Nuprl Lemma : lookup-list-map-isEmpty-prop
∀[Key,Value:Type]. ∀[deqKey:EqDecider(Key)]. ∀[m:lookup-list-map-type(Key;Value)].
(↑lookup-list-map-isEmpty(m) ⇐⇒ ∀k:Key. (¬↑lookup-list-map-inDom(deqKey;k;m)))
Proof
Definitions occuring in Statement :
lookup-list-map-isEmpty: lookup-list-map-isEmpty(m),
lookup-list-map-inDom: lookup-list-map-inDom(deqKey;key;m),
lookup-list-map-type: lookup-list-map-type(Key;Value),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
universe: Type
Definitions unfolded in proof :
lookup-list-map-type: lookup-list-map-type(Key;Value),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
lookup-list-map-inDom: lookup-list-map-inDom(deqKey;key;m),
lookup-list-map-isEmpty: lookup-list-map-isEmpty(m),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
true: True,
subtype_rel: A ⊆r B,
listp: A List+,
or: P ∨ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
eqof: eqof(d)
Latex:
\mforall{}[Key,Value:Type]. \mforall{}[deqKey:EqDecider(Key)]. \mforall{}[m:lookup-list-map-type(Key;Value)].
(\muparrow{}lookup-list-map-isEmpty(m) \mLeftarrow{}{}\mRightarrow{} \mforall{}k:Key. (\mneg{}\muparrow{}lookup-list-map-inDom(deqKey;k;m)))
Date html generated:
2016_05_17-PM-01_51_10
Last ObjectModification:
2015_12_28-PM-08_50_43
Theory : datatype-signatures
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