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1 (* Title: QuotBase.thy |
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2 Author: Cezary Kaliszyk and Christian Urban |
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3 *) |
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4 |
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5 theory QuotBase |
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6 imports Plain ATP_Linkup Predicate |
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7 begin |
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8 |
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9 text {* |
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10 Basic definition for equivalence relations |
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11 that are represented by predicates. |
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12 *} |
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13 |
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14 definition |
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15 "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)" |
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16 |
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17 definition |
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18 "reflp E \<equiv> \<forall>x. E x x" |
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19 |
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20 definition |
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21 "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x" |
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22 |
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23 definition |
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24 "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z" |
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25 |
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26 lemma equivp_reflp_symp_transp: |
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27 shows "equivp E = (reflp E \<and> symp E \<and> transp E)" |
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28 unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq |
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29 by blast |
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30 |
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31 lemma equivp_reflp: |
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32 shows "equivp E \<Longrightarrow> (\<And>x. E x x)" |
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33 by (simp only: equivp_reflp_symp_transp reflp_def) |
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34 |
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35 lemma equivp_symp: |
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36 shows "equivp E \<Longrightarrow> (\<And>x y. E x y \<Longrightarrow> E y x)" |
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37 by (metis equivp_reflp_symp_transp symp_def) |
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38 |
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39 lemma equivp_transp: |
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40 shows "equivp E \<Longrightarrow> (\<And>x y z. E x y \<Longrightarrow> E y z \<Longrightarrow> E x z)" |
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41 by (metis equivp_reflp_symp_transp transp_def) |
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42 |
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43 lemma equivpI: |
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44 assumes "reflp R" "symp R" "transp R" |
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45 shows "equivp R" |
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46 using assms by (simp add: equivp_reflp_symp_transp) |
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47 |
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48 lemma eq_imp_rel: |
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49 shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b" |
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50 by (simp add: equivp_reflp) |
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51 |
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52 text {* Partial equivalences: not yet used anywhere *} |
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53 definition |
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54 "part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))" |
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55 |
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56 lemma equivp_IMP_part_equivp: |
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57 assumes a: "equivp E" |
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58 shows "part_equivp E" |
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59 using a |
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60 unfolding equivp_def part_equivp_def |
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61 by auto |
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62 |
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63 text {* Composition of Relations *} |
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64 abbreviation |
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65 rel_conj (infixr "OOO" 75) |
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66 where |
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67 "r1 OOO r2 \<equiv> r1 OO r2 OO r1" |
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68 |
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69 section {* Quotient Predicate *} |
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70 |
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71 definition |
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72 "Quotient E Abs Rep \<equiv> |
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73 (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and> |
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74 (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))" |
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75 |
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76 lemma Quotient_abs_rep: |
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77 assumes a: "Quotient E Abs Rep" |
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78 shows "Abs (Rep a) = a" |
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79 using a |
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80 unfolding Quotient_def |
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81 by simp |
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82 |
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83 lemma Quotient_rep_reflp: |
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84 assumes a: "Quotient E Abs Rep" |
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85 shows "E (Rep a) (Rep a)" |
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86 using a |
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87 unfolding Quotient_def |
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88 by blast |
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89 |
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90 lemma Quotient_rel: |
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91 assumes a: "Quotient E Abs Rep" |
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92 shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))" |
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93 using a |
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94 unfolding Quotient_def |
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95 by blast |
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96 |
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97 lemma Quotient_rel_rep: |
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98 assumes a: "Quotient R Abs Rep" |
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99 shows "R (Rep a) (Rep b) = (a = b)" |
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100 using a |
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101 unfolding Quotient_def |
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102 by metis |
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103 |
