Nuprl Lemma : finite-decidable-set
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. Dec(P[x])) ⇒ (finite-type({x:T| P[x]} ) ⇐⇒ ∃L:T List. ∀x:T. (P[x] ⇒ (x ∈ L))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
false: False,
not: ¬A,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
exists_wf,
list_wf,
all_wf,
iff_wf,
l_member_wf,
finite-set-type,
sq_stable_from_decidable,
finite-type_wf,
subtype_rel_self,
decidable_wf,
or_wf,
not_wf,
btrue_wf,
bfalse_wf,
equal_wf,
assert_wf,
true_wf,
false_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
set_wf,
assert_witness,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
addLevel,
productElimination,
impliesFunctionality,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
setEquality,
instantiate,
universeEquality,
cumulativity,
dependent_pairFormation,
rename,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
natural_numberEquality,
voidElimination,
independent_isectElimination,
setElimination,
productEquality,
allFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:T. Dec(P[x])) {}\mRightarrow{} (finite-type(\{x:T| P[x]\} ) \mLeftarrow{}{}\mRightarrow{} \mexists{}L:T List. \mforall{}x:T. (P[x] {}\mRightarrow{} (x \mmember{} L))))
Date html generated:
2019_06_20-PM-01_32_45
Last ObjectModification:
2018_08_21-PM-01_55_21
Theory : list_1
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