
397 |
simplified word_size, simplified, THEN eq_reflection, standard]
|
|
398 |
|
|
399 |
lemma msb_shift': "msb (w::'a::len word) <-> (w >> (size w - 1)) ~= 0"
|
|
400 |
apply (unfold shiftr_bl word_msb_alt)
|
|
401 |
apply (simp add: word_size Suc_le_eq take_Suc)
|
|
402 |
apply (cases "hd (to_bl w)")
|
|
403 |
apply (auto simp: word_1_bl word_0_bl
|
|
404 |
of_bl_rep_False [where n=1 and bs="[]", simplified])
|
|
405 |
done
|
|
406 |
|
|
407 |
lemmas msb_shift = msb_shift' [unfolded word_size]
|
|
408 |
|
|
409 |
lemma align_lem_or [rule_format] :
|
|
410 |
"ALL x m. length x = n + m --> length y = n + m -->
|
|
411 |
drop m x = replicate n False --> take m y = replicate m False -->
|
|
412 |
app2 op | x y = take m x @ drop m y"
|
|
413 |
apply (induct_tac y)
|
|
414 |
apply force
|
|
415 |
apply clarsimp
|
|
416 |
apply (case_tac x, force)
|
|
417 |
apply (case_tac m, auto)
|
|
418 |
apply (drule sym)
|
|
419 |
apply auto
|
|
420 |
apply (induct_tac list, auto)
|
|
421 |
done
|
|
422 |
|
|
423 |
lemma align_lem_and [rule_format] :
|
|
424 |
"ALL x m. length x = n + m --> length y = n + m -->
|
|
425 |
drop m x = replicate n False --> take m y = replicate m False -->
|
|
426 |
app2 op & x y = replicate (n + m) False"
|
|
427 |
apply (induct_tac y)
|
|
428 |
apply force
|
|
429 |
apply clarsimp
|
|
430 |
apply (case_tac x, force)
|
|
431 |
apply (case_tac m, auto)
|
|
432 |
apply (drule sym)
|
|
433 |
apply auto
|
|
434 |
apply (induct_tac list, auto)
|
|
435 |
done
|
|
436 |
|
|
437 |
lemma aligned_bl_add_size':
|
|
438 |
"size x - n = m ==> n <= size x ==> drop m (to_bl x) = replicate n False ==>
|
|
439 |
take m (to_bl y) = replicate m False ==>
|
|
440 |
to_bl (x + y) = take m (to_bl x) @ drop m (to_bl y)"
|
|
441 |
apply (subgoal_tac "x AND y = 0")
|
|
442 |
prefer 2
|
|
443 |
apply (rule word_bl.Rep_eqD)
|
|
444 |
apply (simp add: bl_word_and to_bl_0)
|
|
445 |
apply (rule align_lem_and [THEN trans])
|
|
446 |
apply (simp_all add: word_size)[5]
|
|
447 |
apply (rule_tac f = "%n. replicate n False" in arg_cong)
|
|
448 |
apply simp
|
|
449 |
apply (subst word_plus_and_or [symmetric])
|
|
450 |
apply (simp add : bl_word_or)
|
|
451 |
apply (rule align_lem_or)
|
|
452 |
apply (simp_all add: word_size)
|
|
453 |
done
|
|
454 |
|
|
455 |
lemmas aligned_bl_add_size = refl [THEN aligned_bl_add_size']
|
|
456 |
|
|
457 |
subsection "Mask"
|
|
458 |
|
|
459 |
lemma nth_mask': "m = mask n ==> test_bit m i = (i < n & i < size m)"
|
|
460 |
apply (unfold mask_def test_bit_bl)
|
|
461 |
apply (simp only: word_1_bl [symmetric] shiftl_of_bl)
|
|
462 |
apply (clarsimp simp add: word_size)
|
|
463 |
apply (simp only: of_bl_no mask_lem number_of_succ add_diff_cancel2)
|
|
464 |
apply (fold of_bl_no)
|
|
465 |
apply (simp add: word_1_bl)
|
|
466 |
apply (rule test_bit_of_bl [THEN trans, unfolded test_bit_bl word_size])
|
|
467 |
apply auto
|
|
468 |
done
|
|
469 |
|
|
470 |
lemmas nth_mask [simp] = refl [THEN nth_mask']
|
|
471 |
|
|
472 |
lemma mask_bl: "mask n = of_bl (replicate n True)"
|
|
473 |
by (auto simp add : test_bit_of_bl word_size intro: word_eqI)
|
|
474 |
|
|
475 |
lemma mask_bin: "mask n = number_of (bintrunc n Numeral.Min)"
|
|
476 |
by (auto simp add: nth_bintr word_size intro: word_eqI)
|
|
477 |
|
|
478 |
lemma and_mask_bintr: "w AND mask n = number_of (bintrunc n (uint w))"
|
|
479 |
apply (rule word_eqI)
|
|
480 |
apply (simp add: nth_bintr word_size word_ops_nth_size)
|
|
481 |
apply (auto simp add: test_bit_bin)
|
|
482 |
done
|
|
483 |
|
|
484 |
lemma and_mask_no: "number_of i AND mask n = number_of (bintrunc n i)"
|
|
485 |
by (auto simp add : nth_bintr word_size word_ops_nth_size intro: word_eqI)
|
|
486 |
|
|
487 |
lemmas and_mask_wi = and_mask_no [unfolded word_number_of_def]
|
|
488 |
|
|
489 |
lemma bl_and_mask:
|
|
490 |
"to_bl (w AND mask n :: 'a :: len word) =
|
|
491 |
replicate (len_of TYPE('a) - n) False @
|
|
492 |
drop (len_of TYPE('a) - n) (to_bl w)"
|
|
493 |
apply (rule nth_equalityI)
|
|
494 |
apply simp
|
|
495 |
apply (clarsimp simp add: to_bl_nth word_size)
|
|
496 |
apply (simp add: word_size word_ops_nth_size)
|
|
497 |
apply (auto simp add: word_size test_bit_bl nth_append nth_rev)
|
|
498 |
done
|
|
499 |
|
|
500 |
lemmas and_mask_mod_2p =
|
|
501 |
and_mask_bintr [unfolded word_number_of_alt no_bintr_alt]
|
|
502 |
|
|
503 |
lemma and_mask_lt_2p: "uint (w AND mask n) < 2 ^ n"
|
|
504 |
apply (simp add : and_mask_bintr no_bintr_alt)
|
|
505 |
apply (rule xtr8)
|
|
506 |
prefer 2
|
|
507 |
apply (rule pos_mod_bound)
|
|
508 |
apply auto
|
|
509 |
done
|
|
510 |
|
|
511 |
lemmas eq_mod_iff = trans [symmetric, OF int_mod_lem eq_sym_conv]
|
|
512 |
|
|
513 |
lemma mask_eq_iff: "(w AND mask n) = w <-> uint w < 2 ^ n"
|
|
514 |
apply (simp add: and_mask_bintr word_number_of_def)
|
|
515 |
apply (simp add: word_ubin.inverse_norm)
|
|
516 |
apply (simp add: eq_mod_iff bintrunc_mod2p min_def)
|
|
517 |
apply (fast intro!: lt2p_lem)
|
|
518 |
done
|
|
519 |
|
|
520 |
lemma and_mask_dvd: "2 ^ n dvd uint w = (w AND mask n = 0)"
|
|
521 |
apply (simp add: zdvd_iff_zmod_eq_0 and_mask_mod_2p)
|
|
522 |
apply (simp add: word_uint.norm_eq_iff [symmetric] word_of_int_homs)
|
|
523 |
apply (subst word_uint.norm_Rep [symmetric])
|
|
524 |
apply (simp only: bintrunc_bintrunc_min bintrunc_mod2p [symmetric] min_def)
|
|
525 |
apply auto
|
|
526 |
done
|
|
527 |
|
|
528 |
lemma and_mask_dvd_nat: "2 ^ n dvd unat w = (w AND mask n = 0)"
|
|
529 |
apply (unfold unat_def)
|
|
530 |
apply (rule trans [OF _ and_mask_dvd])
|
|
531 |
apply (unfold dvd_def)
|
|
532 |
apply auto
|
|
533 |
apply (drule uint_ge_0 [THEN nat_int.Abs_inverse' [simplified], symmetric])
|
|
534 |
apply (simp add : int_mult int_power)
|
|
535 |
apply (simp add : nat_mult_distrib nat_power_eq)
|
|
536 |
done
|
|
537 |
|
|
538 |
lemma word_2p_lem:
|
|
539 |
"n < size w ==> w < 2 ^ n = (uint (w :: 'a :: len word) < 2 ^ n)"
|
|
540 |
apply (unfold word_size word_less_alt word_number_of_alt)
|
|
541 |
apply (clarsimp simp add: word_of_int_power_hom word_uint.eq_norm
|
|
542 |
int_mod_eq'
|
|
543 |
simp del: word_of_int_bin)
|
|
544 |
done
|
|
545 |
|
|
546 |
lemma less_mask_eq: "x < 2 ^ n ==> x AND mask n = (x :: 'a :: len word)"
