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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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1 |
theory QuotProd
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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2 |
imports QuotScript
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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3 |
begin
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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4 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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5 |
fun
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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6 |
prod_rel
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QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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7 |
where
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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8 |
"prod_rel r1 r2 = (\<lambda>(a,b) (c,d). r1 a c \<and> r2 b d)"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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9 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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10 |
(* prod_fun is a good mapping function *)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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11 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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lemma prod_equivp:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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assumes a: "equivp R1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes b: "equivp R2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "equivp (prod_rel R1 R2)"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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apply(auto simp add: equivp_reflp[OF a] equivp_reflp[OF b])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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apply(simp only: equivp_symp[OF a])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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apply(simp only: equivp_symp[OF b])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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using equivp_transp[OF a] apply blast
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
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using equivp_transp[OF b] apply blast
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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done
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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23 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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lemma prod_quotient:
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QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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changeset
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assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "Quotient (prod_rel R1 R2) (prod_fun Abs1 Abs2) (prod_fun Rep1 Rep2)"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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unfolding Quotient_def
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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apply (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q1] Quotient_rel_rep[OF q2])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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using Quotient_rel[OF q1] Quotient_rel[OF q2] by blast
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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31 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma pair_rsp:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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34 |
assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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35 |
shows "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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36 |
by auto
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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37 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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38 |
lemma pair_prs:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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39 |
assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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40 |
assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "(prod_fun Abs1 Abs2) (Rep1 l, Rep2 r) \<equiv> (l, r)"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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by (simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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43 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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44 |
(* TODO: Is the quotient assumption q1 necessary? *)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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45 |
(* TODO: Aren't there hard to use later? *)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma fst_rsp:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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47 |
assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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48 |
assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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49 |
assumes a: "(prod_rel R1 R2) p1 p2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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50 |
shows "R1 (fst p1) (fst p2)"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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51 |
using a
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(case_tac p1)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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53 |
apply(case_tac p2)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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54 |
apply(auto)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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55 |
done
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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56 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma snd_rsp:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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58 |
assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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59 |
assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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60 |
assumes a: "(prod_rel R1 R2) p1 p2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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61 |
shows "R2 (snd p1) (snd p2)"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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62 |
using a
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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63 |
apply(case_tac p1)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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64 |
apply(case_tac p2)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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65 |
apply(auto)
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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66 |
done
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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67 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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68 |
lemma fst_prs:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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69 |
assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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70 |
assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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71 |
shows "Abs1 (fst ((prod_fun Rep1 Rep2) p)) = fst p"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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72 |
by (case_tac p) (auto simp add: Quotient_abs_rep[OF q1])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
73 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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74 |
lemma snd_prs:
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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75 |
assumes q1: "Quotient R1 Abs1 Rep1"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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76 |
assumes q2: "Quotient R2 Abs2 Rep2"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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77 |
shows "Abs2 (snd ((prod_fun Rep1 Rep2) p)) = snd p"
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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78 |
by (case_tac p) (auto simp add: Quotient_abs_rep[OF q2])
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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79 |
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18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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80 |
end
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