author | wenzelm |
Thu, 13 Aug 2015 11:05:19 +0200 | |
changeset 60924 | 610794dff23c |
parent 60601 | 6e83d94760c4 |
child 61424 | c3658c18b7bc |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Library/Permutations.thy |
2 |
Author: Amine Chaieb, University of Cambridge |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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3 |
*) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
4 |
|
60500 | 5 |
section \<open>Permutations, both general and specifically on finite sets.\<close> |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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6 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
7 |
theory Permutations |
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59474
diff
changeset
|
8 |
imports Binomial |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
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|
9 |
begin |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
10 |
|
60500 | 11 |
subsection \<open>Transpositions\<close> |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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12 |
|
56608 | 13 |
lemma swap_id_idempotent [simp]: |
14 |
"Fun.swap a b id \<circ> Fun.swap a b id = id" |
|
56545 | 15 |
by (rule ext, auto simp add: Fun.swap_def) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
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|
16 |
|
56608 | 17 |
lemma inv_swap_id: |
18 |
"inv (Fun.swap a b id) = Fun.swap a b id" |
|
54681 | 19 |
by (rule inv_unique_comp) simp_all |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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|
20 |
|
56608 | 21 |
lemma swap_id_eq: |
22 |
"Fun.swap a b id x = (if x = a then b else if x = b then a else x)" |
|
56545 | 23 |
by (simp add: Fun.swap_def) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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24 |
|
54681 | 25 |
|
60500 | 26 |
subsection \<open>Basic consequences of the definition\<close> |
54681 | 27 |
|
28 |
definition permutes (infixr "permutes" 41) |
|
29 |
where "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)" |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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30 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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31 |
lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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|
32 |
unfolding permutes_def by metis |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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|
33 |
|
54681 | 34 |
lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S" |
30488 | 35 |
unfolding permutes_def |
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renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
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parents:
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36 |
apply (rule set_eqI) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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37 |
apply (simp add: image_iff) |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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38 |
apply metis |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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|
39 |
done |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
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|
40 |
|
54681 | 41 |
lemma permutes_inj: "p permutes S \<Longrightarrow> inj p" |
30488 | 42 |
unfolding permutes_def inj_on_def by blast |
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Permutations, both general and specifically on finite sets.
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parents:
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|
43 |
|
54681 | 44 |
lemma permutes_surj: "p permutes s \<Longrightarrow> surj p" |
30488 | 45 |
unfolding permutes_def surj_def by metis |
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Permutations, both general and specifically on finite sets.
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parents:
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|
46 |
|
60601 | 47 |
lemma permutes_bij: "p permutes s \<Longrightarrow> bij p" |
48 |
unfolding bij_def by (metis permutes_inj permutes_surj) |
|
49 |
||
59474 | 50 |
lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S" |
60601 | 51 |
by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI) |
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renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
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52 |
|
59474 | 53 |
lemma bij_imp_permutes: "bij_betw p S S \<Longrightarrow> (\<And>x. x \<notin> S \<Longrightarrow> p x = x) \<Longrightarrow> p permutes S" |
54 |
unfolding permutes_def bij_betw_def inj_on_def |
|
55 |
by auto (metis image_iff)+ |
|
56 |
||
54681 | 57 |
lemma permutes_inv_o: |
58 |
assumes pS: "p permutes S" |
|
59 |
shows "p \<circ> inv p = id" |
|
60 |
and "inv p \<circ> p = id" |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
61 |
using permutes_inj[OF pS] permutes_surj[OF pS] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
62 |
unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+ |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
63 |
|
30488 | 64 |
lemma permutes_inverses: |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
65 |
fixes p :: "'a \<Rightarrow> 'a" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
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changeset
|
66 |
assumes pS: "p permutes S" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
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changeset
|
67 |
shows "p (inv p x) = x" |
54681 | 68 |
and "inv p (p x) = x" |
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renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
69 |
using permutes_inv_o[OF pS, unfolded fun_eq_iff o_def] by auto |
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chaieb
parents:
diff
changeset
|
70 |
|
54681 | 71 |
lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> p permutes T" |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
72 |
unfolding permutes_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
73 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
74 |
lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id" |
54681 | 75 |
unfolding fun_eq_iff permutes_def by simp metis |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
76 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
77 |
lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id" |
54681 | 78 |
unfolding fun_eq_iff permutes_def by simp metis |
30488 | 79 |
|
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
80 |
lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
81 |
unfolding permutes_def by simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
