Nuprl Lemma : lookup-list-map-add-prop
∀[Key,Value:Type]. ∀[deqKey:EqDecider(Key)]. ∀[key2:Key]. ∀[val:Value]. ∀[m:lookup-list-map-type(Key;Value)].
∀[key1:Key].
(lookup-list-map-find(deqKey;key1;lookup-list-map-add(deqKey;key2;val;m))
= if (deqKey key1 key2) ∧b (¬blookup-list-map-inDom(deqKey;key2;m))
then inl val
else lookup-list-map-find(deqKey;key1;m)
fi
∈ (Value?))
Proof
Definitions occuring in Statement :
lookup-list-map-add: lookup-list-map-add(deqKey;key;val;m),
lookup-list-map-inDom: lookup-list-map-inDom(deqKey;key;m),
lookup-list-map-find: lookup-list-map-find(deqKey;key;m),
lookup-list-map-type: lookup-list-map-type(Key;Value),
deq: EqDecider(T),
band: p ∧b q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
unit: Unit,
apply: f a,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
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],
deq: EqDecider(T),
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
lookup-list-map-find: lookup-list-map-find(deqKey;key;m),
lookup-list-map-add: lookup-list-map-add(deqKey;key;val;m),
apply-alist: apply-alist(eq;L;x),
update-alist: update-alist(eq;L;x;z;v.f[v]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
pi1: fst(t),
pi2: snd(t),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bnot: ¬bb,
lookup-list-map-inDom: lookup-list-map-inDom(deqKey;key;m),
list_ind: list_ind,
nil: [],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
false: False,
not: ¬A,
bor: p ∨bq
Latex:
\mforall{}[Key,Value:Type]. \mforall{}[deqKey:EqDecider(Key)]. \mforall{}[key2:Key]. \mforall{}[val:Value].
\mforall{}[m:lookup-list-map-type(Key;Value)]. \mforall{}[key1:Key].
(lookup-list-map-find(deqKey;key1;lookup-list-map-add(deqKey;key2;val;m))
= if (deqKey key1 key2) \mwedge{}\msubb{} (\mneg{}\msubb{}lookup-list-map-inDom(deqKey;key2;m))
then inl val
else lookup-list-map-find(deqKey;key1;m)
fi )
Date html generated:
2016_05_17-PM-01_51_29
Last ObjectModification:
2015_12_28-PM-08_51_53
Theory : datatype-signatures
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