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104 lemma Quotient_rep_abs: |
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105 assumes a: "Quotient R Abs Rep" |
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106 shows "R r r \<Longrightarrow> R (Rep (Abs r)) r" |
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107 using a unfolding Quotient_def |
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108 by blast |
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109 |
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110 lemma Quotient_rel_abs: |
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111 assumes a: "Quotient E Abs Rep" |
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112 shows "E r s \<Longrightarrow> Abs r = Abs s" |
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113 using a unfolding Quotient_def |
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114 by blast |
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115 |
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116 lemma Quotient_symp: |
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117 assumes a: "Quotient E Abs Rep" |
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118 shows "symp E" |
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119 using a unfolding Quotient_def symp_def |
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120 by metis |
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121 |
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122 lemma Quotient_transp: |
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123 assumes a: "Quotient E Abs Rep" |
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124 shows "transp E" |
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125 using a unfolding Quotient_def transp_def |
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126 by metis |
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127 |
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128 section {* Lemmas about (op =) *} |
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129 |
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130 lemma identity_equivp: |
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131 shows "equivp (op =)" |
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132 unfolding equivp_def |
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133 by auto |
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134 |
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135 lemma identity_quotient: |
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136 shows "Quotient (op =) id id" |
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137 unfolding Quotient_def id_def |
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138 by blast |
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139 |
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140 section {* Function map and function relation *} |
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141 |
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142 definition |
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143 fun_map (infixr "--->" 55) |
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144 where |
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145 [simp]: "fun_map f g h x = g (h (f x))" |
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146 |
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147 definition |
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148 fun_rel (infixr "===>" 55) |
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149 where |
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150 [simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))" |
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151 |
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152 |
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153 lemma fun_map_id: |
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154 shows "(id ---> id) = id" |
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155 by (simp add: expand_fun_eq id_def) |
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156 |
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157 lemma fun_rel_eq: |
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158 shows "(op =) ===> (op =) \<equiv> (op =)" |
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159 by (rule eq_reflection) (simp add: expand_fun_eq) |
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160 |
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161 lemma fun_quotient: |
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162 assumes q1: "Quotient R1 abs1 rep1" |
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163 and q2: "Quotient R2 abs2 rep2" |
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164 shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)" |
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165 proof - |
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166 have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a" |
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167 apply(simp add: expand_fun_eq) |
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168 using q1 q2 |
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169 apply(simp add: Quotient_def) |
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170 done |
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171 moreover |
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172 have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)" |
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173 apply(auto) |
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174 using q1 q2 unfolding Quotient_def |
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175 apply(metis) |
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176 done |
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177 moreover |
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178 have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and> |
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179 (rep1 ---> abs2) r = (rep1 ---> abs2) s)" |
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180 apply(auto simp add: expand_fun_eq) |
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181 using q1 q2 unfolding Quotient_def |
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182 apply(metis) |
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183 using q1 q2 unfolding Quotient_def |
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184 apply(metis) |
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185 using q1 q2 unfolding Quotient_def |
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186 apply(metis) |
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187 using q1 q2 unfolding Quotient_def |
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188 apply(metis) |
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189 done |
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190 ultimately |
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191 show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)" |
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192 unfolding Quotient_def by blast |
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193 qed |
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194 |
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195 section {* Respects predicate *} |
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196 |
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197 definition |
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198 Respects |
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199 where |
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200 "Respects R x \<equiv> (R x x)" |
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201 |
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202 lemma in_respects: |
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203 shows "(x \<in> Respects R) = R x x" |
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204 unfolding mem_def Respects_def by simp |
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205 |
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206 lemma equals_rsp: |