82 |
|
54681 | 83 |
lemma permutes_inv_eq: "p permutes S \<Longrightarrow> inv p y = x \<longleftrightarrow> p x = y" |
84 |
unfolding permutes_def inv_def |
|
85 |
apply auto |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
86 |
apply (erule allE[where x=y]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
87 |
apply (erule allE[where x=y]) |
54681 | 88 |
apply (rule someI_ex) |
89 |
apply blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
90 |
apply (rule some1_equality) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
91 |
apply blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
92 |
apply blast |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
93 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
94 |
|
54681 | 95 |
lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> Fun.swap a b id permutes S" |
56545 | 96 |
unfolding permutes_def Fun.swap_def fun_upd_def by auto metis |
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chaieb
parents:
diff
changeset
|
97 |
|
54681 | 98 |
lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T" |
99 |
by (simp add: Ball_def permutes_def) metis |
|
100 |
||
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
101 |
|
60500 | 102 |
subsection \<open>Group properties\<close> |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
103 |
|
54681 | 104 |
lemma permutes_id: "id permutes S" |
105 |
unfolding permutes_def by simp |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
106 |
|
54681 | 107 |
lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> p permutes S" |
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
108 |
unfolding permutes_def o_def by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
109 |
|
54681 | 110 |
lemma permutes_inv: |
111 |
assumes pS: "p permutes S" |
|
112 |
shows "inv p permutes S" |
|
30488 | 113 |
using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis |
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
114 |
|
54681 | 115 |
lemma permutes_inv_inv: |
116 |
assumes pS: "p permutes S" |
|
117 |
shows "inv (inv p) = p" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
118 |
unfolding fun_eq_iff permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
119 |
by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
120 |
|
54681 | 121 |
|
60500 | 122 |
subsection \<open>The number of permutations on a finite set\<close> |
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
123 |
|
30488 | 124 |
lemma permutes_insert_lemma: |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
125 |
assumes pS: "p permutes (insert a S)" |
54681 | 126 |
shows "Fun.swap a (p a) id \<circ> p permutes S" |
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
127 |
apply (rule permutes_superset[where S = "insert a S"]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
128 |
apply (rule permutes_compose[OF pS]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
129 |
apply (rule permutes_swap_id, simp) |
54681 | 130 |
using permutes_in_image[OF pS, of a] |
131 |
apply simp |
|
56545 | 132 |
apply (auto simp add: Ball_def Fun.swap_def) |
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
133 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
134 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
135 |
lemma permutes_insert: "{p. p permutes (insert a S)} = |
54681 | 136 |
(\<lambda>(b,p). Fun.swap a b id \<circ> p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}" |
137 |
proof - |
|
138 |
{ |
|
139 |
fix p |
|
140 |
{ |
|
141 |
assume pS: "p permutes insert a S" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
142 |
let ?b = "p a" |
54681 | 143 |
let ?q = "Fun.swap a (p a) id \<circ> p" |
144 |
have th0: "p = Fun.swap a ?b id \<circ> ?q" |
|
145 |
unfolding fun_eq_iff o_assoc by simp |
|
146 |
have th1: "?b \<in> insert a S" |
|
147 |
unfolding permutes_in_image[OF pS] by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
148 |
from permutes_insert_lemma[OF pS] th0 th1 |
54681 | 149 |
have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S" by blast |
150 |
} |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
151 |
moreover |
54681 | 152 |
{ |
153 |
fix b q |
|
154 |
assume bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S" |
|
30488 | 155 |
from permutes_subset[OF bq(3), of "insert a S"] |
54681 | 156 |
have qS: "q permutes insert a S" |
157 |
by auto |
|
158 |
have aS: "a \<in> insert a S" |
|
159 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
160 |
from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]] |
54681 | 161 |
have "p permutes insert a S" |
162 |
by simp |
|
163 |
} |
|
164 |
ultimately have "p permutes insert a S \<longleftrightarrow> |
|
165 |
(\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S)" |
|
166 |
by blast |
|
167 |
} |
|
168 |
then show ?thesis |
|
169 |
by auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
170 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
171 |
|
54681 | 172 |
lemma card_permutations: |
173 |
assumes Sn: "card S = n" |
|
174 |
and fS: "finite S" |
|
33715 | 175 |
shows "card {p. p permutes S} = fact n" |
54681 | 176 |
using fS Sn |
177 |
proof (induct arbitrary: n) |
|
178 |
case empty |
|
179 |
then show ?case by simp |
|
33715 | 180 |
next |
181 |
case (insert x F) |
|
54681 | 182 |
{ |
183 |
fix n |
|
184 |
assume H0: "card (insert x F) = n" |
|
33715 | 185 |
let ?xF = "{p. p permutes insert x F}" |
186 |
let ?pF = "{p. p permutes F}" |
|
187 |
let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}" |
|
188 |
let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)" |
|
189 |
from permutes_insert[of x F] |
|
190 |
have xfgpF': "?xF = ?g ` ?pF'" . |
|
54681 | 191 |
have Fs: "card F = n - 1" |
60500 | 192 |
using \<open>x \<notin> F\<close> H0 \<open>finite F\<close> by auto |
54681 | 193 |
from insert.hyps Fs have pFs: "card ?pF = fact (n - 1)" |
60500 | 194 |
using \<open>finite F\<close> by auto |
54681 | 195 |
then have "finite ?pF" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
196 |
by (auto intro: card_ge_0_finite) |
54681 | 197 |
then have pF'f: "finite ?pF'" |
60500 | 198 |
using H0 \<open>finite F\<close> |
33715 | 199 |
apply (simp only: Collect_split Collect_mem_eq) |
200 |
apply (rule finite_cartesian_product) |
|
201 |
apply simp_all |
|
202 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
203 |
|
33715 | 204 |
have ginj: "inj_on ?g ?pF'" |
54681 | 205 |
proof - |
33715 | 206 |
{ |
54681 | 207 |
fix b p c q |
208 |
assume bp: "(b,p) \<in> ?pF'" |
|
209 |
assume cq: "(c,q) \<in> ?pF'" |
|
210 |
assume eq: "?g (b,p) = ?g (c,q)" |
|
211 |
from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" |
|
212 |
"p permutes F" "q permutes F" |
|
213 |
by auto |
|
60500 | 214 |
from ths(4) \<open>x \<notin> F\<close> eq have "b = ?g (b,p) x" |
54681 | 215 |
unfolding permutes_def |
56545 | 216 |
by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff) |
54681 | 217 |
also have "\<dots> = ?g (c,q) x" |
60500 | 218 |
