Nuprl Lemma : p-lift_wf
∀[A,B:Type]. ∀[P:A ⟶ ℙ]. ∀[d:x:A ⟶ Dec(P[x])]. ∀[f:{x:A| P[x]} ⟶ B]. (p-lift(d;f) ∈ A ⟶ (B + Top))
Proof
Definitions occuring in Statement :
p-lift: p-lift(d;f),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
p-lift: p-lift(d;f),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
top: Top,
prop: ℙ
Lemmas referenced :
decidable_wf,
top_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
applyEquality,
hypothesisEquality,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesis,
lambdaFormation,
unionElimination,
inlEquality,
dependent_set_memberEquality,
because_Cache,
inrEquality,
isect_memberEquality,
voidElimination,
voidEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
functionEquality,
setEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[d:x:A {}\mrightarrow{} Dec(P[x])]. \mforall{}[f:\{x:A| P[x]\} {}\mrightarrow{} B].
(p-lift(d;f) \mmember{} A {}\mrightarrow{} (B + Top))
Date html generated:
2016_05_15-PM-03_29_16
Last ObjectModification:
2015_12_27-PM-01_09_57
Theory : general
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