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1 theory ComputeNumeral |
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2 imports ComputeHOL Float |
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3 begin |
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4 |
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5 (* normalization of bit strings *) |
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6 lemmas bitnorm = Pls_0_eq Min_1_eq |
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7 |
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8 (* neg for bit strings *) |
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9 lemma neg1: "neg Numeral.Pls = False" by (simp add: Numeral.Pls_def) |
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10 lemma neg2: "neg Numeral.Min = True" apply (subst Numeral.Min_def) by auto |
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11 lemma neg3: "neg (x BIT Numeral.B0) = neg x" apply (simp add: neg_def) apply (subst Bit_def) by auto |
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12 lemma neg4: "neg (x BIT Numeral.B1) = neg x" apply (simp add: neg_def) apply (subst Bit_def) by auto |
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13 lemmas bitneg = neg1 neg2 neg3 neg4 |
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14 |
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15 (* iszero for bit strings *) |
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16 lemma iszero1: "iszero Numeral.Pls = True" by (simp add: Numeral.Pls_def iszero_def) |
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17 lemma iszero2: "iszero Numeral.Min = False" apply (subst Numeral.Min_def) apply (subst iszero_def) by simp |
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18 lemma iszero3: "iszero (x BIT Numeral.B0) = iszero x" apply (subst Numeral.Bit_def) apply (subst iszero_def)+ by auto |
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19 lemma iszero4: "iszero (x BIT Numeral.B1) = False" apply (subst Numeral.Bit_def) apply (subst iszero_def)+ apply simp by arith |
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20 lemmas bitiszero = iszero1 iszero2 iszero3 iszero4 |
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21 |
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22 (* lezero for bit strings *) |
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23 constdefs |
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24 "lezero x == (x \<le> 0)" |
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25 lemma lezero1: "lezero Numeral.Pls = True" unfolding Numeral.Pls_def lezero_def by auto |
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26 lemma lezero2: "lezero Numeral.Min = True" unfolding Numeral.Min_def lezero_def by auto |
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27 lemma lezero3: "lezero (x BIT Numeral.B0) = lezero x" unfolding Numeral.Bit_def lezero_def by auto |
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28 lemma lezero4: "lezero (x BIT Numeral.B1) = neg x" unfolding Numeral.Bit_def lezero_def neg_def by auto |
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29 lemmas bitlezero = lezero1 lezero2 lezero3 lezero4 |
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30 |
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31 (* equality for bit strings *) |
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32 lemma biteq1: "(Numeral.Pls = Numeral.Pls) = True" by auto |
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33 lemma biteq2: "(Numeral.Min = Numeral.Min) = True" by auto |
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34 lemma biteq3: "(Numeral.Pls = Numeral.Min) = False" unfolding Pls_def Min_def by auto |
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35 lemma biteq4: "(Numeral.Min = Numeral.Pls) = False" unfolding Pls_def Min_def by auto |
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36 lemma biteq5: "(x BIT Numeral.B0 = y BIT Numeral.B0) = (x = y)" unfolding Bit_def by auto |
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37 lemma biteq6: "(x BIT Numeral.B1 = y BIT Numeral.B1) = (x = y)" unfolding Bit_def by auto |
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38 lemma biteq7: "(x BIT Numeral.B0 = y BIT Numeral.B1) = False" unfolding Bit_def by (simp, arith) |
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39 lemma biteq8: "(x BIT Numeral.B1 = y BIT Numeral.B0) = False" unfolding Bit_def by (simp, arith) |
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40 lemma biteq9: "(Numeral.Pls = x BIT Numeral.B0) = (Numeral.Pls = x)" unfolding Bit_def Pls_def by auto |
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41 lemma biteq10: "(Numeral.Pls = x BIT Numeral.B1) = False" unfolding Bit_def Pls_def by (simp, arith) |
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42 lemma biteq11: "(Numeral.Min = x BIT Numeral.B0) = False" unfolding Bit_def Min_def by (simp, arith) |
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43 lemma biteq12: "(Numeral.Min = x BIT Numeral.B1) = (Numeral.Min = x)" unfolding Bit_def Min_def by auto |
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44 lemma biteq13: "(x BIT Numeral.B0 = Numeral.Pls) = (x = Numeral.Pls)" unfolding Bit_def Pls_def by auto |
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45 lemma biteq14: "(x BIT Numeral.B1 = Numeral.Pls) = False" unfolding Bit_def Pls_def by (simp, arith) |
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46 lemma biteq15: "(x BIT Numeral.B0 = Numeral.Min) = False" unfolding Bit_def Pls_def Min_def by (simp, arith) |
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47 lemma biteq16: "(x BIT Numeral.B1 = Numeral.Min) = (x = Numeral.Min)" unfolding Bit_def Min_def by (simp, arith) |
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48 lemmas biteq = biteq1 biteq2 biteq3 biteq4 biteq5 biteq6 biteq7 biteq8 biteq9 biteq10 biteq11 biteq12 biteq13 biteq14 biteq15 biteq16 |
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49 |
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50 (* x < y for bit strings *) |
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51 lemma bitless1: "(Numeral.Pls < Numeral.Min) = False" unfolding Pls_def Min_def by auto |
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52 lemma bitless2: "(Numeral.Pls < Numeral.Pls) = False" by auto |
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53 lemma bitless3: "(Numeral.Min < Numeral.Pls) = True" unfolding Pls_def Min_def by auto |
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54 lemma bitless4: "(Numeral.Min < Numeral.Min) = False" unfolding Pls_def Min_def by auto |