using ths(5) \<open>x \<notin> F\<close> eq |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
219 |
by (auto simp add: swap_def fun_upd_def fun_eq_iff) |
54681 | 220 |
also have "\<dots> = c" |
60500 | 221 |
using ths(5) \<open>x \<notin> F\<close> |
54681 | 222 |
unfolding permutes_def |
56545 | 223 |
by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff) |
33715 | 224 |
finally have bc: "b = c" . |
54681 | 225 |
then have "Fun.swap x b id = Fun.swap x c id" |
226 |
by simp |
|
227 |
with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q" |
|
228 |
by simp |
|
229 |
then have "Fun.swap x b id \<circ> (Fun.swap x b id \<circ> p) = |
|
230 |
Fun.swap x b id \<circ> (Fun.swap x b id \<circ> q)" |
|
231 |
by simp |
|
232 |
then have "p = q" |
|
233 |
by (simp add: o_assoc) |
|
234 |
with bc have "(b, p) = (c, q)" |
|
235 |
by simp |
|
33715 | 236 |
} |
54681 | 237 |
then show ?thesis |
238 |
unfolding inj_on_def by blast |
|
33715 | 239 |
qed |
60500 | 240 |
from \<open>x \<notin> F\<close> H0 have n0: "n \<noteq> 0" |
241 |
using \<open>finite F\<close> by auto |
|
54681 | 242 |
then have "\<exists>m. n = Suc m" |
243 |
by presburger |
|
244 |
then obtain m where n[simp]: "n = Suc m" |
|
245 |
by blast |
|
33715 | 246 |
from pFs H0 have xFc: "card ?xF = fact n" |
54681 | 247 |
unfolding xfgpF' card_image[OF ginj] |
60500 | 248 |
using \<open>finite F\<close> \<open>finite ?pF\<close> |
33715 | 249 |
apply (simp only: Collect_split Collect_mem_eq card_cartesian_product) |
54681 | 250 |
apply simp |
251 |
done |
|
252 |
from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" |
|
253 |
unfolding xfgpF' by simp |
|
33715 | 254 |
have "card ?xF = fact n" |
255 |
using xFf xFc unfolding xFf by blast |
|
256 |
} |
|
54681 | 257 |
then show ?case |
258 |
using insert by simp |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
259 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
260 |
|
54681 | 261 |
lemma finite_permutations: |
262 |
assumes fS: "finite S" |
|
263 |
shows "finite {p. p permutes S}" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
264 |
using card_permutations[OF refl fS] |
33715 | 265 |
by (auto intro: card_ge_0_finite) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
266 |
|
54681 | 267 |
|
60500 | 268 |
subsection \<open>Permutations of index set for iterated operations\<close> |
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
269 |
|
51489 | 270 |
lemma (in comm_monoid_set) permute: |
271 |
assumes "p permutes S" |
|
54681 | 272 |
shows "F g S = F (g \<circ> p) S" |
51489 | 273 |
proof - |
60500 | 274 |
from \<open>p permutes S\<close> have "inj p" |
54681 | 275 |
by (rule permutes_inj) |
276 |
then have "inj_on p S" |
|
277 |
by (auto intro: subset_inj_on) |
|
278 |
then have "F g (p ` S) = F (g \<circ> p) S" |
|
279 |
by (rule reindex) |
|
60500 | 280 |
moreover from \<open>p permutes S\<close> have "p ` S = S" |
54681 | 281 |
by (rule permutes_image) |
282 |
ultimately show ?thesis |
|
283 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
284 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
285 |
|
54681 | 286 |
|
60500 | 287 |
subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close> |
54681 | 288 |
|
289 |
lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow> |
|
290 |
Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id" |
|
56545 | 291 |
by (simp add: fun_eq_iff Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
292 |
|
54681 | 293 |
lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow> |
294 |
Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id" |
|
56545 | 295 |
by (simp add: fun_eq_iff Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
296 |
|
54681 | 297 |
lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow> |
298 |
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id" |
|
56545 | 299 |
by (simp add: fun_eq_iff Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
300 |
|
54681 | 301 |
|
60500 | 302 |
subsection \<open>Permutations as transposition sequences\<close> |
54681 | 303 |
|
304 |
inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" |
|
305 |
where |
|
306 |
id[simp]: "swapidseq 0 id" |
|
307 |
| comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id \<circ> p)" |
|
308 |
||
309 |
declare id[unfolded id_def, simp] |
|
310 |
||
311 |
definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
312 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
313 |
|
60500 | 314 |
subsection \<open>Some closure properties of the set of permutations, with lengths\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
315 |
|
54681 | 316 |
lemma permutation_id[simp]: "permutation id" |
317 |
unfolding permutation_def by (rule exI[where x=0]) simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
318 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
319 |
declare permutation_id[unfolded id_def, simp] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
320 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
321 |
lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
322 |
apply clarsimp |
54681 | 323 |
using comp_Suc[of 0 id a b] |
324 |
apply simp |
|
325 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
326 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
327 |
lemma permutation_swap_id: "permutation (Fun.swap a b id)" |
54681 | 328 |
apply (cases "a = b") |
329 |
apply simp_all |
|
330 |
unfolding permutation_def |
|
331 |
using swapidseq_swap[of a b] |
|
332 |
apply blast |
|
333 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
334 |
|
54681 | 335 |
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)" |
336 |
proof (induct n p arbitrary: m q rule: swapidseq.induct) |
|
337 |
case (id m q) |
|
338 |
then show ?case by simp |
|
339 |
next |
|
340 |
case (comp_Suc n p a b m q) |
|
341 |
have th: "Suc n + m = Suc (n + m)" |
|
342 |
by arith |
|
343 |
show ?case |
|
344 |
unfolding th comp_assoc |
|
345 |
apply (rule swapidseq.comp_Suc) |
|
346 |
using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3) |
|
347 |
apply blast+ |
|
348 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
349 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
350 |
|
54681 | 351 |
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
352 |
unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
353 |
|
54681 | 354 |
lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
355 |
apply (induct n p rule: swapidseq.induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
356 |
using swapidseq_swap[of a b] |
54681 | 357 |
apply (auto simp add: comp_assoc intro: swapidseq.comp_Suc) |
358 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
359 |
|
54681 | 360 |
lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id" |
361 |
proof (induct n p rule: swapidseq.induct) |
|
362 |
case id |
|
363 |
then show ?case |
|
364 |
by (rule exI[where x=id]) simp |
|
30488 | 365 |
next |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
366 |
case (comp_Suc n p a b) |
54681 | 367 |