|
|
547 |
apply (unfold word_less_alt word_number_of_alt)
|
|
548 |
apply (clarsimp simp add: and_mask_mod_2p word_of_int_power_hom
|
|
549 |
word_uint.eq_norm
|
|
550 |
simp del: word_of_int_bin)
|
|
551 |
apply (drule xtr8 [rotated])
|
|
552 |
apply (rule int_mod_le)
|
|
553 |
apply (auto simp add : mod_pos_pos_trivial)
|
|
554 |
done
|
|
555 |
|
|
556 |
lemmas mask_eq_iff_w2p =
|
|
557 |
trans [OF mask_eq_iff word_2p_lem [symmetric], standard]
|
|
558 |
|
|
559 |
lemmas and_mask_less' =
|
|
560 |
iffD2 [OF word_2p_lem and_mask_lt_2p, simplified word_size, standard]
|
|
561 |
|
|
562 |
lemma and_mask_less_size: "n < size x ==> x AND mask n < 2^n"
|
|
563 |
unfolding word_size by (erule and_mask_less')
|
|
564 |
|
|
565 |
lemma word_mod_2p_is_mask':
|
|
566 |
"c = 2 ^ n ==> c > 0 ==> x mod c = (x :: 'a :: len word) AND mask n"
|
|
567 |
by (clarsimp simp add: word_mod_def uint_2p and_mask_mod_2p)
|
|
568 |
|
|
569 |
lemmas word_mod_2p_is_mask = refl [THEN word_mod_2p_is_mask']
|
|
570 |
|
|
571 |
lemma mask_eqs:
|
|
572 |
"(a AND mask n) + b AND mask n = a + b AND mask n"
|
|
573 |
"a + (b AND mask n) AND mask n = a + b AND mask n"
|
|
574 |
"(a AND mask n) - b AND mask n = a - b AND mask n"
|
|
575 |
"a - (b AND mask n) AND mask n = a - b AND mask n"
|
|
576 |
"a * (b AND mask n) AND mask n = a * b AND mask n"
|
|
577 |
"(b AND mask n) * a AND mask n = b * a AND mask n"
|
|
578 |
"(a AND mask n) + (b AND mask n) AND mask n = a + b AND mask n"
|
|
579 |
"(a AND mask n) - (b AND mask n) AND mask n = a - b AND mask n"
|
|
580 |
"(a AND mask n) * (b AND mask n) AND mask n = a * b AND mask n"
|
|
581 |
"- (a AND mask n) AND mask n = - a AND mask n"
|
|
582 |
"word_succ (a AND mask n) AND mask n = word_succ a AND mask n"
|
|
583 |
"word_pred (a AND mask n) AND mask n = word_pred a AND mask n"
|
|
584 |
using word_of_int_Ex [where x=a] word_of_int_Ex [where x=b]
|
|
585 |
by (auto simp: and_mask_wi bintr_ariths bintr_arith1s new_word_of_int_homs)
|
|
586 |
|
|
587 |
lemma mask_power_eq:
|
|
588 |
"(x AND mask n) ^ k AND mask n = x ^ k AND mask n"
|
|
589 |
using word_of_int_Ex [where x=x]
|
|
590 |
by (clarsimp simp: and_mask_wi word_of_int_power_hom bintr_ariths)
|
|
591 |
|
|
592 |
|
|
593 |
subsection "Revcast"
|
|
594 |
|
|
595 |
lemmas revcast_def' = revcast_def [simplified]
|
|
596 |
lemmas revcast_def'' = revcast_def' [simplified word_size]
|
|
597 |
lemmas revcast_no_def [simp] =
|
|
598 |
revcast_def' [where w="number_of ?w", unfolded word_size]
|
|
599 |
|
|
600 |
lemma to_bl_revcast:
|
|
601 |
"to_bl (revcast w :: 'a :: len0 word) =
|
|
602 |
takefill False (len_of TYPE ('a)) (to_bl w)"
|
|
603 |
apply (unfold revcast_def' word_size)
|
|
604 |
apply (rule word_bl.Abs_inverse)
|
|
605 |
apply simp
|
|
606 |
done
|
|
607 |
|
|
608 |
lemma revcast_rev_ucast':
|
|
609 |
"cs = [rc, uc] ==> rc = revcast (word_reverse w) ==> uc = ucast w ==>
|
|
610 |
rc = word_reverse uc"
|
|
611 |
apply (unfold ucast_def revcast_def' Let_def word_reverse_def)
|
|
612 |
apply (clarsimp simp add : to_bl_of_bin takefill_bintrunc)
|
|
613 |
apply (simp add : word_bl.Abs_inverse word_size)
|
|
614 |
done
|
|
615 |
|
|
616 |
lemmas revcast_rev_ucast = revcast_rev_ucast' [OF refl refl refl]
|
|
617 |
|
|
618 |
lemmas revcast_ucast = revcast_rev_ucast
|
|
619 |
[where w = "word_reverse ?w1", simplified word_rev_rev, standard]
|
|
620 |
|
|
621 |
lemmas ucast_revcast = revcast_rev_ucast [THEN word_rev_gal', standard]
|
|
622 |
lemmas ucast_rev_revcast = revcast_ucast [THEN word_rev_gal', standard]
|
|
623 |
|
|
624 |
|
|
625 |
-- "linking revcast and cast via shift"
|
|
626 |
|
|
627 |
lemmas wsst_TYs = source_size target_size word_size
|
|
628 |
|
|
629 |
lemma revcast_down_uu':
|
|
630 |
"rc = revcast ==> source_size rc = target_size rc + n ==>
|
|
631 |
rc (w :: 'a :: len word) = ucast (w >> n)"
|
|
632 |
apply (simp add: revcast_def')
|
|
633 |
apply (rule word_bl.Rep_inverse')
|
|
634 |
apply (rule trans, rule ucast_down_drop)
|
|
635 |
prefer 2
|
|
636 |
apply (rule trans, rule drop_shiftr)
|
|
637 |
apply (auto simp: takefill_alt wsst_TYs)
|
|
638 |
done
|
|
639 |
|
|
640 |
lemma revcast_down_us':
|
|
641 |
"rc = revcast ==> source_size rc = target_size rc + n ==>
|
|
642 |
rc (w :: 'a :: len word) = ucast (w >>> n)"
|
|
643 |
apply (simp add: revcast_def')
|
|
644 |
apply (rule word_bl.Rep_inverse')
|
|
645 |
apply (rule trans, rule ucast_down_drop)
|
|
646 |
prefer 2
|
|
647 |
apply (rule trans, rule drop_sshiftr)
|
|
648 |
apply (auto simp: takefill_alt wsst_TYs)
|
|
649 |
done
|
|
650 |
|
|
651 |
lemma revcast_down_su':
|
|
652 |
"rc = revcast ==> source_size rc = target_size rc + n ==>
|
|
653 |
rc (w :: 'a :: len word) = scast (w >> n)"
|
|
654 |
apply (simp add: revcast_def')
|
|
655 |
apply (rule word_bl.Rep_inverse')
|
|
656 |
apply (rule trans, rule scast_down_drop)
|
|
657 |
prefer 2
|
|
658 |
apply (rule trans, rule drop_shiftr)
|
|
659 |
apply (auto simp: takefill_alt wsst_TYs)
|
|
660 |
done
|
|
661 |
|
|
662 |
lemma revcast_down_ss':
|
|
663 |
"rc = revcast ==> source_size rc = target_size rc + n ==>
|
|
664 |
rc (w :: 'a :: len word) = scast (w >>> n)"
|
|
665 |
apply (simp add: revcast_def')
|
|
666 |
apply (rule word_bl.Rep_inverse')
|
|
667 |
apply (rule trans, rule scast_down_drop)
|
|
668 |
prefer 2
|
|
669 |
apply (rule trans, rule drop_sshiftr)
|
|
670 |
apply (auto simp: takefill_alt wsst_TYs)
|
|
671 |
done
|
|
672 |
|
|
673 |
lemmas revcast_down_uu = refl [THEN revcast_down_uu']
|
|
674 |
lemmas revcast_down_us = refl [THEN revcast_down_us']
|
|
675 |
lemmas revcast_down_su = refl [THEN revcast_down_su']
|
|
676 |
lemmas revcast_down_ss = refl [THEN revcast_down_ss']
|
|
677 |
|
|
678 |
lemma cast_down_rev:
|
|
679 |
"uc = ucast ==> source_size uc = target_size uc + n ==>
|
|
680 |
uc w = revcast ((w :: 'a :: len word) << n)"
|
|
681 |
apply (unfold shiftl_rev)
|
|
682 |
apply clarify
|
|
683 |
apply (simp add: revcast_rev_ucast)
|
|
684 |
apply (rule word_rev_gal')
|
|
685 |
apply (rule trans [OF _ revcast_rev_ucast])
|
|
686 |
apply (rule revcast_down_uu [symmetric])
|
|
687 |
apply (auto simp add: wsst_TYs)
|
|
688 |
done
|
|
689 |
|
|
690 |
lemma revcast_up':
|
|