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55 lemma bitless5: "(x BIT Numeral.B0 < y BIT Numeral.B0) = (x < y)" unfolding Bit_def by auto |
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56 lemma bitless6: "(x BIT Numeral.B1 < y BIT Numeral.B1) = (x < y)" unfolding Bit_def by auto |
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57 lemma bitless7: "(x BIT Numeral.B0 < y BIT Numeral.B1) = (x \<le> y)" unfolding Bit_def by auto |
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58 lemma bitless8: "(x BIT Numeral.B1 < y BIT Numeral.B0) = (x < y)" unfolding Bit_def by auto |
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59 lemma bitless9: "(Numeral.Pls < x BIT Numeral.B0) = (Numeral.Pls < x)" unfolding Bit_def Pls_def by auto |
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60 lemma bitless10: "(Numeral.Pls < x BIT Numeral.B1) = (Numeral.Pls \<le> x)" unfolding Bit_def Pls_def by auto |
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61 lemma bitless11: "(Numeral.Min < x BIT Numeral.B0) = (Numeral.Pls \<le> x)" unfolding Bit_def Pls_def Min_def by auto |
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62 lemma bitless12: "(Numeral.Min < x BIT Numeral.B1) = (Numeral.Min < x)" unfolding Bit_def Min_def by auto |
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63 lemma bitless13: "(x BIT Numeral.B0 < Numeral.Pls) = (x < Numeral.Pls)" unfolding Bit_def Pls_def by auto |
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64 lemma bitless14: "(x BIT Numeral.B1 < Numeral.Pls) = (x < Numeral.Pls)" unfolding Bit_def Pls_def by auto |
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65 lemma bitless15: "(x BIT Numeral.B0 < Numeral.Min) = (x < Numeral.Pls)" unfolding Bit_def Pls_def Min_def by auto |
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66 lemma bitless16: "(x BIT Numeral.B1 < Numeral.Min) = (x < Numeral.Min)" unfolding Bit_def Min_def by auto |
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67 lemmas bitless = bitless1 bitless2 bitless3 bitless4 bitless5 bitless6 bitless7 bitless8 |
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68 bitless9 bitless10 bitless11 bitless12 bitless13 bitless14 bitless15 bitless16 |
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69 |
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70 (* x \<le> y for bit strings *) |
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71 lemma bitle1: "(Numeral.Pls \<le> Numeral.Min) = False" unfolding Pls_def Min_def by auto |
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72 lemma bitle2: "(Numeral.Pls \<le> Numeral.Pls) = True" by auto |
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73 lemma bitle3: "(Numeral.Min \<le> Numeral.Pls) = True" unfolding Pls_def Min_def by auto |
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74 lemma bitle4: "(Numeral.Min \<le> Numeral.Min) = True" unfolding Pls_def Min_def by auto |
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75 lemma bitle5: "(x BIT Numeral.B0 \<le> y BIT Numeral.B0) = (x \<le> y)" unfolding Bit_def by auto |
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76 lemma bitle6: "(x BIT Numeral.B1 \<le> y BIT Numeral.B1) = (x \<le> y)" unfolding Bit_def by auto |
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77 lemma bitle7: "(x BIT Numeral.B0 \<le> y BIT Numeral.B1) = (x \<le> y)" unfolding Bit_def by auto |
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78 lemma bitle8: "(x BIT Numeral.B1 \<le> y BIT Numeral.B0) = (x < y)" unfolding Bit_def by auto |
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79 lemma bitle9: "(Numeral.Pls \<le> x BIT Numeral.B0) = (Numeral.Pls \<le> x)" unfolding Bit_def Pls_def by auto |
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80 lemma bitle10: "(Numeral.Pls \<le> x BIT Numeral.B1) = (Numeral.Pls \<le> x)" unfolding Bit_def Pls_def by auto |
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81 lemma bitle11: "(Numeral.Min \<le> x BIT Numeral.B0) = (Numeral.Pls \<le> x)" unfolding Bit_def Pls_def Min_def by auto |
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82 lemma bitle12: "(Numeral.Min \<le> x BIT Numeral.B1) = (Numeral.Min \<le> x)" unfolding Bit_def Min_def by auto |
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83 lemma bitle13: "(x BIT Numeral.B0 \<le> Numeral.Pls) = (x \<le> Numeral.Pls)" unfolding Bit_def Pls_def by auto |
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84 lemma bitle14: "(x BIT Numeral.B1 \<le> Numeral.Pls) = (x < Numeral.Pls)" unfolding Bit_def Pls_def by auto |
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85 lemma bitle15: "(x BIT Numeral.B0 \<le> Numeral.Min) = (x < Numeral.Pls)" unfolding Bit_def Pls_def Min_def by auto |
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86 lemma bitle16: "(x BIT Numeral.B1 \<le> Numeral.Min) = (x \<le> Numeral.Min)" unfolding Bit_def Min_def by auto |
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87 lemmas bitle = bitle1 bitle2 bitle3 bitle4 bitle5 bitle6 bitle7 bitle8 |
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88 bitle9 bitle10 bitle11 bitle12 bitle13 bitle14 bitle15 bitle16 |
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89 |
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90 (* succ for bit strings *) |
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91 lemmas bitsucc = succ_Pls succ_Min succ_1 succ_0 |
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92 |
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93 (* pred for bit strings *) |
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94 lemmas bitpred = pred_Pls pred_Min pred_1 pred_0 |
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95 |
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96 (* unary minus for bit strings *) |
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97 lemmas bituminus = minus_Pls minus_Min minus_1 minus_0 |
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98 |
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99 (* addition for bit strings *) |
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100 lemmas bitadd = add_Pls add_Pls_right add_Min add_Min_right add_BIT_11 add_BIT_10 add_BIT_0[where b="Numeral.B0"] add_BIT_0[where b="Numeral.B1"] |
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101 |
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102 (* multiplication for bit strings *) |
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103 lemma mult_Pls_right: "x * Numeral.Pls = Numeral.Pls" by (simp add: Pls_def) |
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104 lemma mult_Min_right: "x * Numeral.Min = - x" by (subst mult_commute, simp add: mult_Min) |
