isabelle: src/HOL/Library/Quotient_Product.thy@b9556a055632 (annotated)

47455 26315a545e26 updated headers; wenzelm parents: 47308 diff changeset	1	(* Title: HOL/Library/Quotient_Product.thy
53012 cb82606b8215 move Lifting/Transfer relevant parts of Library/Quotient_* to Main kuncar parents: 51994 diff changeset	2	Author: Cezary Kaliszyk and Christian Urban
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	3	*)
35788 f1deaca15ca3 observe standard header format; wenzelm parents: 35222 diff changeset	4
58881 b9556a055632 modernized header; wenzelm parents: 55944 diff changeset	5	section {* Quotient infrastructure for the product type *}
35788 f1deaca15ca3 observe standard header format; wenzelm parents: 35222 diff changeset	6
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	7	theory Quotient_Product
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	8	imports Main Quotient_Syntax
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	9	begin
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	10
53012 cb82606b8215 move Lifting/Transfer relevant parts of Library/Quotient_* to Main kuncar parents: 51994 diff changeset	11	subsection {* Rules for the Quotient package *}
51956 a4d81cdebf8b better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal kuncar parents: 51377 diff changeset	12
55932 68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	13	lemma map_prod_id [id_simps]:
68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	14	shows "map_prod id id = id"
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	15	by (simp add: fun_eq_iff)
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	16
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	17	lemma rel_prod_eq [id_simps]:
7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	18	shows "rel_prod (op =) (op =) = (op =)"
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	19	by (simp add: fun_eq_iff)
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	20
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	21	lemma prod_equivp [quot_equiv]:
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	22	assumes "equivp R1"
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	23	assumes "equivp R2"
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	24	shows "equivp (rel_prod R1 R2)"
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	25	using assms by (auto intro!: equivpI reflpI sympI transpI elim!: equivpE elim: reflpE sympE transpE)
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	26
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	27	lemma prod_quotient [quot_thm]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	28	assumes "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	29	assumes "Quotient3 R2 Abs2 Rep2"
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	30	shows "Quotient3 (rel_prod R1 R2) (map_prod Abs1 Abs2) (map_prod Rep1 Rep2)"
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	31	apply (rule Quotient3I)
55932 68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	32	apply (simp add: map_prod.compositionality comp_def map_prod.identity
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	33	Quotient3_abs_rep [OF assms(1)] Quotient3_abs_rep [OF assms(2)])
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	34	apply (simp add: split_paired_all Quotient3_rel_rep [OF assms(1)] Quotient3_rel_rep [OF assms(2)])
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	35	using Quotient3_rel [OF assms(1)] Quotient3_rel [OF assms(2)]
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	36	apply (auto simp add: split_paired_all)
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	37	done
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	38
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	39	declare [[mapQ3 prod = (rel_prod, prod_quotient)]]
47094 1a7ad2601cb5 store the relational theorem for every relator kuncar parents: 45802 diff changeset	40
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	41	lemma Pair_rsp [quot_respect]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	42	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	43	assumes q2: "Quotient3 R2 Abs2 Rep2"
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	44	shows "(R1 ===> R2 ===> rel_prod R1 R2) Pair Pair"
47624 16d433895d2e add new transfer rules and setup for lifting package huffman parents: 47455 diff changeset	45	by (rule Pair_transfer)
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	46
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	47	lemma Pair_prs [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	48	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	49	assumes q2: "Quotient3 R2 Abs2 Rep2"
55932 68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	50	shows "(Rep1 ---> Rep2 ---> (map_prod Abs1 Abs2)) Pair = Pair"
39302 d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI nipkow parents: 39198 diff changeset	51	apply(simp add: fun_eq_iff)
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	52	apply(simp add: Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	53	done
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	54
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	55	lemma fst_rsp [quot_respect]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	56	assumes "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	57	assumes "Quotient3 R2 Abs2 Rep2"
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	58	shows "(rel_prod R1 R2 ===> R1) fst fst"
40465 2989f9f3aa10 more appropriate specification packages; fun_rel_def is no simp rule by default haftmann parents: 39302 diff changeset	59	by auto
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	60
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	61	lemma fst_prs [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	62	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	63	assumes q2: "Quotient3 R2 Abs2 Rep2"
55932 68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	64	shows "(map_prod Rep1 Rep2 ---> Abs1) fst = fst"
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	65	by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1])
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	66
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	67	lemma snd_rsp [quot_respect]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	68	assumes "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	69	assumes "Quotient3 R2 Abs2 Rep2"
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	70	shows "(rel_prod R1 R2 ===> R2) snd snd"
40465 2989f9f3aa10 more appropriate specification packages; fun_rel_def is no simp rule by default haftmann parents: 39302 diff changeset	71	by auto
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	72
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	73	lemma snd_prs [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	74	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	75	assumes q2: "Quotient3 R2 Abs2 Rep2"
55932 68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	76	shows "(map_prod Rep1 Rep2 ---> Abs2) snd = snd"
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	77	by (simp add: fun_eq_iff Quotient3_abs_rep[OF q2])
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	78
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	79	lemma split_rsp [quot_respect]:
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	80	shows "((R1 ===> R2 ===> (op =)) ===> (rel_prod R1 R2) ===> (op =)) split split"
55414 eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...' blanchet parents: 53012 diff changeset	81	by (rule case_prod_transfer)
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	82
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40607 diff changeset	83	lemma split_prs [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	84	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	85	and q2: "Quotient3 R2 Abs2 Rep2"
55932 68c5104d2204 renamed 'map_pair' to 'map_prod' blanchet parents: 55414 diff changeset	86	shows "(((Abs1 ---> Abs2 ---> id) ---> map_prod Rep1 Rep2 ---> id) split) = split"
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	87	by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	88
36695 b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	89	lemma [quot_respect]:
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	90	shows "((R2 ===> R2 ===> op =) ===> (R1 ===> R1 ===> op =) ===>
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	91	rel_prod R2 R1 ===> rel_prod R2 R1 ===> op =) rel_prod rel_prod"
7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	92	by (rule rel_prod_transfer)
36695 b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	93
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	94	lemma [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	95	assumes q1: "Quotient3 R1 abs1 rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	96	and q2: "Quotient3 R2 abs2 rep2"
36695 b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	97	shows "((abs1 ---> abs1 ---> id) ---> (abs2 ---> abs2 ---> id) --->
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	98	map_prod rep1 rep2 ---> map_prod rep1 rep2 ---> id) rel_prod = rel_prod"
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47301 diff changeset	99	by (simp add: fun_eq_iff Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
36695 b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	100
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	101	lemma [quot_preserve]:
55944 7ab8f003fe41 renamed 'prod_rel' to 'rel_prod' blanchet parents: 55932 diff changeset	102	shows"(rel_prod ((rep1 ---> rep1 ---> id) R1) ((rep2 ---> rep2 ---> id) R2)
36695 b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	103	(l1, l2) (r1, r2)) = (R1 (rep1 l1) (rep1 r1) \<and> R2 (rep2 l2) (rep2 r2))"
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	104	by simp
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	105
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	106	declare Pair_eq[quot_preserve]
b434506fb0d4 respectfullness and preservation of prod_rel Cezary Kaliszyk <kaliszyk@in.tum.de> parents: 35788 diff changeset	107
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	108	end

author	wenzelm
	Sun, 02 Nov 2014 17:20:45 +0100
changeset 58881	b9556a055632
parent 55944	7ab8f003fe41
child 58916	229765cc3414
permissions	-rw-r--r--