691 |
"rc = revcast ==> source_size rc + n = target_size rc ==>
|
|
692 |
rc w = (ucast w :: 'a :: len word) << n"
|
|
693 |
apply (simp add: revcast_def')
|
|
694 |
apply (rule word_bl.Rep_inverse')
|
|
695 |
apply (simp add: takefill_alt)
|
|
696 |
apply (rule bl_shiftl [THEN trans])
|
|
697 |
apply (subst ucast_up_app)
|
|
698 |
apply (auto simp add: wsst_TYs)
|
|
699 |
apply (drule sym)
|
|
700 |
apply (simp add: min_def)
|
|
701 |
done
|
|
702 |
|
|
703 |
lemmas revcast_up = refl [THEN revcast_up']
|
|
704 |
|
|
705 |
lemmas rc1 = revcast_up [THEN
|
|
706 |
revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
|
|
707 |
lemmas rc2 = revcast_down_uu [THEN
|
|
708 |
revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]]
|
|
709 |
|
|
710 |
lemmas ucast_up =
|
|
711 |
rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]]
|
|
712 |
lemmas ucast_down =
|
|
713 |
rc2 [simplified rev_shiftr revcast_ucast [symmetric]]
|
|
714 |
|
|
715 |
|
|
716 |
subsection "Slices"
|
|
717 |
|
|
718 |
lemmas slice1_no_bin [simp] =
|
|
719 |
slice1_def [where w="number_of ?w", unfolded to_bl_no_bin]
|
|
720 |
|
|
721 |
lemmas slice_no_bin [simp] =
|
|
722 |
trans [OF slice_def [THEN meta_eq_to_obj_eq]
|
|
723 |
slice1_no_bin [THEN meta_eq_to_obj_eq],
|
|
724 |
unfolded word_size, standard]
|
|
725 |
|
|
726 |
lemma slice1_0 [simp] : "slice1 n 0 = 0"
|
|
727 |
unfolding slice1_def by (simp add : to_bl_0)
|
|
728 |
|
|
729 |
lemma slice_0 [simp] : "slice n 0 = 0"
|
|
730 |
unfolding slice_def by auto
|
|
731 |
|
|
732 |
lemma slice_take': "slice n w = of_bl (take (size w - n) (to_bl w))"
|
|
733 |
unfolding slice_def' slice1_def
|
|
734 |
by (simp add : takefill_alt word_size)
|
|
735 |
|
|
736 |
lemmas slice_take = slice_take' [unfolded word_size]
|
|
737 |
|
|
738 |
-- "shiftr to a word of the same size is just slice,
|
|
739 |
slice is just shiftr then ucast"
|
|
740 |
lemmas shiftr_slice = trans
|
|
741 |
[OF shiftr_bl [THEN meta_eq_to_obj_eq] slice_take [symmetric], standard]
|
|
742 |
|
|
743 |
lemma slice_shiftr: "slice n w = ucast (w >> n)"
|
|
744 |
apply (unfold slice_take shiftr_bl)
|
|
745 |
apply (rule ucast_of_bl_up [symmetric])
|
|
746 |
apply (simp add: word_size)
|
|
747 |
done
|
|
748 |
|
|
749 |
lemma nth_slice:
|
|
750 |
"(slice n w :: 'a :: len0 word) !! m =
|
|
751 |
(w !! (m + n) & m < len_of TYPE ('a))"
|
|
752 |
unfolding slice_shiftr
|
|
753 |
by (simp add : nth_ucast nth_shiftr)
|
|
754 |
|
|
755 |
lemma slice1_down_alt':
|
|
756 |
"sl = slice1 n w ==> fs = size sl ==> fs + k = n ==>
|
|
757 |
to_bl sl = takefill False fs (drop k (to_bl w))"
|
|
758 |
unfolding slice1_def word_size of_bl_def uint_bl
|
|
759 |
by (clarsimp simp: word_ubin.eq_norm bl_bin_bl_rep_drop drop_takefill)
|
|
760 |
|
|
761 |
lemma slice1_up_alt':
|
|
762 |
"sl = slice1 n w ==> fs = size sl ==> fs = n + k ==>
|
|
763 |
to_bl sl = takefill False fs (replicate k False @ (to_bl w))"
|
|
764 |
apply (unfold slice1_def word_size of_bl_def uint_bl)
|
|
765 |
apply (clarsimp simp: word_ubin.eq_norm bl_bin_bl_rep_drop
|
|
766 |
takefill_append [symmetric])
|
|
767 |
apply (rule_tac f = "%k. takefill False (len_of TYPE('a))
|
|
768 |
(replicate k False @ bin_to_bl (len_of TYPE('b)) (uint w))" in arg_cong)
|
|
769 |
apply arith
|
|
770 |
done
|
|
771 |
|
|
772 |
lemmas sd1 = slice1_down_alt' [OF refl refl, unfolded word_size]
|
|
773 |
lemmas su1 = slice1_up_alt' [OF refl refl, unfolded word_size]
|
|
774 |
lemmas slice1_down_alt = le_add_diff_inverse [THEN sd1]
|
|
775 |
lemmas slice1_up_alts =
|
|
776 |
le_add_diff_inverse [symmetric, THEN su1]
|
|
777 |
le_add_diff_inverse2 [symmetric, THEN su1]
|
|
778 |
|
|
779 |
lemma ucast_slice1: "ucast w = slice1 (size w) w"
|
|
780 |
unfolding slice1_def ucast_bl
|
|
781 |
by (simp add : takefill_same' word_size)
|
|
782 |
|
|
783 |
lemma ucast_slice: "ucast w = slice 0 w"
|
|
784 |
unfolding slice_def by (simp add : ucast_slice1)
|
|
785 |
|
|
786 |
lemmas slice_id = trans [OF ucast_slice [symmetric] ucast_id]
|
|
787 |
|
|
788 |
lemma revcast_slice1':
|
|
789 |
"rc = revcast w ==> slice1 (size rc) w = rc"
|
|
790 |
unfolding slice1_def revcast_def' by (simp add : word_size)
|
|
791 |
|
|
792 |
lemmas revcast_slice1 = refl [THEN revcast_slice1']
|
|
793 |
|
|
794 |
lemma slice1_tf_tf':
|
|
795 |
"to_bl (slice1 n w :: 'a :: len0 word) =
|
|
796 |
rev (takefill False (len_of TYPE('a)) (rev (takefill False n (to_bl w))))"
|
|
797 |
unfolding slice1_def by (rule word_rev_tf)
|
|
798 |
|
|
799 |
lemmas slice1_tf_tf = slice1_tf_tf'
|
|
800 |
[THEN word_bl.Rep_inverse', symmetric, standard]
|
|
801 |
|
|
802 |
lemma rev_slice1:
|
|
803 |
"n + k = len_of TYPE('a) + len_of TYPE('b) \<Longrightarrow>
|
|
804 |
slice1 n (word_reverse w :: 'b :: len0 word) =
|
|
805 |
word_reverse (slice1 k w :: 'a :: len0 word)"
|
|
806 |
apply (unfold word_reverse_def slice1_tf_tf)
|
|
807 |
apply (rule word_bl.Rep_inverse')
|
|
808 |
apply (rule rev_swap [THEN iffD1])
|
|
809 |
apply (rule trans [symmetric])
|
|
810 |
apply (rule tf_rev)
|
|
811 |
apply (simp add: word_bl.Abs_inverse)
|
|
812 |
apply (simp add: word_bl.Abs_inverse)
|
|
813 |
done
|
|
814 |
|
|
815 |
lemma rev_slice':
|
|
816 |
"res = slice n (word_reverse w) ==> n + k + size res = size w ==>
|
|
817 |
res = word_reverse (slice k w)"
|
|
818 |
apply (unfold slice_def word_size)
|
|
819 |
apply clarify
|
|
820 |
apply (rule rev_slice1)
|
|
821 |
apply arith
|
|
822 |
done
|
|
823 |
|
|
824 |
lemmas rev_slice = refl [THEN rev_slice', unfolded word_size]
|
|
825 |
|
|
826 |
lemmas sym_notr =
|
|
827 |
not_iff [THEN iffD2, THEN not_sym, THEN not_iff [THEN iffD1]]
|
|
828 |
|
|
829 |
-- {* problem posed by TPHOLs referee:
|
|
830 |
criterion for overflow of addition of signed integers *}
|
|
831 |
|
|
832 |
lemma sofl_test:
|
|
833 |
"(sint (x :: 'a :: len word) + sint y = sint (x + y)) =
|
|
834 |
((((x+y) XOR x) AND ((x+y) XOR y)) >> (size x - 1) = 0)"
|
|
835 |
apply (unfold word_size)
|
|
836 |
apply (cases "len_of TYPE('a)", simp)
|
|
837 |
apply (subst msb_shift [THEN sym_notr])
|
|
838 |
apply (simp add: word_ops_msb)
|
|
839 |
apply (simp add: word_msb_sint)
|
|
840 |
apply safe
|
|
841 |
apply simp_all
|
|
842 |