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207 assumes q: "Quotient R Abs Rep" |
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208 and a: "R xa xb" "R ya yb" |
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209 shows "R xa ya = R xb yb" |
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210 using Quotient_symp[OF q] Quotient_transp[OF q] unfolding symp_def transp_def |
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211 using a by blast |
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212 |
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213 lemma lambda_prs: |
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214 assumes q1: "Quotient R1 Abs1 Rep1" |
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215 and q2: "Quotient R2 Abs2 Rep2" |
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216 shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)" |
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217 unfolding expand_fun_eq |
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218 using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] |
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219 by simp |
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220 |
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221 lemma lambda_prs1: |
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222 assumes q1: "Quotient R1 Abs1 Rep1" |
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223 and q2: "Quotient R2 Abs2 Rep2" |
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224 shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)" |
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225 unfolding expand_fun_eq |
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226 using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] |
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227 by simp |
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228 |
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229 lemma rep_abs_rsp: |
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230 assumes q: "Quotient R Abs Rep" |
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231 and a: "R x1 x2" |
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232 shows "R x1 (Rep (Abs x2))" |
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233 using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]) |
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234 |
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235 lemma rep_abs_rsp_left: |
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236 assumes q: "Quotient R Abs Rep" |
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237 and a: "R x1 x2" |
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238 shows "R (Rep (Abs x1)) x2" |
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239 using q a by (metis Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]) |
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240 |
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241 (* In the following theorem R1 can be instantiated with anything, |
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242 but we know some of the types of the Rep and Abs functions; |
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243 so by solving Quotient assumptions we can get a unique R1 that |
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244 will be provable; which is why we need to use apply_rsp and |
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245 not the primed version *) |
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246 lemma apply_rsp: |
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247 fixes f g::"'a \<Rightarrow> 'c" |
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248 assumes q: "Quotient R1 Abs1 Rep1" |
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249 and a: "(R1 ===> R2) f g" "R1 x y" |
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250 shows "R2 (f x) (g y)" |
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251 using a by simp |
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252 |
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253 lemma apply_rsp': |
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254 assumes a: "(R1 ===> R2) f g" "R1 x y" |
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255 shows "R2 (f x) (g y)" |
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256 using a by simp |
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257 |
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258 (* Set of lemmas for regularisation of ball and bex *) |
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259 |
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260 lemma ball_reg_eqv: |
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261 fixes P :: "'a \<Rightarrow> bool" |
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262 assumes a: "equivp R" |
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263 shows "Ball (Respects R) P = (All P)" |
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264 by (metis equivp_def in_respects a) |
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265 |
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266 lemma bex_reg_eqv: |
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267 fixes P :: "'a \<Rightarrow> bool" |
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268 assumes a: "equivp R" |
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269 shows "Bex (Respects R) P = (Ex P)" |
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270 by (metis equivp_def in_respects a) |
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271 |
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272 lemma ball_reg_right: |
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273 assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x" |
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274 shows "All P \<longrightarrow> Ball R Q" |
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275 by (metis COMBC_def Collect_def Collect_mem_eq a) |
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276 |
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277 lemma bex_reg_left: |
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278 assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x" |
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279 shows "Bex R Q \<longrightarrow> Ex P" |
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280 by (metis COMBC_def Collect_def Collect_mem_eq a) |
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281 |
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282 lemma ball_reg_left: |
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283 assumes a: "equivp R" |
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284 shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P" |
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285 by (metis equivp_reflp in_respects a) |
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286 |
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287 lemma bex_reg_right: |
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288 assumes a: "equivp R" |
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289 shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P" |
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290 by (metis equivp_reflp in_respects a) |
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291 |
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292 lemma ball_reg_eqv_range: |
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293 fixes P::"'a \<Rightarrow> bool" |
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294 and x::"'a" |
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295 assumes a: "equivp R2" |
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296 shows "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))" |
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297 apply(rule iffI) |
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298 apply(rule allI) |
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299 apply(drule_tac x="\<lambda>y. f x" in bspec) |
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300 apply(simp add: in_respects) |
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301 apply(rule impI) |