from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" |
368 |
by blast |
|
369 |
let ?q = "q \<circ> Fun.swap a b id" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
370 |
note H = comp_Suc.hyps |
54681 | 371 |
from swapidseq_swap[of a b] H(3) have th0: "swapidseq 1 (Fun.swap a b id)" |
372 |
by simp |
|
373 |
from swapidseq_comp_add[OF q(1) th0] have th1: "swapidseq (Suc n) ?q" |
|
374 |
by simp |
|
375 |
have "Fun.swap a b id \<circ> p \<circ> ?q = Fun.swap a b id \<circ> (p \<circ> q) \<circ> Fun.swap a b id" |
|
376 |
by (simp add: o_assoc) |
|
377 |
also have "\<dots> = id" |
|
378 |
by (simp add: q(2)) |
|
379 |
finally have th2: "Fun.swap a b id \<circ> p \<circ> ?q = id" . |
|
380 |
have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id \<circ> Fun.swap a b id) \<circ> p" |
|
381 |
by (simp only: o_assoc) |
|
382 |
then have "?q \<circ> (Fun.swap a b id \<circ> p) = id" |
|
383 |
by (simp add: q(3)) |
|
384 |
with th1 th2 show ?case |
|
385 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
386 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
387 |
|
54681 | 388 |
lemma swapidseq_inverse: |
389 |
assumes H: "swapidseq n p" |
|
390 |
shows "swapidseq n (inv p)" |
|
391 |
using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto |
|
392 |
||
393 |
lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)" |
|
394 |
using permutation_def swapidseq_inverse by blast |
|
395 |
||
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
396 |
|
60500 | 397 |
subsection \<open>The identity map only has even transposition sequences\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
398 |
|
54681 | 399 |
lemma symmetry_lemma: |
400 |
assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c" |
|
401 |
and "\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> |
|
402 |
a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<Longrightarrow> |
|
403 |
P a b c d" |
|
404 |
shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> P a b c d" |
|
405 |
using assms by metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
406 |
|
54681 | 407 |
lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> |
408 |
Fun.swap a b id \<circ> Fun.swap c d id = id \<or> |
|
409 |
(\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> |
|
410 |
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)" |
|
411 |
proof - |
|
412 |
assume H: "a \<noteq> b" "c \<noteq> d" |
|
413 |
have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> |
|
414 |
(Fun.swap a b id \<circ> Fun.swap c d id = id \<or> |
|
415 |
(\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> |
|
416 |
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id))" |
|
417 |
apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d]) |
|
56545 | 418 |
apply (simp_all only: swap_commute) |
54681 | 419 |
apply (case_tac "a = c \<and> b = d") |
56608 | 420 |
apply (clarsimp simp only: swap_commute swap_id_idempotent) |
54681 | 421 |
apply (case_tac "a = c \<and> b \<noteq> d") |
422 |
apply (rule disjI2) |
|
423 |
apply (rule_tac x="b" in exI) |
|
424 |
apply (rule_tac x="d" in exI) |
|
425 |
apply (rule_tac x="b" in exI) |
|
56545 | 426 |
apply (clarsimp simp add: fun_eq_iff Fun.swap_def) |
54681 | 427 |
apply (case_tac "a \<noteq> c \<and> b = d") |
428 |
apply (rule disjI2) |
|
429 |
apply (rule_tac x="c" in exI) |
|
430 |
apply (rule_tac x="d" in exI) |
|
431 |
apply (rule_tac x="c" in exI) |
|
56545 | 432 |
apply (clarsimp simp add: fun_eq_iff Fun.swap_def) |
54681 | 433 |
apply (rule disjI2) |
434 |
apply (rule_tac x="c" in exI) |
|
435 |
apply (rule_tac x="d" in exI) |
|
436 |
apply (rule_tac x="b" in exI) |
|
56545 | 437 |
apply (clarsimp simp add: fun_eq_iff Fun.swap_def) |
54681 | 438 |
done |
439 |
with H show ?thesis by metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
440 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
441 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
442 |
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
443 |
using swapidseq.cases[of 0 p "p = id"] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
444 |
by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
445 |
|
54681 | 446 |
lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow> |
447 |
n = 0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id \<circ> q \<and> swapidseq m q \<and> a \<noteq> b)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
448 |
apply (rule iffI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
449 |
apply (erule swapidseq.cases[of n p]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
450 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
451 |
apply (rule disjI2) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
452 |
apply (rule_tac x= "a" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
453 |
apply (rule_tac x= "b" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
454 |
apply (rule_tac x= "pa" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
455 |
apply (rule_tac x= "na" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
456 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
457 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
458 |
apply (rule comp_Suc, simp_all) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
459 |
done |
54681 | 460 |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
461 |
lemma fixing_swapidseq_decrease: |
54681 | 462 |
assumes spn: "swapidseq n p" |
463 |
and ab: "a \<noteq> b" |
|
464 |
and pa: "(Fun.swap a b id \<circ> p) a = a" |
|
465 |
shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
466 |
using spn ab pa |
54681 | 467 |
proof (induct n arbitrary: p a b) |
468 |
case 0 |
|
469 |
then show ?case |
|
56545 | 470 |
by (auto simp add: Fun.swap_def fun_upd_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
471 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
472 |
case (Suc n p a b) |
54681 | 473 |
from Suc.prems(1) swapidseq_cases[of "Suc n" p] |
474 |
obtain c d q m where |
|
475 |
cdqm: "Suc n = Suc m" "p = Fun.swap c d id \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
476 |
by auto |
54681 | 477 |
{ |
478 |
assume H: "Fun.swap a b id \<circ> Fun.swap c d id = id" |
|
479 |
have ?case by (simp only: cdqm o_assoc H) (simp add: cdqm) |
|
480 |
} |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
481 |
moreover |
54681 | 482 |
{ |
483 |
fix x y z |
|
484 |
assume H: "x \<noteq> a" "y \<noteq> a" "z \<noteq> a" "x \<noteq> y" |
|
485 |
"Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id" |
|
486 |
from H have az: "a \<noteq> z" |
|
487 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
488 |
|
54681 | 489 |
{ |
490 |
fix h |
|
491 |
have "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a" |
|
56545 | 492 |
using H by (simp add: Fun.swap_def) |
54681 | 493 |
} |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
494 |
note th3 = this |
54681 | 495 |
from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)" |
496 |
by simp |
|
497 |