apply (unfold sint_word_ariths)
|
|
843 |
apply (unfold word_sbin.set_iff_norm [symmetric] sints_num)
|
|
844 |
apply safe
|
|
845 |
apply (insert sint_range' [where x=x])
|
|
846 |
apply (insert sint_range' [where x=y])
|
|
847 |
defer
|
|
848 |
apply (simp (no_asm), arith)
|
|
849 |
apply (simp (no_asm), arith)
|
|
850 |
defer
|
|
851 |
defer
|
|
852 |
apply (simp (no_asm), arith)
|
|
853 |
apply (simp (no_asm), arith)
|
|
854 |
apply (rule notI [THEN notnotD],
|
|
855 |
drule leI not_leE,
|
|
856 |
drule sbintrunc_inc sbintrunc_dec,
|
|
857 |
simp)+
|
|
858 |
done
|
|
859 |
|
|
860 |
|
|
861 |
section "Split and cat"
|
|
862 |
|
|
863 |
lemmas word_split_bin' = word_split_def [THEN meta_eq_to_obj_eq, standard]
|
|
864 |
lemmas word_cat_bin' = word_cat_def [THEN meta_eq_to_obj_eq, standard]
|
|
865 |
|
|
866 |
lemma word_rsplit_no:
|
|
867 |
"(word_rsplit (number_of bin :: 'b :: len0 word) :: 'a word list) =
|
|
868 |
map number_of (bin_rsplit (len_of TYPE('a :: len))
|
|
869 |
(len_of TYPE('b), bintrunc (len_of TYPE('b)) bin))"
|
|
870 |
apply (unfold word_rsplit_def word_no_wi)
|
|
871 |
apply (simp add: word_ubin.eq_norm)
|
|
872 |
done
|
|
873 |
|
|
874 |
lemmas word_rsplit_no_cl [simp] = word_rsplit_no
|
|
875 |
[unfolded bin_rsplitl_def bin_rsplit_l [symmetric]]
|
|
876 |
|
|
877 |
lemma test_bit_cat:
|
|
878 |
"wc = word_cat a b ==> wc !! n = (n < size wc &
|
|
879 |
(if n < size b then b !! n else a !! (n - size b)))"
|
|
880 |
apply (unfold word_cat_bin' test_bit_bin)
|
|
881 |
apply (auto simp add : word_ubin.eq_norm nth_bintr bin_nth_cat word_size)
|
|
882 |
apply (erule bin_nth_uint_imp)
|
|
883 |
done
|
|
884 |
|
|
885 |
lemma word_cat_bl: "word_cat a b = of_bl (to_bl a @ to_bl b)"
|
|
886 |
apply (unfold of_bl_def to_bl_def word_cat_bin')
|
|
887 |
apply (simp add: bl_to_bin_app_cat)
|
|
888 |
done
|
|
889 |
|
|
890 |
lemma of_bl_append:
|
|
891 |
"(of_bl (xs @ ys) :: 'a :: len word) = of_bl xs * 2^(length ys) + of_bl ys"
|
|
892 |
apply (unfold of_bl_def)
|
|
893 |
apply (simp add: bl_to_bin_app_cat bin_cat_num)
|
|
894 |
apply (simp add: word_of_int_power_hom [symmetric] new_word_of_int_hom_syms)
|
|
895 |
done
|
|
896 |
|
|
897 |
lemma of_bl_False [simp]:
|
|
898 |
"of_bl (False#xs) = of_bl xs"
|
|
899 |
by (rule word_eqI)
|
|
900 |
(auto simp add: test_bit_of_bl nth_append)
|
|
901 |
|
|
902 |
lemma of_bl_True:
|
|
903 |
"(of_bl (True#xs)::'a::len word) = 2^length xs + of_bl xs"
|
|
904 |
by (subst of_bl_append [where xs="[True]", simplified])
|
|
905 |
(simp add: word_1_bl)
|
|
906 |
|
|
907 |
lemma of_bl_Cons:
|
|
908 |
"of_bl (x#xs) = of_bool x * 2^length xs + of_bl xs"
|
|
909 |
by (cases x) (simp_all add: of_bl_True)
|
|
910 |
|
|
911 |
lemma split_uint_lem: "bin_split n (uint (w :: 'a :: len0 word)) = (a, b) ==>
|
|
912 |
a = bintrunc (len_of TYPE('a) - n) a & b = bintrunc (len_of TYPE('a)) b"
|
|
913 |
apply (frule word_ubin.norm_Rep [THEN ssubst])
|
|
914 |
apply (drule bin_split_trunc1)
|
|
915 |
apply (drule sym [THEN trans])
|
|
916 |
apply assumption
|
|
917 |
apply safe
|
|
918 |
done
|
|
919 |
|
|
920 |
lemma word_split_bl':
|
|
921 |
"std = size c - size b ==> (word_split c = (a, b)) ==>
|
|
922 |
(a = of_bl (take std (to_bl c)) & b = of_bl (drop std (to_bl c)))"
|
|
923 |
apply (unfold word_split_bin')
|
|
924 |
apply safe
|
|
925 |
defer
|
|
926 |
apply (clarsimp split: prod.splits)
|
|
927 |
apply (drule word_ubin.norm_Rep [THEN ssubst])
|
|
928 |
apply (drule split_bintrunc)
|
|
929 |
apply (simp add : of_bl_def bl2bin_drop word_size
|
|
930 |
word_ubin.norm_eq_iff [symmetric] min_def del : word_ubin.norm_Rep)
|
|
931 |
apply (clarsimp split: prod.splits)
|
|
932 |
apply (frule split_uint_lem [THEN conjunct1])
|
|
933 |
apply (unfold word_size)
|
|
934 |
apply (cases "len_of TYPE('a) >= len_of TYPE('b)")
|
|
935 |
defer
|
|
936 |
apply (simp add: word_0_bl word_0_wi_Pls)
|
|
937 |
apply (simp add : of_bl_def to_bl_def)
|
|
938 |
apply (subst bin_split_take1 [symmetric])
|
|
939 |
prefer 2
|
|
940 |
apply assumption
|
|
941 |
apply simp
|
|
942 |
apply (erule thin_rl)
|
|
943 |
apply (erule arg_cong [THEN trans])
|
|
944 |
apply (simp add : word_ubin.norm_eq_iff [symmetric])
|
|
945 |
done
|
|
946 |
|
|
947 |
lemma word_split_bl: "std = size c - size b ==>
|
|
948 |
(a = of_bl (take std (to_bl c)) & b = of_bl (drop std (to_bl c))) <->
|
|
949 |
word_split c = (a, b)"
|
|
950 |
apply (rule iffI)
|
|
951 |
defer
|
|
952 |
apply (erule (1) word_split_bl')
|
|
953 |
apply (case_tac "word_split c")
|
|
954 |
apply (auto simp add : word_size)
|
|
955 |
apply (frule word_split_bl' [rotated])
|
|
956 |
apply (auto simp add : word_size)
|
|
957 |
done
|
|
958 |
|
|
959 |
lemma word_split_bl_eq:
|
|
960 |
"(word_split (c::'a::len word) :: ('c :: len0 word * 'd :: len0 word)) =
|
|
961 |
(of_bl (take (len_of TYPE('a::len) - len_of TYPE('d::len0)) (to_bl c)),
|
|
962 |
of_bl (drop (len_of TYPE('a) - len_of TYPE('d)) (to_bl c)))"
|
|
963 |
apply (rule word_split_bl [THEN iffD1])
|
|
964 |
apply (unfold word_size)
|
|
965 |
apply (rule refl conjI)+
|
|
966 |
done
|
|
967 |
|
|
968 |
-- "keep quantifiers for use in simplification"
|
|
969 |
lemma test_bit_split':
|
|
970 |
"word_split c = (a, b) --> (ALL n m. b !! n = (n < size b & c !! n) &
|
|
971 |
a !! m = (m < size a & c !! (m + size b)))"
|
|
972 |
apply (unfold word_split_bin' test_bit_bin)
|
|
973 |
apply (clarify)
|
|
974 |
apply (clarsimp simp: word_ubin.eq_norm nth_bintr word_size split: prod.splits)
|
|
975 |
apply (drule bin_nth_split)
|
|
976 |
apply safe
|
|
977 |
apply (simp_all add: add_commute)
|
|
978 |
apply (erule bin_nth_uint_imp)+
|
|
979 |
done
|
|
980 |
|
|
981 |
lemmas test_bit_split =
|
|
982 |
test_bit_split' [THEN mp, simplified all_simps, standard]
|
|
983 |
|
|
984 |
lemma test_bit_split_eq: "word_split c = (a, b) <->
|
|
985 |
((ALL n::nat. b !! n = (n < size b & c !! n)) &
|
|
986 |
(ALL m::nat. a !! m = (m < size a & c !! (m + size b))))"
|
|
987 |
apply (rule_tac iffI)
|
|
988 |
apply (rule_tac conjI)
|
|
989 |
apply (erule test_bit_split [THEN conjunct1])
|
|
990 |
apply (erule test_bit_split [THEN conjunct2])
|
|
991 |
apply (case_tac "word_split c")
|
|
992 |
apply (frule test_bit_split)
|
|
993 |