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302 using a equivp_reflp_symp_transp[of "R2"] |
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303 apply(simp add: reflp_def) |
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304 apply(simp) |
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305 apply(simp) |
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306 done |
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307 |
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308 lemma bex_reg_eqv_range: |
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309 assumes a: "equivp R2" |
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310 shows "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))" |
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311 apply(auto) |
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312 apply(rule_tac x="\<lambda>y. f x" in bexI) |
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313 apply(simp) |
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314 apply(simp add: Respects_def in_respects) |
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315 apply(rule impI) |
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316 using a equivp_reflp_symp_transp[of "R2"] |
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317 apply(simp add: reflp_def) |
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318 done |
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319 |
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320 lemma all_reg: |
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321 assumes a: "!x :: 'a. (P x --> Q x)" |
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322 and b: "All P" |
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323 shows "All Q" |
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324 using a b by (metis) |
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325 |
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326 lemma ex_reg: |
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327 assumes a: "!x :: 'a. (P x --> Q x)" |
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328 and b: "Ex P" |
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329 shows "Ex Q" |
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330 using a b by (metis) |
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331 |
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332 lemma ball_reg: |
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333 assumes a: "!x :: 'a. (R x --> P x --> Q x)" |
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334 and b: "Ball R P" |
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335 shows "Ball R Q" |
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336 using a b by (metis COMBC_def Collect_def Collect_mem_eq) |
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337 |
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338 lemma bex_reg: |
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339 assumes a: "!x :: 'a. (R x --> P x --> Q x)" |
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340 and b: "Bex R P" |
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341 shows "Bex R Q" |
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342 using a b by (metis COMBC_def Collect_def Collect_mem_eq) |
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343 |
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344 lemma ball_all_comm: |
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345 "(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))" |
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346 by auto |
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347 |
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348 lemma bex_ex_comm: |
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349 "((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))" |
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350 by auto |
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351 |
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352 (* Bounded abstraction *) |
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353 definition |
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354 Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" |
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355 where |
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356 "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)" |
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357 |
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358 definition |
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359 Bexeq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" |
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360 where |
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361 "Bexeq R P \<equiv> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))" |
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362 |
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363 (* 3 lemmas needed for proving repabs_inj *) |
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364 lemma ball_rsp: |
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365 assumes a: "(R ===> (op =)) f g" |
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366 shows "Ball (Respects R) f = Ball (Respects R) g" |
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367 using a by (simp add: Ball_def in_respects) |
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368 |
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369 lemma bex_rsp: |
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370 assumes a: "(R ===> (op =)) f g" |
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371 shows "(Bex (Respects R) f = Bex (Respects R) g)" |
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372 using a by (simp add: Bex_def in_respects) |
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373 |
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374 lemma bex1_rsp: |
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375 assumes a: "(R ===> (op =)) f g" |
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376 shows "(Bex1 (Respects R) f = Bex1 (Respects R) g)" |
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377 using a |
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378 by (simp add: Ex1_def Bex1_def in_respects) auto |
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379 |
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380 (* TODO/FIXME: simplify the repetitions in this proof *) |
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381 lemma bexeq_rsp: |
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382 assumes a: "Quotient R absf repf" |
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383 shows "((R ===> op =) ===> op =) (Bexeq R) (Bexeq R)" |
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384 apply simp |
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385 unfolding Bexeq_def |
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386 apply rule |
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387 apply rule |
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388 apply rule |
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389 apply rule |
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390 apply (erule conjE)+ |
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391 apply (erule bexE) |
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392 apply rule |
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393 apply (rule_tac x="xa" in bexI) |
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394 apply metis |
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395 apply metis |
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396 apply rule+ |
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397 apply (erule_tac x="xb" in ballE) |
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398 prefer 2 |
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399 apply (metis) |
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400 apply (erule_tac x="ya" in ballE) |
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401 prefer 2 |
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402 apply (metis) |
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403 apply (metis in_respects) |
