Nuprl Lemma : do-apply-p-filter
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[f:∀x:T. Dec(P[x])]. ∀[x:T].
do-apply(p-filter(f);x) = x ∈ T supposing ↑can-apply(p-filter(f);x)
Proof
Definitions occuring in Statement :
p-filter: p-filter(f),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
p-filter: p-filter(f),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
false: False
Lemmas referenced :
all_wf,
decidable_wf,
assert_wf,
can-apply_wf,
p-filter_wf,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
true_wf,
false_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
unionEquality,
independent_isectElimination,
lambdaFormation,
voidElimination,
voidEquality,
unionElimination,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[f:\mforall{}x:T. Dec(P[x])]. \mforall{}[x:T].
do-apply(p-filter(f);x) = x supposing \muparrow{}can-apply(p-filter(f);x)
Date html generated:
2016_05_15-PM-03_31_04
Last ObjectModification:
2015_12_27-PM-01_11_05
Theory : general
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