1 (* Author: Clemens Ballarin, started 23 June 1999 |
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2 |
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3 Experimental theory: long division of polynomials. |
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4 *) |
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5 |
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6 theory LongDiv |
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7 imports PolyHomo |
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8 begin |
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9 |
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10 definition |
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11 lcoeff :: "'a::ring up => 'a" where |
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12 "lcoeff p = coeff p (deg p)" |
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13 |
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14 definition |
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15 eucl_size :: "'a::zero up => nat" where |
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16 "eucl_size p = (if p = 0 then 0 else deg p + 1)" |
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17 |
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18 lemma SUM_shrink_below_lemma: |
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19 "!! f::(nat=>'a::ring). (ALL i. i < m --> f i = 0) --> |
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20 setsum (%i. f (i+m)) {..d} = setsum f {..m+d}" |
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21 apply (induct_tac d) |
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22 apply (induct_tac m) |
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23 apply simp |
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24 apply force |
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25 apply (simp add: add_commute [of m]) |
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26 done |
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27 |
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28 lemma SUM_extend_below: |
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29 "!! f::(nat=>'a::ring). |
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30 [| m <= n; !!i. i < m ==> f i = 0; P (setsum (%i. f (i+m)) {..n-m}) |] |
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31 ==> P (setsum f {..n})" |
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32 by (simp add: SUM_shrink_below_lemma add_diff_inverse leD) |
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33 |
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34 lemma up_repr2D: |
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35 "!! p::'a::ring up. |
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36 [| deg p <= n; P (setsum (%i. monom (coeff p i) i) {..n}) |] |
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37 ==> P p" |
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38 by (simp add: up_repr_le) |
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39 |
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40 |
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41 (* Start of LongDiv *) |
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42 |
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43 lemma deg_lcoeff_cancel: |
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44 "!!p::('a::ring up). |
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45 [| deg p <= deg r; deg q <= deg r; |
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46 coeff p (deg r) = - (coeff q (deg r)); deg r ~= 0 |] ==> |
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47 deg (p + q) < deg r" |
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48 apply (rule le_less_trans [of _ "deg r - 1"]) |
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49 prefer 2 |
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50 apply arith |
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51 apply (rule deg_aboveI) |
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52 apply (case_tac "deg r = m") |
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53 apply clarify |
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54 apply simp |
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55 (* case "deg q ~= m" *) |
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56 apply (subgoal_tac "deg p < m & deg q < m") |
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57 apply (simp (no_asm_simp) add: deg_aboveD) |
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58 apply arith |
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59 done |
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60 |
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61 lemma deg_lcoeff_cancel2: |
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62 "!!p::('a::ring up). |
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63 [| deg p <= deg r; deg q <= deg r; |
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64 p ~= -q; coeff p (deg r) = - (coeff q (deg r)) |] ==> |
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65 deg (p + q) < deg r" |
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66 apply (rule deg_lcoeff_cancel) |
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67 apply assumption+ |
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68 apply (rule classical) |
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69 apply clarify |
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70 apply (erule notE) |
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71 apply (rule_tac p = p in up_repr2D, assumption) |
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72 apply (rule_tac p = q in up_repr2D, assumption) |
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73 apply (rotate_tac -1) |
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74 apply (simp add: smult_l_minus) |
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75 done |
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76 |
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77 lemma long_div_eucl_size: |
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78 "!!g::('a::ring up). g ~= 0 ==> |
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79 Ex (% (q, r, k). |
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80 (lcoeff g)^k *s f = q * g + r & (eucl_size r < eucl_size g))" |
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81 apply (rule_tac P = "%f. Ex (% (q, r, k) . (lcoeff g) ^k *s f = q * g + r & (eucl_size r < eucl_size g))" in wf_induct) |
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82 (* TO DO: replace by measure_induct *) |
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83 apply (rule_tac f = eucl_size in wf_measure) |
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84 apply (case_tac "eucl_size x < eucl_size g") |
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85 apply (rule_tac x = "(0, x, 0)" in exI) |
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86 apply (simp (no_asm_simp)) |
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87 (* case "eucl_size x >= eucl_size g" *) |
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88 apply (drule_tac x = "lcoeff g *s x - (monom (lcoeff x) (deg x - deg g)) * g" in spec) |
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89 apply (erule impE) |
