593
18eac4596ef1
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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1 |
theory QuotProd
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648
830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
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2 |
imports QuotMain
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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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changeset
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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:
diff
changeset
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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:
diff
changeset
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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:
diff
changeset
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6 |
prod_rel
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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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7 |
where
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922
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"prod_rel R1 R2 = (\<lambda>(a,b) (c,d). R1 a c \<and> R2 b d)"
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593
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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922
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10 |
declare [[map * = (prod_fun, prod_rel)]]
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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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922
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lemma prod_equivp[quot_equiv]:
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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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14 |
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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15 |
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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16 |
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:
diff
changeset
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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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changeset
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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:
diff
changeset
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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:
diff
changeset
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20 |
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:
diff
changeset
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21 |
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
changeset
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22 |
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:
diff
changeset
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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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922
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lemma prod_quotient[quot_thm]:
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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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26 |
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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27 |
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:
diff
changeset
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unfolding Quotient_def
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699
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using q1 q2
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apply (simp add: Quotient_abs_rep Quotient_abs_rep Quotient_rel_rep Quotient_rel_rep)
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using Quotient_rel[OF q1] Quotient_rel[OF q2]
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by blast
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QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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922
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lemma pair_rsp[quot_respect]:
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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 |
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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37 |
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 "(R1 ===> R2 ===> prod_rel R1 R2) Pair Pair"
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922
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by simp
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593
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 |
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648
830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
41 |
lemma pair_prs[quot_preserve]:
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830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
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42 |
assumes q1: "Quotient R1 Abs1 Rep1"
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830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
43 |
assumes q2: "Quotient R2 Abs2 Rep2"
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830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
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44 |
shows "(Rep1 ---> Rep2 ---> (prod_fun Abs1 Abs2)) Pair = Pair"
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922
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apply(simp add: expand_fun_eq)
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apply(simp add: Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
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830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
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47 |
done
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830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
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922
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lemma fst_rsp[quot_respect]:
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assumes "Quotient R1 Abs1 Rep1"
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assumes "Quotient R2 Abs2 Rep2"
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shows "(prod_rel R1 R2 ===> R1) fst fst"
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by simp
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830b58c2fa94
decoupled QuotProd from QuotMain and also started new cleaning strategy
Christian Urban <urbanc@in.tum.de>
diff
changeset
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922
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lemma fst_prs[quot_preserve]:
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593
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 |
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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922
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shows "(prod_fun Rep1 Rep2 ---> Abs1) fst = fst"
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apply(simp add: expand_fun_eq)
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apply(simp add: Quotient_abs_rep[OF q1])
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done
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593
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[quot_respect]:
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assumes "Quotient R1 Abs1 Rep1"
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assumes "Quotient R2 Abs2 Rep2"
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shows "(prod_rel R1 R2 ===> R2) snd snd"
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by simp
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lemma snd_prs[quot_preserve]:
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593
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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 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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922
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shows "(prod_fun Rep1 Rep2 ---> Abs2) snd = snd"
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apply(simp add: expand_fun_eq)
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apply(simp add: Quotient_abs_rep[OF q2])
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done
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593
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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922
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lemma prod_fun_id[id_simps]:
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shows "prod_fun id id \<equiv> id"
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by (rule eq_reflection)
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(simp add: prod_fun_def)
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593
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 prod_rel_eq[id_simps]:
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925
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shows "prod_rel (op =) (op =) \<equiv> (op =)"
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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84 |
apply (rule eq_reflection)
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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85 |
apply (rule ext)+
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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86 |
apply auto
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
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87 |
done
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
88 |
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593
18eac4596ef1
QuotProd with product_quotient and a 3 respects and preserves lemmas.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
89 |
end
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