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404 apply (erule conjE)+ |
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405 apply (erule bexE) |
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406 apply rule |
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407 apply (rule_tac x="xa" in bexI) |
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408 apply metis |
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409 apply metis |
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410 apply rule+ |
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411 apply (erule_tac x="xb" in ballE) |
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412 prefer 2 |
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413 apply (metis) |
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414 apply (erule_tac x="ya" in ballE) |
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415 prefer 2 |
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416 apply (metis) |
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417 apply (metis in_respects) |
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418 done |
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419 |
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420 lemma babs_rsp: |
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421 assumes q: "Quotient R1 Abs1 Rep1" |
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422 and a: "(R1 ===> R2) f g" |
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423 shows "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)" |
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424 apply (auto simp add: Babs_def) |
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425 apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1") |
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426 using a apply (simp add: Babs_def) |
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427 apply (simp add: in_respects) |
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428 using Quotient_rel[OF q] |
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429 by metis |
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430 |
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431 lemma babs_prs: |
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432 assumes q1: "Quotient R1 Abs1 Rep1" |
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433 and q2: "Quotient R2 Abs2 Rep2" |
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434 shows "(Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f)) \<equiv> f" |
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435 apply(rule eq_reflection) |
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436 apply(rule ext) |
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437 apply simp |
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438 apply (subgoal_tac "Rep1 x \<in> Respects R1") |
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439 apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]) |
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440 apply (simp add: in_respects Quotient_rel_rep[OF q1]) |
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441 done |
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442 |
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443 lemma babs_simp: |
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444 assumes q: "Quotient R1 Abs Rep" |
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445 shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)" |
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446 apply(rule iffI) |
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447 apply(simp_all only: babs_rsp[OF q]) |
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448 apply(auto simp add: Babs_def) |
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449 apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1") |
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450 apply(metis Babs_def) |
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451 apply (simp add: in_respects) |
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452 using Quotient_rel[OF q] |
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453 by metis |
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454 |
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455 (* If a user proves that a particular functional relation |
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456 is an equivalence this may be useful in regularising *) |
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457 lemma babs_reg_eqv: |
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458 shows "equivp R \<Longrightarrow> Babs (Respects R) P = P" |
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459 by (simp add: expand_fun_eq Babs_def in_respects equivp_reflp) |
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460 |
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461 (* 3 lemmas needed for cleaning of quantifiers *) |
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462 lemma all_prs: |
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463 assumes a: "Quotient R absf repf" |
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464 shows "Ball (Respects R) ((absf ---> id) f) = All f" |
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465 using a unfolding Quotient_def Ball_def in_respects fun_map_def id_apply |
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466 by metis |
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467 |
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468 lemma ex_prs: |
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469 assumes a: "Quotient R absf repf" |
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470 shows "Bex (Respects R) ((absf ---> id) f) = Ex f" |
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471 using a unfolding Quotient_def Bex_def in_respects fun_map_def id_apply |
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472 by metis |
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473 |
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474 lemma ex1_prs: |
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475 assumes a: "Quotient R absf repf" |
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476 shows "((absf ---> id) ---> id) (Bexeq R) f = Ex1 f" |
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477 apply simp |
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478 apply (subst Bexeq_def) |
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479 apply (subst Bex_def) |
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480 apply (subst Ex1_def) |
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481 apply simp |
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482 apply rule |
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483 apply (erule conjE)+ |
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484 apply (erule_tac exE) |
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485 apply (erule conjE) |
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486 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y") |
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487 apply (rule_tac x="absf x" in exI) |
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488 apply (thin_tac "\<forall>x\<in>Respects R. \<forall>y\<in>Respects R. f (absf x) \<and> f (absf y) \<longrightarrow> R x y") |
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489 apply (simp) |
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490 apply rule+ |
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491 using a unfolding Quotient_def |
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492 apply metis |
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493 apply rule+ |
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494 apply (erule_tac x="x" in ballE) |
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495 apply (erule_tac x="y" in ballE) |
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496 apply simp |
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497 apply (simp add: in_respects) |
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498 apply (simp add: in_respects) |