then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)" |
|
498 |
by (simp add: o_assoc H) |
|
499 |
then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a" |
|
500 |
by simp |
|
501 |
then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a" |
|
502 |
unfolding Suc by metis |
|
503 |
then have th1: "(Fun.swap a z id \<circ> q) a = a" |
|
504 |
unfolding th3 . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
505 |
from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1] |
54681 | 506 |
have th2: "swapidseq (n - 1) (Fun.swap a z id \<circ> q)" "n \<noteq> 0" |
507 |
by blast+ |
|
508 |
have th: "Suc n - 1 = Suc (n - 1)" |
|
509 |
using th2(2) by auto |
|
510 |
have ?case |
|
511 |
unfolding cdqm(2) H o_assoc th |
|
49739 | 512 |
apply (simp only: Suc_not_Zero simp_thms comp_assoc) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
513 |
apply (rule comp_Suc) |
54681 | 514 |
using th2 H |
515 |
apply blast+ |
|
516 |
done |
|
517 |
} |
|
518 |
ultimately show ?case |
|
519 |
using swap_general[OF Suc.prems(2) cdqm(4)] by metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
520 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
521 |
|
30488 | 522 |
lemma swapidseq_identity_even: |
54681 | 523 |
assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" |
524 |
shows "even n" |
|
60500 | 525 |
using \<open>swapidseq n id\<close> |
54681 | 526 |
proof (induct n rule: nat_less_induct) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
527 |
fix n |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
528 |
assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)" |
54681 | 529 |
{ |
530 |
assume "n = 0" |
|
531 |
then have "even n" by presburger |
|
532 |
} |
|
30488 | 533 |
moreover |
54681 | 534 |
{ |
535 |
fix a b :: 'a and q m |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
536 |
assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
537 |
from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]] |
54681 | 538 |
have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)" |
539 |
by auto |
|
540 |
from h m have mn: "m - 1 < n" |
|
541 |
by arith |
|
542 |
from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n" |
|
543 |
by presburger |
|
544 |
} |
|
545 |
ultimately show "even n" |
|
546 |
using H(2)[unfolded swapidseq_cases[of n id]] by auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
547 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
548 |
|
54681 | 549 |
|
60500 | 550 |
subsection \<open>Therefore we have a welldefined notion of parity\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
551 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
552 |
definition "evenperm p = even (SOME n. swapidseq n p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
553 |
|
54681 | 554 |
lemma swapidseq_even_even: |
555 |
assumes m: "swapidseq m p" |
|
556 |
and n: "swapidseq n p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
557 |
shows "even m \<longleftrightarrow> even n" |
54681 | 558 |
proof - |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
559 |
from swapidseq_inverse_exists[OF n] |
54681 | 560 |
obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" |
561 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
562 |
from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] |
54681 | 563 |
show ?thesis |
564 |
by arith |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
565 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
566 |
|
54681 | 567 |
lemma evenperm_unique: |
568 |
assumes p: "swapidseq n p" |
|
569 |
and n:"even n = b" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
570 |
shows "evenperm p = b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
571 |
unfolding n[symmetric] evenperm_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
572 |
apply (rule swapidseq_even_even[where p = p]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
573 |
apply (rule someI[where x = n]) |
54681 | 574 |
using p |
575 |
apply blast+ |
|
576 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
577 |
|
54681 | 578 |
|
60500 | 579 |
subsection \<open>And it has the expected composition properties\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
580 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
581 |
lemma evenperm_id[simp]: "evenperm id = True" |
54681 | 582 |
by (rule evenperm_unique[where n = 0]) simp_all |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
583 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
584 |
lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)" |
54681 | 585 |
by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
586 |
|
30488 | 587 |
lemma evenperm_comp: |
54681 | 588 |
assumes p: "permutation p" |
589 |
and q:"permutation q" |
|
590 |
shows "evenperm (p \<circ> q) = (evenperm p = evenperm q)" |
|
591 |
proof - |
|
592 |
from p q obtain n m where n: "swapidseq n p" and m: "swapidseq m q" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
593 |
unfolding permutation_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
594 |
note nm = swapidseq_comp_add[OF n m] |
54681 | 595 |
have th: "even (n + m) = (even n \<longleftrightarrow> even m)" |
596 |
by arith |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
597 |
from evenperm_unique[OF n refl] evenperm_unique[OF m refl] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
598 |
evenperm_unique[OF nm th] |
54681 | 599 |
show ?thesis |
600 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
601 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
602 |
|
54681 | 603 |
lemma evenperm_inv: |
604 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
605 |
shows "evenperm (inv p) = evenperm p" |
54681 | 606 |
proof - |
607 |
from p obtain n where n: "swapidseq n p" |
|
608 |
unfolding permutation_def by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
609 |
from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
610 |
show ?thesis . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
611 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
612 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
613 |
|
60500 | 614 |
subsection \<open>A more abstract characterization of permutations\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
615 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
616 |
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
617 |
unfolding bij_def inj_on_def surj_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
618 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
619 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
620 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
621 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
622 |
|
30488 | 623 |
lemma permutation_bijective: |
624 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
625 |
shows "bij p" |
54681 | 626 |
proof - |
627 |
from p obtain n where n: "swapidseq n p" |
|
628 |
unfolding permutation_def by blast |
|
629 |
from swapidseq_inverse_exists[OF n] |
|
630 |
obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" |