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90 apply (simp (no_asm_use) add: inv_image_def measure_def lcoeff_def) |
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91 apply (case_tac "x = 0") |
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92 apply (rotate_tac -1) |
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93 apply (simp add: eucl_size_def) |
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94 (* case "x ~= 0 *) |
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95 apply (rotate_tac -1) |
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96 apply (simp add: eucl_size_def) |
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97 apply (rule impI) |
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98 apply (rule deg_lcoeff_cancel2) |
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99 (* replace by linear arithmetic??? *) |
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100 apply (rule_tac [2] le_trans) |
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101 apply (rule_tac [2] deg_smult_ring) |
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102 prefer 2 |
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103 apply simp |
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104 apply (simp (no_asm)) |
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105 apply (rule le_trans) |
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106 apply (rule deg_mult_ring) |
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107 apply (rule le_trans) |
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108 (**) |
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109 apply (rule add_le_mono) |
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110 apply (rule le_refl) |
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111 (* term order forces to use this instead of add_le_mono1 *) |
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112 apply (rule deg_monom_ring) |
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113 apply (simp (no_asm_simp)) |
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114 apply force |
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115 apply (simp (no_asm)) |
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116 (**) |
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117 (* This change is probably caused by application of commutativity *) |
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118 apply (rule_tac m = "deg g" and n = "deg x" in SUM_extend) |
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119 apply (simp (no_asm)) |
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120 apply (simp (no_asm_simp)) |
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121 apply arith |
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122 apply (rule_tac m = "deg g" and n = "deg g" in SUM_extend_below) |
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123 apply (rule le_refl) |
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124 apply (simp (no_asm_simp)) |
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125 apply arith |
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126 apply (simp (no_asm)) |
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127 (**) |
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128 (* end of subproof deg f1 < deg f *) |
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129 apply (erule exE) |
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130 apply (rule_tac x = "((% (q,r,k) . (monom (lcoeff g ^ k * lcoeff x) (deg x - deg g) + q)) xa, (% (q,r,k) . r) xa, (% (q,r,k) . Suc k) xa) " in exI) |
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131 apply clarify |
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132 apply (drule sym) |
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133 using [[simproc del: ring]] |
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134 apply (simp (no_asm_use) add: l_distr a_assoc) |
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135 apply (simp (no_asm_simp)) |
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136 apply (simp (no_asm_use) add: minus_def smult_r_distr smult_r_minus |
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137 monom_mult_smult smult_assoc2) |
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138 using [[simproc ring]] |
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139 apply (simp add: smult_assoc1 [symmetric]) |
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140 done |
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141 |
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142 lemma long_div_ring_aux: |
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143 "(g :: 'a::ring up) ~= 0 ==> |
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144 Ex (\<lambda>(q, r, k). lcoeff g ^ k *s f = q * g + r \<and> |
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145 (if r = 0 then 0 else deg r + 1) < (if g = 0 then 0 else deg g + 1))" |
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146 proof - |
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147 note [[simproc del: ring]] |
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148 assume "g ~= 0" |
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149 then show ?thesis |
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150 by (rule long_div_eucl_size [simplified eucl_size_def]) |
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151 qed |
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152 |
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153 lemma long_div_ring: |
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154 "!!g::('a::ring up). g ~= 0 ==> |
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155 Ex (% (q, r, k). |
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156 (lcoeff g)^k *s f = q * g + r & (r = 0 | deg r < deg g))" |
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157 apply (frule_tac f = f in long_div_ring_aux) |
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158 using [[simproc del: ring]] |
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159 apply auto |
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160 apply (case_tac "aa = 0") |
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161 apply blast |
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162 (* case "aa ~= 0 *) |
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163 apply (rotate_tac -1) |
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164 apply auto |
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165 done |
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166 |
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167 (* Next one fails *) |
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168 lemma long_div_unit: |
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169 "!!g::('a::ring up). [| g ~= 0; (lcoeff g) dvd 1 |] ==> |
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170 Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))" |
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171 apply (frule_tac f = "f" in long_div_ring) |
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172 apply (erule exE) |
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173 apply (rule_tac x = "((% (q,r,k) . (inverse (lcoeff g ^k) *s q)) x, (% (q,r,k) . inverse (lcoeff g ^k) *s r) x) " in exI) |