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499 apply (erule_tac exE) |
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500 apply rule |
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501 apply (rule_tac x="repf x" in exI) |
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502 apply (simp only: in_respects) |
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503 apply rule |
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504 apply (metis Quotient_rel_rep[OF a]) |
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505 using a unfolding Quotient_def apply (simp) |
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506 apply rule+ |
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507 using a unfolding Quotient_def in_respects |
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508 apply metis |
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509 done |
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510 |
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511 lemma fun_rel_id: |
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512 assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)" |
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513 shows "(R1 ===> R2) f g" |
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514 using a by simp |
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515 |
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516 lemma fun_rel_id_asm: |
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517 assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))" |
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518 shows "A \<longrightarrow> (R1 ===> R2) f g" |
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519 using a by auto |
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520 |
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521 lemma quot_rel_rsp: |
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522 assumes a: "Quotient R Abs Rep" |
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523 shows "(R ===> R ===> op =) R R" |
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524 apply(rule fun_rel_id)+ |
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525 apply(rule equals_rsp[OF a]) |
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526 apply(assumption)+ |
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527 done |
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528 |
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529 lemma o_prs: |
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530 assumes q1: "Quotient R1 Abs1 Rep1" |
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531 and q2: "Quotient R2 Abs2 Rep2" |
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532 and q3: "Quotient R3 Abs3 Rep3" |
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533 shows "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g" |
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534 using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3] |
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535 unfolding o_def expand_fun_eq by simp |
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536 |
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537 lemma o_rsp: |
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538 assumes q1: "Quotient R1 Abs1 Rep1" |
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539 and q2: "Quotient R2 Abs2 Rep2" |
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540 and q3: "Quotient R3 Abs3 Rep3" |
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541 and a1: "(R2 ===> R3) f1 f2" |
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542 and a2: "(R1 ===> R2) g1 g2" |
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543 shows "(R1 ===> R3) (f1 o g1) (f2 o g2)" |
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544 using a1 a2 unfolding o_def expand_fun_eq |
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545 by (auto) |
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546 |
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547 lemma cond_prs: |
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548 assumes a: "Quotient R absf repf" |
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549 shows "absf (if a then repf b else repf c) = (if a then b else c)" |
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550 using a unfolding Quotient_def by auto |
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551 |
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552 lemma if_prs: |
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553 assumes q: "Quotient R Abs Rep" |
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554 shows "Abs (If a (Rep b) (Rep c)) = If a b c" |
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555 using Quotient_abs_rep[OF q] by auto |
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556 |
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557 (* q not used *) |
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558 lemma if_rsp: |
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559 assumes q: "Quotient R Abs Rep" |
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560 and a: "a1 = a2" "R b1 b2" "R c1 c2" |
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561 shows "R (If a1 b1 c1) (If a2 b2 c2)" |
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562 using a by auto |
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563 |
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564 lemma let_prs: |
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565 assumes q1: "Quotient R1 Abs1 Rep1" |
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566 and q2: "Quotient R2 Abs2 Rep2" |
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567 shows "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f" |
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568 using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto |
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569 |
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570 lemma let_rsp: |
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571 assumes q1: "Quotient R1 Abs1 Rep1" |
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572 and a1: "(R1 ===> R2) f g" |
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573 and a2: "R1 x y" |
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574 shows "R2 ((Let x f)::'c) ((Let y g)::'c)" |
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575 using apply_rsp[OF q1 a1] a2 by auto |
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576 |
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577 |
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578 |
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579 |
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580 (******************************************) |
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581 (* REST OF THE FILE IS UNUSED (until now) *) |
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582 (******************************************) |
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583 |
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584 lemma in_fun: |
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585 shows "x \<in> ((f ---> g) s) = g (f x \<in> s)" |
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586 by (simp add: mem_def) |
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587 |
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588 lemma respects_thm: |
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589 shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))" |
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590 unfolding Respects_def |
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591 by (simp add: expand_fun_eq) |
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592 |
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593 lemma respects_rep_abs: |
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594 assumes a: "Quotient R1 Abs1 Rep1" |
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595 and b: "Respects (R1 ===> R2) f" |
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596 and c: "R1 x x" |
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597 shows "R2 (f (Rep1 (Abs1 x))) (f x)" |