apply (erule trans)
|
|
994 |
apply (fastsimp intro ! : word_eqI simp add : word_size)
|
|
995 |
done
|
|
996 |
|
|
997 |
-- {* this odd result is analogous to ucast\_id,
|
|
998 |
result to the length given by the result type *}
|
|
999 |
|
|
1000 |
lemma word_cat_id: "word_cat a b = b"
|
|
1001 |
unfolding word_cat_bin' by (simp add: word_ubin.inverse_norm)
|
|
1002 |
|
|
1003 |
-- "limited hom result"
|
|
1004 |
lemma word_cat_hom:
|
|
1005 |
"len_of TYPE('a::len0) <= len_of TYPE('b::len0) + len_of TYPE ('c::len0)
|
|
1006 |
==>
|
|
1007 |
(word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) =
|
|
1008 |
word_of_int (bin_cat w (size b) (uint b))"
|
|
1009 |
apply (unfold word_cat_def word_size)
|
|
1010 |
apply (clarsimp simp add : word_ubin.norm_eq_iff [symmetric]
|
|
1011 |
word_ubin.eq_norm bintr_cat min_def)
|
|
1012 |
apply arith
|
|
1013 |
done
|
|
1014 |
|
|
1015 |
lemma word_cat_split_alt:
|
|
1016 |
"size w <= size u + size v ==> word_split w = (u, v) ==> word_cat u v = w"
|
|
1017 |
apply (rule word_eqI)
|
|
1018 |
apply (drule test_bit_split)
|
|
1019 |
apply (clarsimp simp add : test_bit_cat word_size)
|
|
1020 |
apply safe
|
|
1021 |
apply arith
|
|
1022 |
done
|
|
1023 |
|
|
1024 |
lemmas word_cat_split_size =
|
|
1025 |
sym [THEN [2] word_cat_split_alt [symmetric], standard]
|
|
1026 |
|
|
1027 |
|
|
1028 |
subsection "Split and slice"
|
|
1029 |
|
|
1030 |
lemma split_slices:
|
|
1031 |
"word_split w = (u, v) ==> u = slice (size v) w & v = slice 0 w"
|
|
1032 |
apply (drule test_bit_split)
|
|
1033 |
apply (rule conjI)
|
|
1034 |
apply (rule word_eqI, clarsimp simp: nth_slice word_size)+
|
|
1035 |
done
|
|
1036 |
|
|
1037 |
lemma slice_cat1':
|
|
1038 |
"wc = word_cat a b ==> size wc >= size a + size b ==> slice (size b) wc = a"
|
|
1039 |
apply safe
|
|
1040 |
apply (rule word_eqI)
|
|
1041 |
apply (simp add: nth_slice test_bit_cat word_size)
|
|
1042 |
done
|
|
1043 |
|
|
1044 |
lemmas slice_cat1 = refl [THEN slice_cat1']
|
|
1045 |
lemmas slice_cat2 = trans [OF slice_id word_cat_id]
|
|
1046 |
|
|
1047 |
lemma cat_slices:
|
|
1048 |
"a = slice n c ==> b = slice 0 c ==> n = size b ==> \
|
|
1049 |
size a + size b >= size c ==> word_cat a b = c"
|
|
1050 |
apply safe
|
|
1051 |
apply (rule word_eqI)
|
|
1052 |
apply (simp add: nth_slice test_bit_cat word_size)
|
|
1053 |
apply safe
|
|
1054 |
apply arith
|
|
1055 |
done
|
|
1056 |
|
|
1057 |
lemma word_split_cat_alt:
|
|
1058 |
"w = word_cat u v ==> size u + size v <= size w ==> word_split w = (u, v)"
|
|
1059 |
apply (case_tac "word_split ?w")
|
|
1060 |
apply (rule trans, assumption)
|
|
1061 |
apply (drule test_bit_split)
|
|
1062 |
apply safe
|
|
1063 |
apply (rule word_eqI, clarsimp simp: test_bit_cat word_size)+
|
|
1064 |
done
|
|
1065 |
|
|
1066 |
lemmas word_cat_bl_no_bin [simp] =
|
|
1067 |
word_cat_bl [where a="number_of ?a"
|
|
1068 |
and b="number_of ?b",
|
|
1069 |
unfolded to_bl_no_bin]
|
|
1070 |
|
|
1071 |
lemmas word_split_bl_no_bin [simp] =
|
|
1072 |
word_split_bl_eq [where c="number_of ?c", unfolded to_bl_no_bin]
|
|
1073 |
|
|
1074 |
-- {* this odd result arises from the fact that the statement of the
|
|
1075 |
result implies that the decoded words are of the same type,
|
|
1076 |
and therefore of the same length, as the original word *}
|
|
1077 |
|
|
1078 |
lemma word_rsplit_same: "word_rsplit w = [w]"
|
|
1079 |
unfolding word_rsplit_def by (simp add : bin_rsplit_all)
|
|
1080 |
|
|
1081 |
lemma word_rsplit_empty_iff_size:
|
|
1082 |
"(word_rsplit w = []) = (size w = 0)"
|
|
1083 |
unfolding word_rsplit_def bin_rsplit_def word_size
|
|
1084 |
by (simp add: bin_rsplit_aux_simp_alt Let_def split: split_split)
|
|
1085 |
|
|
1086 |
lemma test_bit_rsplit:
|
|
1087 |
"sw = word_rsplit w ==> m < size (hd sw :: 'a :: len word) ==>
|
|
1088 |
k < length sw ==> (rev sw ! k) !! m = (w !! (k * size (hd sw) + m))"
|
|
1089 |
apply (unfold word_rsplit_def word_test_bit_def)
|
|
1090 |
apply (rule trans)
|
|
1091 |
apply (rule_tac f = "%x. bin_nth x m" in arg_cong)
|
|
1092 |
apply (rule nth_map [symmetric])
|
|
1093 |
apply simp
|
|
1094 |
apply (rule bin_nth_rsplit)
|
|
1095 |
apply simp_all
|
|
1096 |
apply (simp add : word_size rev_map map_compose [symmetric])
|
|
1097 |
apply (rule trans)
|
|
1098 |
defer
|
|
1099 |
apply (rule map_ident [THEN fun_cong])
|
|
1100 |
apply (rule refl [THEN map_cong])
|
|
1101 |
apply (simp add : word_ubin.eq_norm)
|
|
1102 |
apply (erule bin_rsplit_size_sign [OF len_gt_0 refl])
|
|
1103 |
done
|
|
1104 |
|
|
1105 |
lemma word_rcat_bl: "word_rcat wl == of_bl (concat (map to_bl wl))"
|
|
1106 |
unfolding word_rcat_def to_bl_def' of_bl_def
|
|
1107 |
by (clarsimp simp add : bin_rcat_bl map_compose)
|
|
1108 |
|
|
1109 |
lemma size_rcat_lem':
|
|
1110 |
"size (concat (map to_bl wl)) = length wl * size (hd wl)"
|
|
1111 |
unfolding word_size by (induct wl) auto
|
|
1112 |
|
|
1113 |
lemmas size_rcat_lem = size_rcat_lem' [unfolded word_size]
|
|
1114 |
|
|
1115 |
lemmas td_gal_lt_len = len_gt_0 [THEN td_gal_lt, standard]
|
|
1116 |
|
|
1117 |
lemma nth_rcat_lem' [rule_format] :
|
|
1118 |
"sw = size (hd wl :: 'a :: len word) ==> (ALL n. n < size wl * sw -->
|
|
1119 |
rev (concat (map to_bl wl)) ! n =
|
|
1120 |
rev (to_bl (rev wl ! (n div sw))) ! (n mod sw))"
|
|
1121 |
apply (unfold word_size)
|
|
1122 |
apply (induct "wl")
|
|
1123 |
apply clarsimp
|
|
1124 |
apply (clarsimp simp add : nth_append size_rcat_lem)
|
|
1125 |
apply (simp (no_asm_use) only: mult_Suc [symmetric]
|
|
1126 |
td_gal_lt_len less_Suc_eq_le mod_div_equality')
|
|
1127 |
apply clarsimp
|
|
1128 |
done
|
|
1129 |
|
|
1130 |
lemmas nth_rcat_lem = refl [THEN nth_rcat_lem', unfolded word_size]
|
|
1131 |
|
|
1132 |
lemma test_bit_rcat:
|
|
1133 |
"sw = size (hd wl :: 'a :: len word) ==> rc = word_rcat wl ==> rc !! n =
|
|
1134 |
(n < size rc & n div sw < size wl & (rev wl) ! (n div sw) !! (n mod sw))"
|
|
1135 |
apply (unfold word_rcat_bl word_size)
|
|
1136 |
apply (clarsimp simp add :
|
|
1137 |
test_bit_of_bl size_rcat_lem word_size td_gal_lt_len)
|
|
1138 |
apply safe
|
|
1139 |
apply (auto simp add :
|
|
1140 |
test_bit_bl word_size td_gal_lt_len [THEN iffD2, THEN nth_rcat_lem])
|
|
1141 |
done
|