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105 lemma multb0x: "(x BIT Numeral.B0) * y = (x * y) BIT Numeral.B0" unfolding Bit_def by simp |
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106 lemma multxb0: "x * (y BIT Numeral.B0) = (x * y) BIT Numeral.B0" unfolding Bit_def by simp |
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107 lemma multb1: "(x BIT Numeral.B1) * (y BIT Numeral.B1) = (((x * y) BIT Numeral.B0) + x + y) BIT Numeral.B1" |
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108 unfolding Bit_def by (simp add: ring_simps) |
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109 lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1 |
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110 |
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111 lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul |
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112 |
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113 constdefs |
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114 "nat_norm_number_of (x::nat) == x" |
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115 |
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116 lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)" |
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117 apply (simp add: nat_norm_number_of_def) |
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118 unfolding lezero_def iszero_def neg_def |
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119 apply (simp add: number_of_is_id) |
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120 done |
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121 |
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122 (* Normalization of nat literals *) |
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123 lemma natnorm0: "(0::nat) = number_of (Numeral.Pls)" by auto |
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124 lemma natnorm1: "(1 :: nat) = number_of (Numeral.Pls BIT Numeral.B1)" by auto |
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125 lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of |
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126 |
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127 (* Suc *) |
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128 lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Numeral.succ x))" by (auto simp add: number_of_is_id) |
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129 |
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130 (* Addition for nat *) |
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131 lemma natadd: "number_of x + ((number_of y)::nat) = (if neg x then (number_of y) else (if neg y then number_of x else (number_of (x + y))))" |
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132 by (auto simp add: number_of_is_id) |
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133 |
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134 (* Subtraction for nat *) |
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135 lemma natsub: "(number_of x) - ((number_of y)::nat) = |
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136 (if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + (- y))))))" |
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137 unfolding nat_norm_number_of |
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138 by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def) |
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139 |
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140 (* Multiplication for nat *) |
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141 lemma natmul: "(number_of x) * ((number_of y)::nat) = |
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142 (if neg x then 0 else (if neg y then 0 else number_of (x * y)))" |
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143 apply (auto simp add: number_of_is_id neg_def iszero_def) |
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144 apply (case_tac "x > 0") |
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145 apply auto |
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146 apply (simp add: mult_strict_left_mono[where a=y and b=0 and c=x, simplified]) |
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147 done |
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148 |
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149 lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x \<and> lezero y) \<or> (x = y))" |
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150 by (auto simp add: iszero_def lezero_def neg_def number_of_is_id) |
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151 |
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152 lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) \<and> (\<not> (lezero y)))" |
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153 by (auto simp add: number_of_is_id neg_def lezero_def) |
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154 |
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155 lemma natle: "(((number_of x)::nat) \<le> (number_of y)) = (y < x \<longrightarrow> lezero x)" |
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156 by (auto simp add: number_of_is_id lezero_def nat_number_of_def) |
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157 |
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158 fun natfac :: "nat \<Rightarrow> nat" |
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159 where |
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160 "natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))" |
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161 |
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162 lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps |
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163 |
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164 lemma number_eq: "(((number_of x)::'a::{number_ring, ordered_idom}) = (number_of y)) = (x = y)" |
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165 unfolding number_of_eq |
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166 apply simp |
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167 done |
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168 |
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169 lemma number_le: "(((number_of x)::'a::{number_ring, ordered_idom}) \<le> (number_of y)) = (x \<le> y)" |
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170 unfolding number_of_eq |
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171 apply simp |
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172 done |
