Nuprl Lemma : p-restrict_wf
∀[A,B:Type]. ∀[f:A ⟶ (B + Top)]. ∀[P:A ⟶ ℙ]. ∀[p:∀x:A. Dec(P[x])]. (p-restrict(f;p) ∈ A ⟶ (B + Top))
Proof
Definitions occuring in Statement :
p-restrict: p-restrict(f;p),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
p-restrict: p-restrict(f;p),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
p-compose_wf,
p-filter_wf,
all_wf,
decidable_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
unionEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} (B + Top)]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[p:\mforall{}x:A. Dec(P[x])].
(p-restrict(f;p) \mmember{} A {}\mrightarrow{} (B + Top))
Date html generated:
2016_05_15-PM-03_31_14
Last ObjectModification:
2015_12_27-PM-01_10_56
Theory : general
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