|
1142 |
|
|
1143 |
lemma foldl_eq_foldr [rule_format] :
|
|
1144 |
"ALL x. foldl op + x xs = foldr op + (x # xs) (0 :: 'a :: comm_monoid_add)"
|
|
1145 |
by (induct xs) (auto simp add : add_assoc)
|
|
1146 |
|
|
1147 |
lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong]
|
|
1148 |
|
|
1149 |
lemmas test_bit_rsplit_alt =
|
|
1150 |
trans [OF nth_rev_alt [THEN test_bit_cong]
|
|
1151 |
test_bit_rsplit [OF refl asm_rl diff_Suc_less]]
|
|
1152 |
|
|
1153 |
-- "lazy way of expressing that u and v, and su and sv, have same types"
|
|
1154 |
lemma word_rsplit_len_indep':
|
|
1155 |
"[u,v] = p ==> [su,sv] = q ==> word_rsplit u = su ==>
|
|
1156 |
word_rsplit v = sv ==> length su = length sv"
|
|
1157 |
apply (unfold word_rsplit_def)
|
|
1158 |
apply (auto simp add : bin_rsplit_len_indep)
|
|
1159 |
done
|
|
1160 |
|
|
1161 |
lemmas word_rsplit_len_indep = word_rsplit_len_indep' [OF refl refl refl refl]
|
|
1162 |
|
|
1163 |
lemma length_word_rsplit_size:
|
|
1164 |
"n = len_of TYPE ('a :: len) ==>
|
|
1165 |
(length (word_rsplit w :: 'a word list) <= m) = (size w <= m * n)"
|
|
1166 |
apply (unfold word_rsplit_def word_size)
|
|
1167 |
apply (clarsimp simp add : bin_rsplit_len_le)
|
|
1168 |
done
|
|
1169 |
|
|
1170 |
lemmas length_word_rsplit_lt_size =
|
|
1171 |
length_word_rsplit_size [unfolded Not_eq_iff linorder_not_less [symmetric]]
|
|
1172 |
|
|
1173 |
lemma length_word_rsplit_exp_size:
|
|
1174 |
"n = len_of TYPE ('a :: len) ==>
|
|
1175 |
length (word_rsplit w :: 'a word list) = (size w + n - 1) div n"
|
|
1176 |
unfolding word_rsplit_def by (clarsimp simp add : word_size bin_rsplit_len)
|
|
1177 |
|
|
1178 |
lemma length_word_rsplit_even_size:
|
|
1179 |
"n = len_of TYPE ('a :: len) ==> size w = m * n ==>
|
|
1180 |
length (word_rsplit w :: 'a word list) = m"
|
|
1181 |
by (clarsimp simp add : length_word_rsplit_exp_size given_quot_alt)
|
|
1182 |
|
|
1183 |
lemmas length_word_rsplit_exp_size' = refl [THEN length_word_rsplit_exp_size]
|
|
1184 |
|
|
1185 |
(* alternative proof of word_rcat_rsplit *)
|
|
1186 |
lemmas tdle = iffD2 [OF split_div_lemma refl, THEN conjunct1]
|
|
1187 |
lemmas dtle = xtr4 [OF tdle mult_commute]
|
|
1188 |
|
|
1189 |
lemma word_rcat_rsplit: "word_rcat (word_rsplit w) = w"
|
|
1190 |
apply (rule word_eqI)
|
|
1191 |
apply (clarsimp simp add : test_bit_rcat word_size)
|
|
1192 |
apply (subst refl [THEN test_bit_rsplit])
|
|
1193 |
apply (simp_all add: word_size
|
|
1194 |
refl [THEN length_word_rsplit_size [simplified le_def, simplified]])
|
|
1195 |
apply safe
|
|
1196 |
apply (erule xtr7, rule len_gt_0 [THEN dtle])+
|
|
1197 |
done
|
|
1198 |
|
|
1199 |
lemma size_word_rsplit_rcat_size':
|
|
1200 |
"word_rcat (ws :: 'a :: len word list) = frcw ==>
|
|
1201 |
size frcw = length ws * len_of TYPE ('a) ==>
|
|
1202 |
size (hd [word_rsplit frcw, ws]) = size ws"
|
|
1203 |
apply (clarsimp simp add : word_size length_word_rsplit_exp_size')
|
|
1204 |
apply (fast intro: given_quot_alt)
|
|
1205 |
done
|
|
1206 |
|
|
1207 |
lemmas size_word_rsplit_rcat_size =
|
|
1208 |
size_word_rsplit_rcat_size' [simplified]
|
|
1209 |
|
|
1210 |
lemma msrevs:
|
|
1211 |
fixes n::nat
|
|
1212 |
shows "0 < n \<Longrightarrow> (k * n + m) div n = m div n + k"
|
|
1213 |
and "(k * n + m) mod n = m mod n"
|
|
1214 |
by (auto simp: add_commute)
|
|
1215 |
|
|
1216 |
lemma word_rsplit_rcat_size':
|
|
1217 |
"word_rcat (ws :: 'a :: len word list) = frcw ==>
|
|
1218 |
size frcw = length ws * len_of TYPE ('a) ==> word_rsplit frcw = ws"
|
|
1219 |
apply (frule size_word_rsplit_rcat_size, assumption)
|
|
1220 |
apply (clarsimp simp add : word_size)
|
|
1221 |
apply (rule nth_equalityI, assumption)
|
|
1222 |
apply clarsimp
|
|
1223 |
apply (rule word_eqI)
|
|
1224 |
apply (rule trans)
|
|
1225 |
apply (rule test_bit_rsplit_alt)
|
|
1226 |
apply (clarsimp simp: word_size)+
|
|
1227 |
apply (rule trans)
|
|
1228 |
apply (rule test_bit_rcat [OF refl refl])
|
|
1229 |
apply (simp add : word_size msrevs)
|
|
1230 |
apply (subst nth_rev)
|
|
1231 |
apply arith
|
|
1232 |
apply (simp add : le0 [THEN [2] xtr7, THEN diff_Suc_less])
|
|
1233 |
apply safe
|
|
1234 |
apply (simp add : diff_mult_distrib)
|
|
1235 |
apply (rule mpl_lem)
|
|
1236 |
apply (cases "size ws")
|
|
1237 |
apply simp_all
|
|
1238 |
done
|
|
1239 |
|
|
1240 |
lemmas word_rsplit_rcat_size = refl [THEN word_rsplit_rcat_size']
|
|
1241 |
|
|
1242 |
|
|
1243 |
section "Rotation"
|
|
1244 |
|
|
1245 |
lemmas rotater_0' [simp] = rotater_def [where n = "0", simplified]
|
|
1246 |
|
|
1247 |
lemmas word_rot_defs = word_roti_def word_rotr_def word_rotl_def
|
|
1248 |
|
|
1249 |
lemma rotate_eq_mod:
|
|
1250 |
"m mod length xs = n mod length xs ==> rotate m xs = rotate n xs"
|
|
1251 |
apply (rule box_equals)
|
|
1252 |
defer
|
|
1253 |
apply (rule rotate_conv_mod [symmetric])+
|
|
1254 |
apply simp
|
|
1255 |
done
|
|
1256 |
|
|
1257 |
lemmas rotate_eqs [standard] =
|
|
1258 |
trans [OF rotate0 [THEN fun_cong] id_apply]
|
|
1259 |
rotate_rotate [symmetric]
|
|
1260 |
rotate_id
|
|
1261 |
rotate_conv_mod
|
|
1262 |
rotate_eq_mod
|
|
1263 |
|
|
1264 |
|
|
1265 |
subsection "Rotation of list to right"
|
|
1266 |
|
|
1267 |
lemma rotate1_rl': "rotater1 (l @ [a]) = a # l"
|
|
1268 |
unfolding rotater1_def by (cases l) auto
|
|
1269 |
|
|
1270 |
lemma rotate1_rl [simp] : "rotater1 (rotate1 l) = l"
|
|
1271 |
apply (unfold rotater1_def)
|
|
1272 |
apply (cases "l")
|
|
1273 |
apply (case_tac [2] "list")
|
|
1274 |
apply auto
|
|
1275 |
done
|
|
1276 |
|
|
1277 |
lemma rotate1_lr [simp] : "rotate1 (rotater1 l) = l"
|
|
1278 |
unfolding rotater1_def by (cases l) auto
|
|
1279 |
|
|
1280 |
lemma rotater1_rev': "rotater1 (rev xs) = rev (rotate1 xs)"
|
|
1281 |
apply (cases "xs")
|
|
1282 |
apply (simp add : rotater1_def)
|
|
1283 |
apply (simp add : rotate1_rl')
|
|
1284 |
done
|
|
1285 |
|
|
1286 |
lemma rotater_rev': "rotater n (rev xs) = rev (rotate n xs)"
|
|
1287 |
unfolding rotater_def by (induct n) (auto intro: rotater1_rev')
|
|
1288 |
|
|
1289 |
lemmas rotater_rev = rotater_rev' [where xs = "rev ?ys", simplified]
|
|
1290 |
|
|
1291 |
lemma rotater_drop_take:
|
|
1292 |
"rotater n xs =
|
|
1293 |
drop (length xs - n mod length xs) xs @
|