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598 using a b[simplified respects_thm] c unfolding Quotient_def |
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599 by blast |
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600 |
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601 lemma respects_mp: |
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602 assumes a: "Respects (R1 ===> R2) f" |
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603 and b: "R1 x y" |
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604 shows "R2 (f x) (f y)" |
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605 using a b unfolding Respects_def |
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606 by simp |
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607 |
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608 lemma respects_o: |
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609 assumes a: "Respects (R2 ===> R3) f" |
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610 and b: "Respects (R1 ===> R2) g" |
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611 shows "Respects (R1 ===> R3) (f o g)" |
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612 using a b unfolding Respects_def |
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613 by simp |
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614 |
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615 lemma abs_o_rep: |
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616 assumes a: "Quotient r absf repf" |
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617 shows "absf o repf = id" |
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618 apply(rule ext) |
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619 apply(simp add: Quotient_abs_rep[OF a]) |
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620 done |
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621 |
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622 lemma eq_comp_r: "op = OO R OO op = \<equiv> R" |
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623 apply (rule eq_reflection) |
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624 apply (rule ext)+ |
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625 apply auto |
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626 done |
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627 |
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628 lemma fun_rel_eq_rel: |
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629 assumes q1: "Quotient R1 Abs1 Rep1" |
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630 and q2: "Quotient R2 Abs2 Rep2" |
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631 shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g) |
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632 \<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))" |
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633 using fun_quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq |
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634 by blast |
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635 |
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636 lemma let_babs: |
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637 "v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam" |
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638 by (simp add: Babs_def) |
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639 |
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640 lemma fun_rel_equals: |
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641 assumes q1: "Quotient R1 Abs1 Rep1" |
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642 and q2: "Quotient R2 Abs2 Rep2" |
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643 and r1: "Respects (R1 ===> R2) f" |
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644 and r2: "Respects (R1 ===> R2) g" |
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645 shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))" |
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646 apply(rule_tac iffI) |
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647 apply(rule)+ |
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648 apply (rule apply_rsp'[of "R1" "R2"]) |
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649 apply(subst Quotient_rel[OF fun_quotient[OF q1 q2]]) |
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650 apply auto |
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651 using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def |
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652 apply (metis let_rsp q1) |
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653 apply (metis fun_rel_eq_rel let_rsp q1 q2 r2) |
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654 using r1 unfolding Respects_def expand_fun_eq |
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655 apply(simp (no_asm_use)) |
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656 apply(metis Quotient_rel[OF q2] Quotient_rel_rep[OF q1]) |
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657 done |
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658 |
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659 (* ask Peter: fun_rel_IMP used twice *) |
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660 lemma fun_rel_IMP2: |
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661 assumes q1: "Quotient R1 Abs1 Rep1" |
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662 and q2: "Quotient R2 Abs2 Rep2" |
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663 and r1: "Respects (R1 ===> R2) f" |
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664 and r2: "Respects (R1 ===> R2) g" |
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665 and a: "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g" |
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666 shows "R1 x y \<Longrightarrow> R2 (f x) (g y)" |
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667 using q1 q2 r1 r2 a |
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668 by (simp add: fun_rel_equals) |
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669 |
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670 lemma lambda_rep_abs_rsp: |
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671 assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))" |
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672 and r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))" |
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673 shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))" |
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674 using r1 r2 by auto |
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675 |
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676 (* ask peter what are literal_case *) |
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677 (* literal_case_PRS *) |
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678 (* literal_case_RSP *) |
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679 (* Cez: !f x. literal_case f x = f x *) |
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680 |
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681 (* We use id_simps which includes id_apply; so these 2 theorems can be removed *) |
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682 lemma id_prs: |
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683 assumes q: "Quotient R Abs Rep" |
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684 shows "Abs (id (Rep e)) = id e" |
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685 using Quotient_abs_rep[OF q] by auto |
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686 |
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687 lemma id_rsp: |
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688 assumes q: "Quotient R Abs Rep" |
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689 and a: "R e1 e2" |
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690 shows "R (id e1) (id e2)" |
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691 using a by auto |
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692 |
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693 end |
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694 |