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174 apply clarify |
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175 apply (rule conjI) |
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176 apply (drule sym) |
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177 using [[simproc del: ring]] |
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178 apply (simp (no_asm_simp) add: smult_r_distr [symmetric] smult_assoc2) |
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179 using [[simproc ring]] |
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180 apply (simp (no_asm_simp) add: l_inverse_ring unit_power smult_assoc1 [symmetric]) |
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181 (* degree property *) |
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182 apply (erule disjE) |
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183 apply (simp (no_asm_simp)) |
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184 apply (rule disjI2) |
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185 apply (rule le_less_trans) |
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186 apply (rule deg_smult_ring) |
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187 apply (simp (no_asm_simp)) |
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188 done |
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189 |
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190 lemma long_div_theorem: |
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191 "!!g::('a::field up). g ~= 0 ==> |
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192 Ex (% (q, r). f = q * g + r & (r = 0 | deg r < deg g))" |
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193 apply (rule long_div_unit) |
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194 apply assumption |
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195 apply (simp (no_asm_simp) add: lcoeff_def lcoeff_nonzero field_ax) |
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196 done |
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197 |
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198 lemma uminus_zero: "- (0::'a::ring) = 0" |
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199 by simp |
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200 |
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201 lemma diff_zero_imp_eq: "!!a::'a::ring. a - b = 0 ==> a = b" |
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202 apply (rule_tac s = "a - (a - b) " in trans) |
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203 apply simp |
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204 apply (simp (no_asm)) |
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205 done |
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206 |
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207 lemma eq_imp_diff_zero: "!!a::'a::ring. a = b ==> a + (-b) = 0" |
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208 by simp |
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209 |
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210 lemma long_div_quo_unique: |
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211 "!!g::('a::field up). [| g ~= 0; |
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212 f = q1 * g + r1; (r1 = 0 | deg r1 < deg g); |
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213 f = q2 * g + r2; (r2 = 0 | deg r2 < deg g) |] ==> q1 = q2" |
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214 apply (subgoal_tac "(q1 - q2) * g = r2 - r1") (* 1 *) |
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215 apply (erule_tac V = "f = ?x" in thin_rl) |
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216 apply (erule_tac V = "f = ?x" in thin_rl) |
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217 apply (rule diff_zero_imp_eq) |
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218 apply (rule classical) |
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219 apply (erule disjE) |
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220 (* r1 = 0 *) |
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221 apply (erule disjE) |
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222 (* r2 = 0 *) |
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223 using [[simproc del: ring]] |
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224 apply (simp add: integral_iff minus_def l_zero uminus_zero) |
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225 (* r2 ~= 0 *) |
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226 apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong) |
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227 apply (simp add: minus_def l_zero uminus_zero) |
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228 (* r1 ~=0 *) |
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229 apply (erule disjE) |
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230 (* r2 = 0 *) |
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231 apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong) |
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232 apply (simp add: minus_def l_zero uminus_zero) |
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233 (* r2 ~= 0 *) |
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234 apply (drule_tac f = "deg" and y = "r2 - r1" in arg_cong) |
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235 apply (simp add: minus_def) |
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236 apply (drule order_eq_refl [THEN add_leD2]) |
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237 apply (drule leD) |
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238 apply (erule notE, rule deg_add [THEN le_less_trans]) |
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239 apply (simp (no_asm_simp)) |
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240 (* proof of 1 *) |
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241 apply (rule diff_zero_imp_eq) |
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242 apply hypsubst |
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243 apply (drule_tac a = "?x+?y" in eq_imp_diff_zero) |
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244 using [[simproc ring]] |
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245 apply simp |
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246 done |
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247 |
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248 lemma long_div_rem_unique: |
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249 "!!g::('a::field up). [| g ~= 0; |
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250 f = q1 * g + r1; (r1 = 0 | deg r1 < deg g); |
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251 f = q2 * g + r2; (r2 = 0 | deg r2 < deg g) |] ==> r1 = r2" |
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252 apply (subgoal_tac "q1 = q2") |
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253 apply (metis a_comm a_lcancel m_comm) |
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254 apply (metis a_comm l_zero long_div_quo_unique m_comm conc) |
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255 done |
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256 |
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257 end |
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