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This instance will be upgraded to Heptapod 17.9.2 on 2025-04-24 between 17:00 and 18:00 UTC+2
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isa-afp
afp-devel
Commits
8c2b5b3c
Commit
8c2b5b3c
authored
15 years ago
by
kleing
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fix equinumerous definition (by Brian Huffman)
parent
8f950ec8
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thys/Category/Yoneda.thy
+15
-4
15 additions, 4 deletions
thys/Category/Yoneda.thy
with
15 additions
and
4 deletions
thys/Category/Yoneda.thy
+
15
−
4
View file @
8c2b5b3c
...
...
@@ -336,8 +336,13 @@
definition
equinumerous
::
"['a set, 'b set]
\<Rightarrow>
bool"
(
infix
"
\<cong>
"
40
)
where
"equinumerous A B
\<longleftrightarrow>
(
\<exists>
f. bij_on f A B)"
"equinumerous A B
\<longleftrightarrow>
(
\<exists>
f. bij_betw f A B)"
lemma
bij_betw_eq
:
"bij_betw f A B
\<longleftrightarrow>
inj_on f A
\<and>
(
\<forall>
y
\<in>
B.
\<exists>
x
\<in>
A. f(x)=y)
\<and>
(
\<forall>
x
\<in>
A. f x
\<in>
B)"
unfolding
bij_betw_def
by
auto
theorem
(
in
Yoneda
)
Yoneda
:
assumes
1
:
"A
\<in>
Ob"
shows
"F\<^sub>
\<o>
A
\<cong>
{u. u : Hom(A,_)
\<Rightarrow>
F in Func(AA,Set)}"
...
...
@@ -340,10 +345,9 @@
theorem
(
in
Yoneda
)
Yoneda
:
assumes
1
:
"A
\<in>
Ob"
shows
"F\<^sub>
\<o>
A
\<cong>
{u. u : Hom(A,_)
\<Rightarrow>
F in Func(AA,Set)}"
apply
(
unfold
equinumerous_def
bij_on_def
surj_on_def
inj_on_def
)
apply
(
intro
exI
conjI
bexI
ballI
impI
)
proof
-
unfolding
equinumerous_def
bij_betw_eq
inj_on_def
proof
(
intro
exI
conjI
bexI
ballI
impI
)
--
"Sandwich is injective"
fix
x
and
y
assume
2
:
"x
\<in>
F\<^sub>
\<o>
A"
and
3
:
"y
\<in>
F\<^sub>
\<o>
A"
...
...
@@ -373,6 +377,13 @@
with
uAfuncset
show
"
\<sigma>
\<^sup>
\<leftarrow>
(A,u)
\<in>
F\<^sub>
\<o>
A"
by
(
simp
add
:
unsandwich_def
,
rule
funcset_mem
)
next
fix
x
assume
"x
\<in>
F \<^sub>
\<o>
A"
with
1
have
"
\<sigma>
(A,x) : Hom(A,_)
\<Rightarrow>
F in Func (AA,Set)"
by
(
rule
sandwich_natural
)
thus
"
\<sigma>
(A,x)
\<in>
{y. y : Hom(A,_)
\<Rightarrow>
F in Func (AA,Set)}"
by
simp
qed
end
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