|
631 |
by blast |
|
632 |
then show ?thesis unfolding bij_iff |
|
633 |
apply (auto simp add: fun_eq_iff) |
|
634 |
apply metis |
|
635 |
done |
|
30488 | 636 |
qed |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
637 |
|
54681 | 638 |
lemma permutation_finite_support: |
639 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
640 |
shows "finite {x. p x \<noteq> x}" |
54681 | 641 |
proof - |
642 |
from p obtain n where n: "swapidseq n p" |
|
643 |
unfolding permutation_def by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
644 |
from n show ?thesis |
54681 | 645 |
proof (induct n p rule: swapidseq.induct) |
646 |
case id |
|
647 |
then show ?case by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
648 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
649 |
case (comp_Suc n p a b) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
650 |
let ?S = "insert a (insert b {x. p x \<noteq> x})" |
54681 | 651 |
from comp_Suc.hyps(2) have fS: "finite ?S" |
652 |
by simp |
|
60500 | 653 |
from \<open>a \<noteq> b\<close> have th: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S" |
56545 | 654 |
by (auto simp add: Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
655 |
from finite_subset[OF th fS] show ?case . |
54681 | 656 |
qed |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
657 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
658 |
|
54681 | 659 |
lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y" |
660 |
using surj_f_inv_f[of p] by (auto simp add: bij_def) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
661 |
|
30488 | 662 |
lemma bij_swap_comp: |
54681 | 663 |
assumes bp: "bij p" |
664 |
shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
665 |
using surj_f_inv_f[OF bij_is_surj[OF bp]] |
56545 | 666 |
by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF bp]) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
667 |
|
54681 | 668 |
lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id \<circ> p)" |
669 |
proof - |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
670 |
assume H: "bij p" |
30488 | 671 |
show ?thesis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
672 |
unfolding bij_swap_comp[OF H] bij_swap_iff |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
673 |
using H . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
674 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
675 |
|
30488 | 676 |
lemma permutation_lemma: |
54681 | 677 |
assumes fS: "finite S" |
678 |
and p: "bij p" |
|
679 |
and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
680 |
shows "permutation p" |
54681 | 681 |
using fS p pS |
682 |
proof (induct S arbitrary: p rule: finite_induct) |
|
683 |
case (empty p) |
|
684 |
then show ?case by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
685 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
686 |
case (insert a F p) |
54681 | 687 |
let ?r = "Fun.swap a (p a) id \<circ> p" |
688 |
let ?q = "Fun.swap a (p a) id \<circ> ?r" |
|
689 |
have raa: "?r a = a" |
|
56545 | 690 |
by (simp add: Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
691 |
from bij_swap_ompose_bij[OF insert(4)] |
30488 | 692 |
have br: "bij ?r" . |
693 |
||
694 |
from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x" |
|
56545 | 695 |
apply (clarsimp simp add: Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
696 |
apply (erule_tac x="x" in allE) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
697 |
apply auto |
54681 | 698 |
unfolding bij_iff |
699 |
apply metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
700 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
701 |
from insert(3)[OF br th] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
702 |
have rp: "permutation ?r" . |
54681 | 703 |
have "permutation ?q" |
704 |
by (simp add: permutation_compose permutation_swap_id rp) |
|
705 |
then show ?case |
|
706 |
by (simp add: o_assoc) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
707 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
708 |
|
30488 | 709 |
lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
710 |
(is "?lhs \<longleftrightarrow> ?b \<and> ?f") |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
711 |
proof |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
712 |
assume p: ?lhs |
54681 | 713 |
from p permutation_bijective permutation_finite_support show "?b \<and> ?f" |
714 |
by auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
715 |
next |
54681 | 716 |
assume "?b \<and> ?f" |
717 |
then have "?f" "?b" by blast+ |
|
718 |
from permutation_lemma[OF this] show ?lhs |
|
719 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
720 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
721 |
|
54681 | 722 |
lemma permutation_inverse_works: |
723 |
assumes p: "permutation p" |
|
724 |
shows "inv p \<circ> p = id" |
|
725 |
and "p \<circ> inv p = id" |
|
44227
78e033e8ba05
get Library/Permutations.thy compiled and working again
huffman
parents:
41959
diff
changeset
|
726 |
using permutation_bijective [OF p] |
78e033e8ba05
get Library/Permutations.thy compiled and working again
huffman
parents:
41959
diff
changeset
|
727 |
unfolding bij_def inj_iff surj_iff by auto |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
728 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
729 |
lemma permutation_inverse_compose: |
54681 | 730 |
assumes p: "permutation p" |
731 |
and q: "permutation q" |
|
732 |
shows "inv (p \<circ> q) = inv q \<circ> inv p" |
|
733 |
proof - |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
734 |
note ps = permutation_inverse_works[OF p] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
735 |
note qs = permutation_inverse_works[OF q] |
54681 | 736 |
have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p" |
737 |
by (simp add: o_assoc) |
|
738 |
also have "\<dots> = id" |
|
739 |
by (simp add: ps qs) |
|
740 |
finally have th0: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" . |
|
741 |
have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q" |
|
742 |
by (simp add: o_assoc) |
|
743 |
also have "\<dots> = id" |
|
744 |
by (simp add: ps qs) |
|
745 |
finally have th1: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
746 |
from inv_unique_comp[OF th0 th1] show ?thesis . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
747 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
748 |
|
54681 | 749 |
|
60500 | 750 |
subsection \<open>Relation to "permutes"\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
751 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
752 |
lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)" |
54681 | 753 |
unfolding permutation permutes_def bij_iff[symmetric] |
754 |
apply (rule iffI, clarify) |
|
755 |
apply (rule exI[where x="{x. p x \<noteq> x}"]) |
|
756 |
apply simp |
|
757 |
apply clarsimp |
|
758 |
apply (rule_tac B="S" in finite_subset) |
|
759 |
apply auto |
|
760 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
