theory Support
imports "../Nominal"
begin
text {*
An example showing that in general
x\<sharp>(A \<union> B) does not imply x\<sharp>A and x\<sharp>B
For this we set A to the set of even atoms and B to
the set of odd atoms. Then A \<union> B, that is the set of
all atoms, has empty support. The sets A, respectively B,
however have the set of all atoms as their support.
*}
atom_decl atom
text {* The set of even atoms. *}
abbreviation
EVEN :: "atom set"
where
"EVEN \<equiv> {atom n | n. \<exists>i. n=2*i}"
text {* The set of odd atoms: *}
abbreviation
ODD :: "atom set"
where
"ODD \<equiv> {atom n | n. \<exists>i. n=2*i+1}"
text {* An atom is either even or odd. *}
lemma even_or_odd:
fixes n :: nat
shows "\<exists>i. (n = 2*i) \<or> (n=2*i+1)"
by (induct n) (presburger)+
text {*
The union of even and odd atoms is the set of all atoms.
(Unfortunately I do not know a simpler proof of this fact.) *}
lemma EVEN_union_ODD:
shows "EVEN \<union> ODD = UNIV"
using even_or_odd
proof -
have "EVEN \<union> ODD = (\<lambda>n. atom n) ` {n. \<exists>i. n = 2*i} \<union> (\<lambda>n. atom n) ` {n. \<exists>i. n = 2*i+1}" by auto
also have "\<dots> = (\<lambda>n. atom n) ` ({n. \<exists>i. n = 2*i} \<union> {n. \<exists>i. n = 2*i+1})" by auto
also have "\<dots> = (\<lambda>n. atom n) ` ({n. \<exists>i. n = 2*i \<or> n = 2*i+1})" by auto
also have "\<dots> = (\<lambda>n. atom n) ` (UNIV::nat set)" using even_or_odd by auto
also have "\<dots> = (UNIV::atom set)" using atom.exhaust
by (auto simp add: surj_def)
finally show "EVEN \<union> ODD = UNIV" by simp
qed
text {* The sets of even and odd atoms are disjunct. *}
lemma EVEN_intersect_ODD:
shows "EVEN \<inter> ODD = {}"
using even_or_odd
by (auto) (presburger)
text {*
The preceeding two lemmas help us to prove
the following two useful equalities: *}
lemma UNIV_subtract:
shows "UNIV - EVEN = ODD"
and "UNIV - ODD = EVEN"
using EVEN_union_ODD EVEN_intersect_ODD
by (blast)+
text {* The sets EVEN and ODD are infinite. *}
lemma EVEN_ODD_infinite:
shows "infinite EVEN"
and "infinite ODD"
unfolding infinite_iff_countable_subset
proof -
let ?f = "\<lambda>n. atom (2*n)"
have "inj ?f \<and> range ?f \<subseteq> EVEN" by (auto simp add: inj_on_def)
then show "\<exists>f::nat\<Rightarrow>atom. inj f \<and> range f \<subseteq> EVEN" by (rule_tac exI)
next
let ?f = "\<lambda>n. atom (2*n+1)"
have "inj ?f \<and> range ?f \<subseteq> ODD" by (auto simp add: inj_on_def)
then show "\<exists>f::nat\<Rightarrow>atom. inj f \<and> range f \<subseteq> ODD" by (rule_tac exI)
qed
text {*
A general fact about a set S of atoms that is both infinite and
coinfinite. Then S has all atoms as its support. Steve Zdancewic
helped with proving this fact. *}
lemma supp_infinite_coinfinite:
fixes S::"atom set"
assumes asm1: "infinite S"
and asm2: "infinite (UNIV-S)"
shows "(supp S) = (UNIV::atom set)"
proof -
have "\<forall>(x::atom). x\<in>(supp S)"
proof
fix x::"atom"
show "x\<in>(supp S)"
proof (cases "x\<in>S")
case True
have "x\<in>S" by fact
hence "\<forall>b\<in>(UNIV-S). [(x,b)]\<bullet>S\<noteq>S" by (auto simp add: perm_set_eq calc_atm)
with asm2 have "infinite {b\<in>(UNIV-S). [(x,b)]\<bullet>S\<noteq>S}" by (rule infinite_Collection)
hence "infinite {b. [(x,b)]\<bullet>S\<noteq>S}" by (rule_tac infinite_super, auto)
then show "x\<in>(supp S)" by (simp add: supp_def)
next
case False
have "x\<notin>S" by fact
hence "\<forall>b\<in>S. [(x,b)]\<bullet>S\<noteq>S" by (auto simp add: perm_set_eq calc_atm)
with asm1 have "infinite {b\<in>S. [(x,b)]\<bullet>S\<noteq>S}" by (rule infinite_Collection)
hence "infinite {b. [(x,b)]\<bullet>S\<noteq>S}" by (rule_tac infinite_super, auto)
then show "x\<in>(supp S)" by (simp add: supp_def)
qed
qed
then show "(supp S) = (UNIV::atom set)" by auto
qed
text {* As a corollary we get that EVEN and ODD have infinite support. *}
lemma EVEN_ODD_supp:
shows "supp EVEN = (UNIV::atom set)"
and "supp ODD = (UNIV::atom set)"
using supp_infinite_coinfinite UNIV_subtract EVEN_ODD_infinite
by simp_all
text {*
The set of all atoms has empty support, since any swappings leaves
this set unchanged. *}
lemma UNIV_supp:
shows "supp (UNIV::atom set) = ({}::atom set)"
proof -
have "\<forall>(x::atom) (y::atom). [(x,y)]\<bullet>UNIV = (UNIV::atom set)"
by (auto simp add: perm_set_eq calc_atm)
then show "supp (UNIV::atom set) = ({}::atom set)" by (simp add: supp_def)
qed
text {* Putting everything together. *}
theorem EVEN_ODD_freshness:
fixes x::"atom"
shows "x\<sharp>(EVEN \<union> ODD)"
and "\<not>x\<sharp>EVEN"
and "\<not>x\<sharp>ODD"
by (auto simp only: fresh_def EVEN_union_ODD EVEN_ODD_supp UNIV_supp)
text {* Moral: support is a sublte notion. *}
end