author | haftmann |
Fri, 11 Jun 2010 17:14:02 +0200 | |
changeset 37407 | 61dd8c145da7 |
parent 35788 | f1deaca15ca3 |
child 37678 | 0040bafffdef |
permissions | -rw-r--r-- |
35788 | 1 |
(* Title: HOL/Library/Quotient_Sum.thy |
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Initial version of HOL quotient package.
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parents:
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Author: Cezary Kaliszyk and Christian Urban |
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Initial version of HOL quotient package.
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parents:
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*) |
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header {* Quotient infrastructure for the sum type *} |
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Initial version of HOL quotient package.
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parents:
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theory Quotient_Sum |
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Initial version of HOL quotient package.
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parents:
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imports Main Quotient_Syntax |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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begin |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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fun |
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Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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sum_rel |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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changeset
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where |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
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"sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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| "sum_rel R1 R2 (Inl a1) (Inr b2) = False" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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| "sum_rel R1 R2 (Inr a2) (Inl b1) = False" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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fun |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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sum_map |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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where |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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"sum_map f1 f2 (Inl a) = Inl (f1 a)" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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| "sum_map f1 f2 (Inr a) = Inr (f2 a)" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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declare [[map "+" = (sum_map, sum_rel)]] |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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|
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Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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changeset
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text {* should probably be in @{theory Sum_Type} *} |
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Initial version of HOL quotient package.
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parents:
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lemma split_sum_all: |
4f1fba00f66d
Initial version of HOL quotient package.
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shows "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))" |
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Initial version of HOL quotient package.
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parents:
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apply(auto) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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apply(case_tac x) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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apply(simp_all) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_equivp[quot_equiv]: |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
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assumes a: "equivp R1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes b: "equivp R2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "equivp (sum_rel R1 R2)" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(rule equivpI) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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unfolding reflp_def symp_def transp_def |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(simp_all add: split_sum_all) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(blast intro: equivp_reflp[OF a] equivp_reflp[OF b]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(blast intro: equivp_symp[OF a] equivp_symp[OF b]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(blast intro: equivp_transp[OF a] equivp_transp[OF b]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_quotient[quot_thm]: |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes q1: "Quotient R1 Abs1 Rep1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
50 |
assumes q2: "Quotient R2 Abs2 Rep2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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unfolding Quotient_def |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(simp add: split_sum_all) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(simp_all add: Quotient_abs_rep[OF q1] Quotient_rel_rep[OF q1]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(simp_all add: Quotient_abs_rep[OF q2] Quotient_rel_rep[OF q2]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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using q1 q2 |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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unfolding Quotient_def |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(blast)+ |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_Inl_rsp[quot_respect]: |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes q1: "Quotient R1 Abs1 Rep1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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assumes q2: "Quotient R2 Abs2 Rep2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "(R1 ===> sum_rel R1 R2) Inl Inl" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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by simp |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_Inr_rsp[quot_respect]: |
4f1fba00f66d
Initial version of HOL quotient package.
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parents:
diff
changeset
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assumes q1: "Quotient R1 Abs1 Rep1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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assumes q2: "Quotient R2 Abs2 Rep2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "(R2 ===> sum_rel R1 R2) Inr Inr" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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by simp |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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|
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Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_Inl_prs[quot_preserve]: |
4f1fba00f66d
Initial version of HOL quotient package.
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parents:
diff
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assumes q1: "Quotient R1 Abs1 Rep1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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75 |
assumes q2: "Quotient R2 Abs2 Rep2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(simp add: expand_fun_eq) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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apply(simp add: Quotient_abs_rep[OF q1]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_Inr_prs[quot_preserve]: |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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82 |
assumes q1: "Quotient R1 Abs1 Rep1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
83 |
assumes q2: "Quotient R2 Abs2 Rep2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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84 |
shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
85 |
apply(simp add: expand_fun_eq) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
86 |
apply(simp add: Quotient_abs_rep[OF q2]) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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87 |
done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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lemma sum_map_id[id_simps]: |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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90 |
shows "sum_map id id = id" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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91 |
by (simp add: expand_fun_eq split_sum_all) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
92 |
|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
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lemma sum_rel_eq[id_simps]: |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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94 |
shows "sum_rel (op =) (op =) = (op =)" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
95 |
by (simp add: expand_fun_eq split_sum_all) |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
96 |
|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
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end |