761 |
|
54681 | 762 |
|
60500 | 763 |
subsection \<open>Hence a sort of induction principle composing by swaps\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
764 |
|
54681 | 765 |
lemma permutes_induct: "finite S \<Longrightarrow> P id \<Longrightarrow> |
766 |
(\<And> a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p \<Longrightarrow> P (Fun.swap a b id \<circ> p)) \<Longrightarrow> |
|
767 |
(\<And>p. p permutes S \<Longrightarrow> P p)" |
|
768 |
proof (induct S rule: finite_induct) |
|
769 |
case empty |
|
770 |
then show ?case by auto |
|
30488 | 771 |
next |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
772 |
case (insert x F p) |
54681 | 773 |
let ?r = "Fun.swap x (p x) id \<circ> p" |
774 |
let ?q = "Fun.swap x (p x) id \<circ> ?r" |
|
775 |
have qp: "?q = p" |
|
776 |
by (simp add: o_assoc) |
|
777 |
from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" |
|
778 |
by blast |
|
30488 | 779 |
from permutes_in_image[OF insert.prems(3), of x] |
54681 | 780 |
have pxF: "p x \<in> insert x F" |
781 |
by simp |
|
782 |
have xF: "x \<in> insert x F" |
|
783 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
784 |
have rp: "permutation ?r" |
30488 | 785 |
unfolding permutation_permutes using insert.hyps(1) |
54681 | 786 |
permutes_insert_lemma[OF insert.prems(3)] |
787 |
by blast |
|
30488 | 788 |
from insert.prems(2)[OF xF pxF Pr Pr rp] |
54681 | 789 |
show ?case |
790 |
unfolding qp . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
791 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
792 |
|
54681 | 793 |
|
60500 | 794 |
subsection \<open>Sign of a permutation as a real number\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
795 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
796 |
definition "sign p = (if evenperm p then (1::int) else -1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
797 |
|
54681 | 798 |
lemma sign_nz: "sign p \<noteq> 0" |
799 |
by (simp add: sign_def) |
|
800 |
||
801 |
lemma sign_id: "sign id = 1" |
|
802 |
by (simp add: sign_def) |
|
803 |
||
804 |
lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
805 |
by (simp add: sign_def evenperm_inv) |
54681 | 806 |
|
807 |
lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q" |
|
808 |
by (simp add: sign_def evenperm_comp) |
|
809 |
||
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
810 |
lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
811 |
by (simp add: sign_def evenperm_swap) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
812 |
|
54681 | 813 |
lemma sign_idempotent: "sign p * sign p = 1" |
814 |
by (simp add: sign_def) |
|
815 |
||
816 |
||
60500 | 817 |
subsection \<open>More lemmas about permutations\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
818 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
819 |
lemma permutes_natset_le: |
54681 | 820 |
fixes S :: "'a::wellorder set" |
821 |
assumes p: "p permutes S" |
|
822 |
and le: "\<forall>i \<in> S. p i \<le> i" |
|
823 |
shows "p = id" |
|
824 |
proof - |
|
825 |
{ |
|
826 |
fix n |
|
30488 | 827 |
have "p n = n" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
828 |
using p le |
54681 | 829 |
proof (induct n arbitrary: S rule: less_induct) |
830 |
fix n S |
|
831 |
assume H: |
|
832 |
"\<And>m S. m < n \<Longrightarrow> p permutes S \<Longrightarrow> \<forall>i\<in>S. p i \<le> i \<Longrightarrow> p m = m" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
833 |
"p permutes S" "\<forall>i \<in>S. p i \<le> i" |
54681 | 834 |
{ |
835 |
assume "n \<notin> S" |
|
836 |
with H(2) have "p n = n" |
|
837 |
unfolding permutes_def by metis |
|
838 |
} |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
839 |
moreover |
54681 | 840 |
{ |
841 |
assume ns: "n \<in> S" |
|
842 |
from H(3) ns have "p n < n \<or> p n = n" |
|
843 |
by auto |
|
844 |
moreover { |
|
845 |
assume h: "p n < n" |
|
846 |
from H h have "p (p n) = p n" |
|
847 |
by metis |
|
848 |
with permutes_inj[OF H(2)] have "p n = n" |
|
849 |
unfolding inj_on_def by blast |
|
850 |
with h have False |
|
851 |
by simp |
|
852 |
} |
|
853 |
ultimately have "p n = n" |
|
854 |
by blast |
|
855 |
} |
|
856 |
ultimately show "p n = n" |
|
857 |
by blast |
|
858 |
qed |
|
859 |
} |
|
860 |
then show ?thesis |
|
861 |
by (auto simp add: fun_eq_iff) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
862 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
863 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
864 |
lemma permutes_natset_ge: |
54681 | 865 |
fixes S :: "'a::wellorder set" |
866 |
assumes p: "p permutes S" |
|
867 |
and le: "\<forall>i \<in> S. p i \<ge> i" |
|
868 |
shows "p = id" |
|
869 |
proof - |
|
870 |
{ |
|
871 |
fix i |
|
872 |
assume i: "i \<in> S" |
|
873 |
from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S" |
|
874 |
by simp |
|
875 |
with le have "p (inv p i) \<ge> inv p i" |
|
876 |
by blast |
|
877 |
with permutes_inverses[OF p] have "i \<ge> inv p i" |
|
878 |
by simp |
|
879 |
} |
|
880 |
then have th: "\<forall>i\<in>S. inv p i \<le> i" |
|
881 |
by blast |
|
30488 | 882 |
from permutes_natset_le[OF permutes_inv[OF p] th] |
54681 | 883 |
have "inv p = inv id" |
884 |
by simp |
|
30488 | 885 |
then show ?thesis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
886 |
apply (subst permutes_inv_inv[OF p, symmetric]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
887 |
apply (rule inv_unique_comp) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
888 |
apply simp_all |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
889 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
890 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
891 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
892 |
lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}" |
54681 | 893 |
apply (rule set_eqI) |
894 |
apply auto |
|
895 |
using permutes_inv_inv permutes_inv |
|
896 |
apply auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
897 |
apply (rule_tac x="inv x" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
898 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
899 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
900 |
|
30488 | 901 |
lemma image_compose_permutations_left: |
54681 | 902 |
assumes q: "q permutes S" |
903 |
shows "{q \<circ> p | p. p permutes S} = {p . p permutes S}" |
|
904 |
apply (rule set_eqI) |
|
905 |
apply auto |
|
906 |
apply (rule permutes_compose) |
|
907 |
using q |
|
908 |
apply auto |
|
909 |
apply (rule_tac x = "inv q \<circ> x" in exI) |
|
910 |
apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o) |
|
911 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
912 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
913 |
lemma image_compose_permutations_right: |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
914 |
assumes q: "q permutes S" |
54681 | 915 |
shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}" |
916 |
apply (rule set_eqI) |
|
917 |
apply auto |
|
918 |
apply (rule permutes_compose) |
|
