isabelle: src/HOL/Library/Quotient_Sum.thy@7e741e22d7fc (annotated)

47455 26315a545e26 updated headers; wenzelm parents: 47308 diff changeset	1	(* Title: HOL/Library/Quotient_Sum.thy
53012 cb82606b8215 move Lifting/Transfer relevant parts of Library/Quotient_* to Main kuncar parents: 53010 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: 35243 diff changeset	4
60500 903bb1495239 isabelle update_cartouches; wenzelm parents: 58916 diff changeset	5	section \<open>Quotient infrastructure for the sum type\<close>
35788 f1deaca15ca3 observe standard header format; wenzelm parents: 35243 diff changeset	6
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	7	theory Quotient_Sum
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
60500 903bb1495239 isabelle update_cartouches; wenzelm parents: 58916 diff changeset	11	subsection \<open>Rules for the Quotient package\<close>
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
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	13	lemma rel_sum_map1:
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	14	"rel_sum R1 R2 (map_sum f1 f2 x) y \<longleftrightarrow> rel_sum (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y"
58916 229765cc3414 more complete fp_sugars for sum and prod; traytel parents: 58881 diff changeset	15	by (rule sum.rel_map(1))
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	16
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	17	lemma rel_sum_map2:
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	18	"rel_sum R1 R2 x (map_sum f1 f2 y) \<longleftrightarrow> rel_sum (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y"
58916 229765cc3414 more complete fp_sugars for sum and prod; traytel parents: 58881 diff changeset	19	by (rule sum.rel_map(2))
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	20
55931 62156e694f3d renamed 'map_sum' to 'sum_map' blanchet parents: 55564 diff changeset	21	lemma map_sum_id [id_simps]:
62156e694f3d renamed 'map_sum' to 'sum_map' blanchet parents: 55564 diff changeset	22	"map_sum id id = id"
62156e694f3d renamed 'map_sum' to 'sum_map' blanchet parents: 55564 diff changeset	23	by (simp add: id_def map_sum.identity fun_eq_iff)
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	24
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	25	lemma rel_sum_eq [id_simps]:
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	26	"rel_sum (op =) (op =) = (op =)"
58916 229765cc3414 more complete fp_sugars for sum and prod; traytel parents: 58881 diff changeset	27	by (rule sum.rel_eq)
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	28
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	29	lemma reflp_rel_sum:
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	30	"reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (rel_sum R1 R2)"
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	31	unfolding reflp_def split_sum_all rel_sum_simps by fast
55564 e81ee43ab290 delete or move now not necessary reflexivity rules due to 1726f46d2aa8 kuncar parents: 53026 diff changeset	32
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	33	lemma sum_symp:
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	34	"symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (rel_sum R1 R2)"
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	35	unfolding symp_def split_sum_all rel_sum_simps by fast
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	36
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	37	lemma sum_transp:
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	38	"transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (rel_sum R1 R2)"
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	39	unfolding transp_def split_sum_all rel_sum_simps by fast
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	40
fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	41	lemma sum_equivp [quot_equiv]:
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	42	"equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (rel_sum R1 R2)"
5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	43	by (blast intro: equivpI reflp_rel_sum sum_symp sum_transp elim: equivpE)
47624 16d433895d2e add new transfer rules and setup for lifting package huffman parents: 47455 diff changeset	44
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	45	lemma sum_quotient [quot_thm]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	46	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	47	assumes q2: "Quotient3 R2 Abs2 Rep2"
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	48	shows "Quotient3 (rel_sum R1 R2) (map_sum Abs1 Abs2) (map_sum Rep1 Rep2)"
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	49	apply (rule Quotient3I)
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	50	apply (simp_all add: map_sum.compositionality comp_def map_sum.identity rel_sum_eq rel_sum_map1 rel_sum_map2
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	51	Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	52	using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
58916 229765cc3414 more complete fp_sugars for sum and prod; traytel parents: 58881 diff changeset	53	apply (fastforce elim!: rel_sum.cases simp add: comp_def split: sum.split)
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	54	done
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	55
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	56	declare [[mapQ3 sum = (rel_sum, sum_quotient)]]
47094 1a7ad2601cb5 store the relational theorem for every relator kuncar parents: 45802 diff changeset	57
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	58	lemma sum_Inl_rsp [quot_respect]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	59	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	60	assumes q2: "Quotient3 R2 Abs2 Rep2"
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	61	shows "(R1 ===> rel_sum R1 R2) Inl Inl"
40465 2989f9f3aa10 more appropriate specification packages; fun_rel_def is no simp rule by default haftmann parents: 39302 diff changeset	62	by auto
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	63
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	64	lemma sum_Inr_rsp [quot_respect]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	65	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	66	assumes q2: "Quotient3 R2 Abs2 Rep2"
55943 5c2df04e97d1 renamed 'sum_rel' to 'rel_sum' blanchet parents: 55931 diff changeset	67	shows "(R2 ===> rel_sum R1 R2) Inr Inr"
40465 2989f9f3aa10 more appropriate specification packages; fun_rel_def is no simp rule by default haftmann parents: 39302 diff changeset	68	by auto
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	69
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	70	lemma sum_Inl_prs [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	71	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	72	assumes q2: "Quotient3 R2 Abs2 Rep2"
55931 62156e694f3d renamed 'map_sum' to 'sum_map' blanchet parents: 55564 diff changeset	73	shows "(Rep1 ---> map_sum Abs1 Abs2) Inl = Inl"
39302 d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI nipkow parents: 39198 diff changeset	74	apply(simp add: fun_eq_iff)
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	75	apply(simp add: Quotient3_abs_rep[OF q1])
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	76	done
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	77
40820 fd9c98ead9a9 more systematic and compact proofs on type relation operators using natural deduction rules haftmann parents: 40610 diff changeset	78	lemma sum_Inr_prs [quot_preserve]:
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	79	assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	80	assumes q2: "Quotient3 R2 Abs2 Rep2"
55931 62156e694f3d renamed 'map_sum' to 'sum_map' blanchet parents: 55564 diff changeset	81	shows "(Rep2 ---> map_sum Abs1 Abs2) Inr = Inr"
39302 d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI nipkow parents: 39198 diff changeset	82	apply(simp add: fun_eq_iff)
47308 9caab698dbe4 new package Lifting - initial commit kuncar parents: 47094 diff changeset	83	apply(simp add: Quotient3_abs_rep[OF q2])
35222 4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	84	done
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	85
4f1fba00f66d Initial version of HOL quotient package. Cezary Kaliszyk <kaliszyk@in.tum.de> parents: diff changeset	86	end

author	wenzelm
	Thu, 25 Jun 2015 23:33:47 +0200
changeset 60580	7e741e22d7fc
parent 60500	903bb1495239
child 62954	c5d0fdc260fa
permissions	-rw-r--r--