Nuprl Lemma : can-apply-p-co-filter
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[f:∀x:T. Dec(P[x])]. ∀[x:T]. uiff(↑can-apply(p-co-filter(f);x);¬P[x])
Proof
Definitions occuring in Statement :
p-co-filter: p-co-filter(f),
can-apply: can-apply(f;x),
assert: ↑b,
decidable: Dec(P),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
p-co-filter: p-co-filter(f),
can-apply: can-apply(f;x),
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
decidable: Dec(P),
or: P ∨ Q,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
true: True
Lemmas referenced :
assert_wf,
can-apply_wf,
p-co-filter_wf,
subtype_rel_dep_function,
top_wf,
subtype_rel_union,
not_wf,
all_wf,
decidable_wf,
assert_witness,
false_wf,
true_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
applyEquality,
hypothesisEquality,
hypothesis,
thin,
lambdaEquality,
sqequalHypSubstitution,
universeEquality,
lemma_by_obid,
isectElimination,
unionEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
cumulativity,
isect_memberFormation,
introduction,
productElimination,
independent_pairEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionElimination,
independent_pairFormation,
natural_numberEquality,
axiomEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[f:\mforall{}x:T. Dec(P[x])]. \mforall{}[x:T]. uiff(\muparrow{}can-apply(p-co-filter(f);x);\mneg{}P[x])
Date html generated:
2016_05_15-PM-03_30_59
Last ObjectModification:
2015_12_27-PM-01_11_00
Theory : general
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