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173 |
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174 lemma number_less: "(((number_of x)::'a::{number_ring, ordered_idom}) < (number_of y)) = (x < y)" |
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175 unfolding number_of_eq |
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176 apply simp |
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177 done |
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178 |
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179 lemma number_diff: "((number_of x)::'a::{number_ring, ordered_idom}) - number_of y = number_of (x + (- y))" |
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180 apply (subst diff_number_of_eq) |
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181 apply simp |
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182 done |
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183 |
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184 lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric] |
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185 |
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186 lemmas compute_numberarith = number_of_minus[symmetric] number_of_add[symmetric] number_diff number_of_mult[symmetric] number_norm number_eq number_le number_less |
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187 |
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188 lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)" |
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189 by (simp only: real_of_nat_number_of number_of_is_id) |
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190 |
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191 lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)" |
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192 by simp |
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193 |
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194 lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of |
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195 |
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196 lemmas zpowerarith = zpower_number_of_even |
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197 zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] |
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198 zpower_Pls zpower_Min |
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199 |
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200 (* div, mod *) |
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201 |
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202 lemma adjust: "adjust b (q, r) = (if 0 \<le> r - b then (2 * q + 1, r - b) else (2 * q, r))" |
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203 by (auto simp only: adjust_def) |
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204 |
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205 lemma negateSnd: "negateSnd (q, r) = (q, -r)" |
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206 by (auto simp only: negateSnd_def) |
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207 |
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208 lemma divAlg: "divAlg (a, b) = (if 0\<le>a then |
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209 if 0\<le>b then posDivAlg a b |
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210 else if a=0 then (0, 0) |
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211 else negateSnd (negDivAlg (-a) (-b)) |
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212 else |
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213 if 0<b then negDivAlg a b |
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214 else negateSnd (posDivAlg (-a) (-b)))" |
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215 by (auto simp only: divAlg_def) |
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216 |
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217 lemmas compute_div_mod = div_def mod_def divAlg adjust negateSnd posDivAlg.simps negDivAlg.simps |
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218 |
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219 |
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220 |
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221 (* collecting all the theorems *) |
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222 |
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223 lemma even_Pls: "even (Numeral.Pls) = True" |
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224 apply (unfold Pls_def even_def) |
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225 by simp |
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226 |
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227 lemma even_Min: "even (Numeral.Min) = False" |
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228 apply (unfold Min_def even_def) |
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229 by simp |
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230 |
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231 lemma even_B0: "even (x BIT Numeral.B0) = True" |
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232 apply (unfold Bit_def) |
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233 by simp |
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234 |
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235 lemma even_B1: "even (x BIT Numeral.B1) = False" |
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236 apply (unfold Bit_def) |
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237 by simp |
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238 |
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239 lemma even_number_of: "even ((number_of w)::int) = even w" |
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240 by (simp only: number_of_is_id) |
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241 |
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242 lemmas compute_even = even_Pls even_Min even_B0 even_B1 even_number_of |
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243 |
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244 lemmas compute_numeral = compute_if compute_let compute_pair compute_bool |
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245 compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even |
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246 |
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247 end |
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248 |
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249 |
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250 |