919 |
using q |
|
920 |
apply auto |
|
921 |
apply (rule_tac x = "x \<circ> inv q" in exI) |
|
922 |
apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc) |
|
923 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
924 |
|
54681 | 925 |
lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} \<Longrightarrow> 1 \<le> p i \<and> p i \<le> n" |
926 |
by (simp add: permutes_def) metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
927 |
|
54681 | 928 |
lemma setsum_permutations_inverse: |
929 |
"setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" |
|
930 |
(is "?lhs = ?rhs") |
|
931 |
proof - |
|
30036 | 932 |
let ?S = "{p . p permutes S}" |
54681 | 933 |
have th0: "inj_on inv ?S" |
934 |
proof (auto simp add: inj_on_def) |
|
935 |
fix q r |
|
936 |
assume q: "q permutes S" |
|
937 |
and r: "r permutes S" |
|
938 |
and qr: "inv q = inv r" |
|
939 |
then have "inv (inv q) = inv (inv r)" |
|
940 |
by simp |
|
941 |
with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r" |
|
942 |
by metis |
|
943 |
qed |
|
944 |
have th1: "inv ` ?S = ?S" |
|
945 |
using image_inverse_permutations by blast |
|
946 |
have th2: "?rhs = setsum (f \<circ> inv) ?S" |
|
947 |
by (simp add: o_def) |
|
57418 | 948 |
from setsum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 . |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
949 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
950 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
951 |
lemma setum_permutations_compose_left: |
30036 | 952 |
assumes q: "q permutes S" |
54681 | 953 |
shows "setsum f {p. p permutes S} = setsum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}" |
954 |
(is "?lhs = ?rhs") |
|
955 |
proof - |
|
30036 | 956 |
let ?S = "{p. p permutes S}" |
54681 | 957 |
have th0: "?rhs = setsum (f \<circ> (op \<circ> q)) ?S" |
958 |
by (simp add: o_def) |
|
959 |
have th1: "inj_on (op \<circ> q) ?S" |
|
960 |
proof (auto simp add: inj_on_def) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
961 |
fix p r |
54681 | 962 |
assume "p permutes S" |
963 |
and r: "r permutes S" |
|
964 |
and rp: "q \<circ> p = q \<circ> r" |
|
965 |
then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r" |
|
966 |
by (simp add: comp_assoc) |
|
967 |
with permutes_inj[OF q, unfolded inj_iff] show "p = r" |
|
968 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
969 |
qed |
54681 | 970 |
have th3: "(op \<circ> q) ` ?S = ?S" |
971 |
using image_compose_permutations_left[OF q] by auto |
|
57418 | 972 |
from setsum.reindex[OF th1, of f] show ?thesis unfolding th0 th1 th3 . |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
973 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
974 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
975 |
lemma sum_permutations_compose_right: |
30036 | 976 |
assumes q: "q permutes S" |
54681 | 977 |
shows "setsum f {p. p permutes S} = setsum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}" |
978 |
(is "?lhs = ?rhs") |
|
979 |
proof - |
|
30036 | 980 |
let ?S = "{p. p permutes S}" |
54681 | 981 |
have th0: "?rhs = setsum (f \<circ> (\<lambda>p. p \<circ> q)) ?S" |
982 |
by (simp add: o_def) |
|
983 |
have th1: "inj_on (\<lambda>p. p \<circ> q) ?S" |
|
984 |
proof (auto simp add: inj_on_def) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
985 |
fix p r |
54681 | 986 |
assume "p permutes S" |
987 |
and r: "r permutes S" |
|
988 |
and rp: "p \<circ> q = r \<circ> q" |
|
989 |
then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)" |
|
990 |
by (simp add: o_assoc) |
|
991 |
with permutes_surj[OF q, unfolded surj_iff] show "p = r" |
|
992 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
993 |
qed |
54681 | 994 |
have th3: "(\<lambda>p. p \<circ> q) ` ?S = ?S" |
995 |
using image_compose_permutations_right[OF q] by auto |
|
57418 | 996 |
from setsum.reindex[OF th1, of f] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
997 |
show ?thesis unfolding th0 th1 th3 . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
998 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
999 |
|
54681 | 1000 |
|
60500 | 1001 |
subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1002 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1003 |
lemma setsum_over_permutations_insert: |
54681 | 1004 |
assumes fS: "finite S" |
1005 |
and aS: "a \<notin> S" |
|
1006 |
shows "setsum f {p. p permutes (insert a S)} = |
|
1007 |
setsum (\<lambda>b. setsum (\<lambda>q. f (Fun.swap a b id \<circ> q)) {p. p permutes S}) (insert a S)" |
|
1008 |
proof - |
|
1009 |
have th0: "\<And>f a b. (\<lambda>(b,p). f (Fun.swap a b id \<circ> p)) = f \<circ> (\<lambda>(b,p). Fun.swap a b id \<circ> p)" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1010 |
by (simp add: fun_eq_iff) |
54681 | 1011 |
have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" |
1012 |
by blast |
|
1013 |
have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" |
|
1014 |
by blast |
|
30488 | 1015 |
show ?thesis |
1016 |
unfolding permutes_insert |
|
57418 | 1017 |
unfolding setsum.cartesian_product |
57129
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56608
diff
changeset
|
1018 |
unfolding th1[symmetric] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1019 |
unfolding th0 |
57418 | 1020 |
proof (rule setsum.reindex) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1021 |
let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1022 |
let ?P = "{p. p permutes S}" |
54681 | 1023 |
{ |
1024 |
fix b c p q |
|
1025 |
assume b: "b \<in> insert a S" |
|
1026 |
assume c: "c \<in> insert a S" |
|
1027 |
assume p: "p permutes S" |
|
1028 |
assume q: "q permutes S" |
|
1029 |
assume eq: "Fun.swap a b id \<circ> p = Fun.swap a c id \<circ> q" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1030 |
from p q aS have pa: "p a = a" and qa: "q a = a" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
1031 |
unfolding permutes_def by metis+ |
54681 | 1032 |
from eq have "(Fun.swap a b id \<circ> p) a = (Fun.swap a c id \<circ> q) a" |
1033 |
by simp |
|
1034 |
then have bc: "b = c" |
|
56545 | 1035 |
by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def |
54681 | 1036 |
cong del: if_weak_cong split: split_if_asm) |
1037 |
from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> p) = |
|
1038 |
(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> q)" by simp |
|
1039 |
then have "p = q" |
|
1040 |
unfolding o_assoc swap_id_idempotent |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
1041 |
by (simp add: o_def) |
54681 | 1042 |
with bc have "b = c \<and> p = q" |
1043 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1044 |
} |
30488 | 1045 |
then show "inj_on ?f (insert a S \<times> ?P)" |
54681 | 1046 |
unfolding inj_on_def by clarify metis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1047 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1048 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1049 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1050 |
end |
51489 | 1051 |