|
1294 |
take (length xs - n mod length xs) xs"
|
|
1295 |
by (clarsimp simp add : rotater_rev rotate_drop_take rev_take rev_drop)
|
|
1296 |
|
|
1297 |
lemma rotater_Suc [simp] :
|
|
1298 |
"rotater (Suc n) xs = rotater1 (rotater n xs)"
|
|
1299 |
unfolding rotater_def by auto
|
|
1300 |
|
|
1301 |
lemma rotate_inv_plus [rule_format] :
|
|
1302 |
"ALL k. k = m + n --> rotater k (rotate n xs) = rotater m xs &
|
|
1303 |
rotate k (rotater n xs) = rotate m xs &
|
|
1304 |
rotater n (rotate k xs) = rotate m xs &
|
|
1305 |
rotate n (rotater k xs) = rotater m xs"
|
|
1306 |
unfolding rotater_def rotate_def
|
|
1307 |
by (induct n) (auto intro: funpow_swap1 [THEN trans])
|
|
1308 |
|
|
1309 |
lemmas rotate_inv_rel = le_add_diff_inverse2 [symmetric, THEN rotate_inv_plus]
|
|
1310 |
|
|
1311 |
lemmas rotate_inv_eq = order_refl [THEN rotate_inv_rel, simplified]
|
|
1312 |
|
|
1313 |
lemmas rotate_lr [simp] = rotate_inv_eq [THEN conjunct1, standard]
|
|
1314 |
lemmas rotate_rl [simp] =
|
|
1315 |
rotate_inv_eq [THEN conjunct2, THEN conjunct1, standard]
|
|
1316 |
|
|
1317 |
lemma rotate_gal: "(rotater n xs = ys) = (rotate n ys = xs)"
|
|
1318 |
by auto
|
|
1319 |
|
|
1320 |
lemma rotate_gal': "(ys = rotater n xs) = (xs = rotate n ys)"
|
|
1321 |
by auto
|
|
1322 |
|
|
1323 |
lemma length_rotater [simp]:
|
|
1324 |
"length (rotater n xs) = length xs"
|
|
1325 |
by (simp add : rotater_rev)
|
|
1326 |
|
|
1327 |
lemmas rrs0 = rotate_eqs [THEN restrict_to_left,
|
|
1328 |
simplified rotate_gal [symmetric] rotate_gal' [symmetric], standard]
|
|
1329 |
lemmas rrs1 = rrs0 [THEN refl [THEN rev_iffD1]]
|
|
1330 |
lemmas rotater_eqs = rrs1 [simplified length_rotater, standard]
|
|
1331 |
lemmas rotater_0 = rotater_eqs (1)
|
|
1332 |
lemmas rotater_add = rotater_eqs (2)
|
|
1333 |
|
|
1334 |
|
|
1335 |
subsection "map, app2, commuting with rotate(r)"
|
|
1336 |
|
|
1337 |
lemma last_map: "xs ~= [] ==> last (map f xs) = f (last xs)"
|
|
1338 |
by (induct xs) auto
|
|
1339 |
|
|
1340 |
lemma butlast_map:
|
|
1341 |
"xs ~= [] ==> butlast (map f xs) = map f (butlast xs)"
|
|
1342 |
by (induct xs) auto
|
|
1343 |
|
|
1344 |
lemma rotater1_map: "rotater1 (map f xs) = map f (rotater1 xs)"
|
|
1345 |
unfolding rotater1_def
|
|
1346 |
by (cases xs) (auto simp add: last_map butlast_map)
|
|
1347 |
|
|
1348 |
lemma rotater_map:
|
|
1349 |
"rotater n (map f xs) = map f (rotater n xs)"
|
|
1350 |
unfolding rotater_def
|
|
1351 |
by (induct n) (auto simp add : rotater1_map)
|
|
1352 |
|
|
1353 |
lemma but_last_zip [rule_format] :
|
|
1354 |
"ALL ys. length xs = length ys --> xs ~= [] -->
|
|
1355 |
last (zip xs ys) = (last xs, last ys) &
|
|
1356 |
butlast (zip xs ys) = zip (butlast xs) (butlast ys)"
|
|
1357 |
apply (induct "xs")
|
|
1358 |
apply auto
|
|
1359 |
apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+
|
|
1360 |
done
|
|
1361 |
|
|
1362 |
lemma but_last_app2 [rule_format] :
|
|
1363 |
"ALL ys. length xs = length ys --> xs ~= [] -->
|
|
1364 |
last (app2 f xs ys) = f (last xs) (last ys) &
|
|
1365 |
butlast (app2 f xs ys) = app2 f (butlast xs) (butlast ys)"
|
|
1366 |
apply (induct "xs")
|
|
1367 |
apply auto
|
|
1368 |
apply (unfold app2_def)
|
|
1369 |
apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+
|
|
1370 |
done
|
|
1371 |
|
|
1372 |
lemma rotater1_zip:
|
|
1373 |
"length xs = length ys ==>
|
|
1374 |
rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)"
|
|
1375 |
apply (unfold rotater1_def)
|
|
1376 |
apply (cases "xs")
|
|
1377 |
apply auto
|
|
1378 |
apply ((case_tac ys, auto simp: neq_Nil_conv but_last_zip)[1])+
|
|
1379 |
done
|
|
1380 |
|
|
1381 |
lemma rotater1_app2:
|
|
1382 |
"length xs = length ys ==>
|
|
1383 |
rotater1 (app2 f xs ys) = app2 f (rotater1 xs) (rotater1 ys)"
|
|
1384 |
unfolding app2_def by (simp add: rotater1_map rotater1_zip)
|
|
1385 |
|
|
1386 |
lemmas lrth =
|
|
1387 |
box_equals [OF asm_rl length_rotater [symmetric]
|
|
1388 |
length_rotater [symmetric],
|
|
1389 |
THEN rotater1_app2]
|
|
1390 |
|
|
1391 |
lemma rotater_app2:
|
|
1392 |
"length xs = length ys ==>
|
|
1393 |
rotater n (app2 f xs ys) = app2 f (rotater n xs) (rotater n ys)"
|
|
1394 |
by (induct n) (auto intro!: lrth)
|
|
1395 |
|
|
1396 |
lemma rotate1_app2:
|
|
1397 |
"length xs = length ys ==>
|
|
1398 |
rotate1 (app2 f xs ys) = app2 f (rotate1 xs) (rotate1 ys)"
|
|
1399 |
apply (unfold app2_def)
|
|
1400 |
apply (cases xs)
|
|
1401 |
apply (cases ys, auto simp add : rotate1_def)+
|
|
1402 |
done
|
|
1403 |
|
|
1404 |
lemmas lth = box_equals [OF asm_rl length_rotate [symmetric]
|
|
1405 |
length_rotate [symmetric], THEN rotate1_app2]
|
|
1406 |
|
|
1407 |
lemma rotate_app2:
|
|
1408 |
"length xs = length ys ==>
|
|
1409 |
rotate n (app2 f xs ys) = app2 f (rotate n xs) (rotate n ys)"
|
|
1410 |
by (induct n) (auto intro!: lth)
|
|
1411 |
|
|
1412 |
|
|
1413 |
-- "corresponding equalities for word rotation"
|
|
1414 |
|
|
1415 |
lemma to_bl_rotl:
|
|
1416 |
"to_bl (word_rotl n w) = rotate n (to_bl w)"
|
|
1417 |
by (simp add: word_bl.Abs_inverse' word_rotl_def)
|
|
1418 |
|
|
1419 |
lemmas blrs0 = rotate_eqs [THEN to_bl_rotl [THEN trans]]
|
|
1420 |
|
|
1421 |
lemmas word_rotl_eqs =
|
|
1422 |
blrs0 [simplified word_bl.Rep' word_bl.Rep_inject to_bl_rotl [symmetric]]
|
|
1423 |
|
|
1424 |
lemma to_bl_rotr:
|
|
1425 |
"to_bl (word_rotr n w) = rotater n (to_bl w)"
|
|
1426 |
by (simp add: word_bl.Abs_inverse' word_rotr_def)
|
|
1427 |
|
|
1428 |
lemmas brrs0 = rotater_eqs [THEN to_bl_rotr [THEN trans]]
|
|
1429 |
|
|
1430 |
lemmas word_rotr_eqs =
|
|
1431 |
brrs0 [simplified word_bl.Rep' word_bl.Rep_inject to_bl_rotr [symmetric]]
|
|
1432 |
|
|
1433 |
declare word_rotr_eqs (1) [simp]
|
|
1434 |
declare word_rotl_eqs (1) [simp]
|
|
1435 |
|
|
1436 |
lemma
|
|
1437 |
word_rot_rl [simp]:
|
|
1438 |
"word_rotl k (word_rotr k v) = v" and
|
|
1439 |
word_rot_lr [simp]:
|
|
1440 |
"word_rotr k (word_rotl k v) = v"
|
|
1441 |
by (auto simp add: to_bl_rotr to_bl_rotl word_bl.Rep_inject [symmetric])
|
|
1442 |
|
|
1443 |
lemma
|
|
1444 |
word_rot_gal:
|
|
1445 |
"(word_rotr n v = w) = (word_rotl n w = v)" and
|
|
1446 |
word_rot_gal':
|
|
1447 |
"(w = word_rotr n v) = (v = word_rotl n w)"
|
|
1448 |
