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1 (* Title: Quotient_Sum.thy |
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2 Author: Cezary Kaliszyk and Christian Urban |
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3 *) |
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4 theory Quotient_Sum |
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5 imports Main Quotient_Syntax |
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6 begin |
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7 |
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8 section {* Quotient infrastructure for the sum type. *} |
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9 |
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10 fun |
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11 sum_rel |
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12 where |
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13 "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1" |
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14 | "sum_rel R1 R2 (Inl a1) (Inr b2) = False" |
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15 | "sum_rel R1 R2 (Inr a2) (Inl b1) = False" |
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16 | "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2" |
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17 |
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18 fun |
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19 sum_map |
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20 where |
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21 "sum_map f1 f2 (Inl a) = Inl (f1 a)" |
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22 | "sum_map f1 f2 (Inr a) = Inr (f2 a)" |
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23 |
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24 declare [[map "+" = (sum_map, sum_rel)]] |
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25 |
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26 |
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27 text {* should probably be in Sum_Type.thy *} |
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28 lemma split_sum_all: |
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29 shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))" |
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30 apply(auto) |
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31 apply(case_tac x) |
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32 apply(simp_all) |
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33 done |
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34 |
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35 lemma sum_equivp[quot_equiv]: |
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36 assumes a: "equivp R1" |
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37 assumes b: "equivp R2" |
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38 shows "equivp (sum_rel R1 R2)" |
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39 apply(rule equivpI) |
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40 unfolding reflp_def symp_def transp_def |
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41 apply(simp_all add: split_sum_all) |
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42 apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b]) |
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43 apply(blast intro: equivp_symp[OF a] equivp_symp[OF b]) |
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44 apply(blast intro: equivp_transp[OF a] equivp_transp[OF b]) |
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45 done |
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46 |
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47 lemma sum_quotient[quot_thm]: |
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48 assumes q1: "Quotient R1 Abs1 Rep1" |
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49 assumes q2: "Quotient R2 Abs2 Rep2" |
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50 shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)" |
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51 unfolding Quotient_def |
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52 apply(simp add: split_sum_all) |
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53 apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1]) |
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54 apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2]) |
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55 using q1 q2 |
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56 unfolding Quotient_def |
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57 apply(blast)+ |
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58 done |
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59 |
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60 lemma sum_Inl_rsp[quot_respect]: |
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61 assumes q1: "Quotient R1 Abs1 Rep1" |
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62 assumes q2: "Quotient R2 Abs2 Rep2" |
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63 shows "(R1 ===> sum_rel R1 R2) Inl Inl" |
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64 by simp |
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65 |
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66 lemma sum_Inr_rsp[quot_respect]: |
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67 assumes q1: "Quotient R1 Abs1 Rep1" |
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68 assumes q2: "Quotient R2 Abs2 Rep2" |
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69 shows "(R2 ===> sum_rel R1 R2) Inr Inr" |
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70 by simp |
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71 |
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72 lemma sum_Inl_prs[quot_preserve]: |
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73 assumes q1: "Quotient R1 Abs1 Rep1" |
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74 assumes q2: "Quotient R2 Abs2 Rep2" |
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75 shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl" |
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76 apply(simp add: expand_fun_eq) |
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77 apply(simp add: Quotient_abs_rep[OF q1]) |
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78 done |
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79 |
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80 lemma sum_Inr_prs[quot_preserve]: |
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81 assumes q1: "Quotient R1 Abs1 Rep1" |
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82 assumes q2: "Quotient R2 Abs2 Rep2" |
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83 shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr" |
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84 apply(simp add: expand_fun_eq) |
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85 apply(simp add: Quotient_abs_rep[OF q2]) |
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86 done |
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87 |
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88 lemma sum_map_id[id_simps]: |
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89 shows "sum_map id id = id" |
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90 by (simp add: expand_fun_eq split_sum_all) |
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91 |
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92 lemma sum_rel_eq[id_simps]: |
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93 shows "sum_rel (op =) (op =) = (op =)" |
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94 by (simp add: expand_fun_eq split_sum_all) |
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95 |
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96 end |