Nuprl Lemma : member-values-for-distinct2
∀[A,V:Type]. ∀eq:EqDecider(A). ∀L:(A × V) List. ∀v:V. ((v ∈ values-for-distinct(eq;L)) ⇒ (∃a:A. (<a, v> ∈ L)))
Proof
Definitions occuring in Statement :
values-for-distinct: values-for-distinct(eq;L),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
values-for-distinct: values-for-distinct(eq;L),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
isl: isl(x),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
istype-universe,
deq_wf,
list_wf,
values-for-distinct_wf,
apply-alist_wf,
unit_wf2,
outl_wf,
list-subtype,
pi1_wf,
map_wf,
remove-repeats_wf,
l_member_wf,
member_map,
member-remove-repeats,
isl-apply-alist,
sq_stable_from_decidable,
assert_wf,
btrue_wf,
bfalse_wf,
decidable__assert
Rules used in proof :
universeEquality,
instantiate,
setIsType,
independent_isectElimination,
because_Cache,
rename,
setElimination,
universeIsType,
dependent_functionElimination,
productIsType,
hypothesis,
inhabitedIsType,
sqequalRule,
lambdaEquality_alt,
productEquality,
hypothesisEquality,
setEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
independent_pairEquality,
unionElimination,
equalityIstype,
equalityTransitivity,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[A,V:Type].
\mforall{}eq:EqDecider(A). \mforall{}L:(A \mtimes{} V) List. \mforall{}v:V.
((v \mmember{} values-for-distinct(eq;L)) {}\mRightarrow{} (\mexists{}a:A. (<a, v> \mmember{} L)))
Date html generated:
2019_10_15-AM-10_24_20
Last ObjectModification:
2019_08_05-PM-02_03_45
Theory : decidable!equality
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