by (auto simp: to_bl_rotr to_bl_rotl word_bl.Rep_inject [symmetric]
|
|
1449 |
dest: sym)
|
|
1450 |
|
|
1451 |
lemma word_rotr_rev:
|
|
1452 |
"word_rotr n w = word_reverse (word_rotl n (word_reverse w))"
|
|
1453 |
by (simp add: word_bl.Rep_inject [symmetric] to_bl_word_rev
|
|
1454 |
to_bl_rotr to_bl_rotl rotater_rev)
|
|
1455 |
|
|
1456 |
lemma word_roti_0 [simp]: "word_roti 0 w = w"
|
|
1457 |
by (unfold word_rot_defs) auto
|
|
1458 |
|
|
1459 |
lemmas abl_cong = arg_cong [where f = "of_bl"]
|
|
1460 |
|
|
1461 |
lemma word_roti_add:
|
|
1462 |
"word_roti (m + n) w = word_roti m (word_roti n w)"
|
|
1463 |
proof -
|
|
1464 |
have rotater_eq_lem:
|
|
1465 |
"\<And>m n xs. m = n ==> rotater m xs = rotater n xs"
|
|
1466 |
by auto
|
|
1467 |
|
|
1468 |
have rotate_eq_lem:
|
|
1469 |
"\<And>m n xs. m = n ==> rotate m xs = rotate n xs"
|
|
1470 |
by auto
|
|
1471 |
|
|
1472 |
note rpts [symmetric, standard] =
|
|
1473 |
rotate_inv_plus [THEN conjunct1]
|
|
1474 |
rotate_inv_plus [THEN conjunct2, THEN conjunct1]
|
|
1475 |
rotate_inv_plus [THEN conjunct2, THEN conjunct2, THEN conjunct1]
|
|
1476 |
rotate_inv_plus [THEN conjunct2, THEN conjunct2, THEN conjunct2]
|
|
1477 |
|
|
1478 |
note rrp = trans [symmetric, OF rotate_rotate rotate_eq_lem]
|
|
1479 |
note rrrp = trans [symmetric, OF rotater_add [symmetric] rotater_eq_lem]
|
|
1480 |
|
|
1481 |
show ?thesis
|
|
1482 |
apply (unfold word_rot_defs)
|
|
1483 |
apply (simp only: split: split_if)
|
|
1484 |
apply (safe intro!: abl_cong)
|
|
1485 |
apply (simp_all only: to_bl_rotl [THEN word_bl.Rep_inverse']
|
|
1486 |
to_bl_rotl
|
|
1487 |
to_bl_rotr [THEN word_bl.Rep_inverse']
|
|
1488 |
to_bl_rotr)
|
|
1489 |
apply (rule rrp rrrp rpts,
|
|
1490 |
simp add: nat_add_distrib [symmetric]
|
|
1491 |
nat_diff_distrib [symmetric])+
|
|
1492 |
done
|
|
1493 |
qed
|
|
1494 |
|
|
1495 |
lemma word_roti_conv_mod': "word_roti n w = word_roti (n mod int (size w)) w"
|
|
1496 |
apply (unfold word_rot_defs)
|
|
1497 |
apply (cut_tac y="size w" in gt_or_eq_0)
|
|
1498 |
apply (erule disjE)
|
|
1499 |
apply simp_all
|
|
1500 |
apply (safe intro!: abl_cong)
|
|
1501 |
apply (rule rotater_eqs)
|
|
1502 |
apply (simp add: word_size nat_mod_distrib)
|
|
1503 |
apply (simp add: rotater_add [symmetric] rotate_gal [symmetric])
|
|
1504 |
apply (rule rotater_eqs)
|
|
1505 |
apply (simp add: word_size nat_mod_distrib)
|
|
1506 |
apply (rule int_eq_0_conv [THEN iffD1])
|
|
1507 |
apply (simp only: zmod_int zadd_int [symmetric])
|
|
1508 |
apply (simp add: rdmods)
|
|
1509 |
done
|
|
1510 |
|
|
1511 |
lemmas word_roti_conv_mod = word_roti_conv_mod' [unfolded word_size]
|
|
1512 |
|
|
1513 |
|
|
1514 |
subsection "Word rotation commutes with bit-wise operations"
|
|
1515 |
|
|
1516 |
(* using locale to not pollute lemma namespace *)
|
|
1517 |
locale word_rotate
|
|
1518 |
|
|
1519 |
context word_rotate
|
|
1520 |
begin
|
|
1521 |
|
|
1522 |
lemmas word_rot_defs' = to_bl_rotl to_bl_rotr
|
|
1523 |
|
|
1524 |
lemmas blwl_syms [symmetric] = bl_word_not bl_word_and bl_word_or bl_word_xor
|
|
1525 |
|
|
1526 |
lemmas lbl_lbl = trans [OF word_bl.Rep' word_bl.Rep' [symmetric]]
|
|
1527 |
|
|
1528 |
lemmas ths_app2 [OF lbl_lbl] = rotate_app2 rotater_app2
|
|
1529 |
|
|
1530 |
lemmas ths_map [where xs = "to_bl v"] = rotate_map rotater_map
|
|
1531 |
|
|
1532 |
lemmas th1s [simplified word_rot_defs' [symmetric]] = ths_app2 ths_map
|
|
1533 |
|
|
1534 |
lemma word_rot_logs:
|
|
1535 |
"word_rotl n (NOT v) = NOT word_rotl n v"
|
|
1536 |
"word_rotr n (NOT v) = NOT word_rotr n v"
|
|
1537 |
"word_rotl n (x AND y) = word_rotl n x AND word_rotl n y"
|
|
1538 |
"word_rotr n (x AND y) = word_rotr n x AND word_rotr n y"
|
|
1539 |
"word_rotl n (x OR y) = word_rotl n x OR word_rotl n y"
|
|
1540 |
"word_rotr n (x OR y) = word_rotr n x OR word_rotr n y"
|
|
1541 |
"word_rotl n (x XOR y) = word_rotl n x XOR word_rotl n y"
|
|
1542 |
"word_rotr n (x XOR y) = word_rotr n x XOR word_rotr n y"
|
|
1543 |
by (rule word_bl.Rep_eqD,
|
|
1544 |
rule word_rot_defs' [THEN trans],
|
|
1545 |
simp only: blwl_syms [symmetric],
|
|
1546 |
rule th1s [THEN trans],
|
|
1547 |
rule refl)+
|
|
1548 |
end
|
|
1549 |
|
|
1550 |
lemmas word_rot_logs = word_rotate.word_rot_logs
|
|
1551 |
|
|
1552 |
lemmas bl_word_rotl_dt = trans [OF to_bl_rotl rotate_drop_take,
|
|
1553 |
simplified word_bl.Rep', standard]
|
|
1554 |
|
|
1555 |
lemmas bl_word_rotr_dt = trans [OF to_bl_rotr rotater_drop_take,
|
|
1556 |
simplified word_bl.Rep', standard]
|
|
1557 |
|
|
1558 |
lemma bl_word_roti_dt':
|
|
1559 |
"n = nat ((- i) mod int (size (w :: 'a :: len word))) ==>
|
|
1560 |
to_bl (word_roti i w) = drop n (to_bl w) @ take n (to_bl w)"
|
|
1561 |
apply (unfold word_roti_def)
|
|
1562 |
apply (simp add: bl_word_rotl_dt bl_word_rotr_dt word_size)
|
|
1563 |
apply safe
|
|
1564 |
apply (simp add: zmod_zminus1_eq_if)
|
|
1565 |
apply safe
|
|
1566 |
apply (simp add: nat_mult_distrib)
|
|
1567 |
apply (simp add: nat_diff_distrib [OF pos_mod_sign pos_mod_conj
|
|
1568 |
[THEN conjunct2, THEN order_less_imp_le]]
|
|
1569 |
nat_mod_distrib)
|
|
1570 |
apply (simp add: nat_mod_distrib)
|
|
1571 |
done
|
|
1572 |
|
|
1573 |
lemmas bl_word_roti_dt = bl_word_roti_dt' [unfolded word_size]
|
|
1574 |
|
|
1575 |
lemmas word_rotl_dt = bl_word_rotl_dt
|
|
1576 |
[THEN word_bl.Rep_inverse' [symmetric], standard]
|
|
1577 |
lemmas word_rotr_dt = bl_word_rotr_dt
|
|
1578 |
[THEN word_bl.Rep_inverse' [symmetric], standard]
|
|
1579 |
lemmas word_roti_dt = bl_word_roti_dt
|
|
1580 |
[THEN word_bl.Rep_inverse' [symmetric], standard]
|
|
1581 |
|
|
1582 |
lemma word_rotx_0 [simp] : "word_rotr i 0 = 0 & word_rotl i 0 = 0"
|
|
1583 |
by (simp add : word_rotr_dt word_rotl_dt to_bl_0 replicate_add [symmetric])
|
|
1584 |
|
|
1585 |
lemma word_roti_0' [simp] : "word_roti n 0 = 0"
|
|
1586 |
unfolding word_roti_def by auto
|
|
1587 |
|
|
1588 |
lemmas word_rotr_dt_no_bin' [simp] =
|
|
1589 |
word_rotr_dt [where w="number_of ?w", unfolded to_bl_no_bin]
|
|
1590 |
|
|
1591 |
lemmas word_rotl_dt_no_bin' [simp] =
|
|
1592 |
word_rotl_dt [where w="number_of ?w", unfolded to_bl_no_bin]
|
|
1593 |
|
|
1594 |
declare word_roti_def [simp]
|
|
1595 |
|
|
1596 |
end
|
|
1597 |
|