Nuprl Lemma : finite-subtype
∀[B:Type]. ∀P:B ⟶ 𝔹. (finite-type(B) ⇒ finite-type({b:B| ↑P[b]} ))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T}
Lemmas referenced :
set_wf,
decidable__assert,
sq_stable_from_decidable,
member_filter,
l_member_set2,
filter_type,
bool_wf,
finite-type_wf,
assert_wf,
finite-type-iff-list,
l_member_wf,
all_wf,
list_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
addLevel,
impliesFunctionality,
independent_functionElimination,
setEquality,
applyEquality,
functionEquality,
cumulativity,
because_Cache,
setElimination,
rename,
dependent_set_memberEquality,
universeEquality,
dependent_pairFormation,
dependent_functionElimination,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[B:Type]. \mforall{}P:B {}\mrightarrow{} \mBbbB{}. (finite-type(B) {}\mRightarrow{} finite-type(\{b:B| \muparrow{}P[b]\} ))
Date html generated:
2016_05_14-PM-01_52_58
Last ObjectModification:
2016_01_15-AM-08_14_53
Theory : list_1
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