Nuprl Lemma : list-to-set-cons
∀[T:Type]
∀eq:EqDecider(T). ∀L:T List. ∀a:T.
(list-to-set(eq;[a / L]) ~ if a ∈b list-to-set(eq;L) then list-to-set(eq;L) else [a / list-to-set(eq;L)] fi )
Proof
Definitions occuring in Statement :
list-to-set: list-to-set(eq;L),
deq-member: x ∈b L,
cons: [a / b],
list: T List,
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
list-to-set: list-to-set(eq;L),
l-union: as ⋃ bs,
top: Top,
insert: insert(a;L),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
has-value: (a)↓
Lemmas referenced :
reduce_cons_lemma,
eval_list_sq,
reduce_wf,
list_wf,
insert_wf,
nil_wf,
subtype_rel_list,
top_wf,
value-type-has-value,
list-value-type,
list-to-set_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
cumulativity,
hypothesisEquality,
because_Cache,
lambdaEquality,
applyEquality,
independent_isectElimination,
callbyvalueReduce,
sqequalAxiom,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}L:T List. \mforall{}a:T.
(list-to-set(eq;[a / L]) \msim{} if a \mmember{}\msubb{} list-to-set(eq;L)
then list-to-set(eq;L)
else [a / list-to-set(eq;L)]
fi )
Date html generated:
2016_05_14-PM-03_25_46
Last ObjectModification:
2015_12_26-PM-06_22_44
Theory : decidable!equality
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