Nuprl Lemma : do-apply-p-lift
∀[A,B:Type]. ∀[P:A ⟶ ℙ]. ∀[d:x:A ⟶ Dec(P[x])]. ∀[f:{x:A| P[x]} ⟶ B]. ∀[x:A].
do-apply(p-lift(d;f);x) = (f x) ∈ B supposing ↑can-apply(p-lift(d;f);x)
Proof
Definitions occuring in Statement :
p-lift: p-lift(d;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],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
p-lift: p-lift(d;f),
do-apply: do-apply(f;x),
can-apply: can-apply(f;x),
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
top: Top,
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 :
assert_wf,
can-apply_wf,
p-lift_wf,
top_wf,
subtype_rel_dep_function,
set_wf,
decidable_wf,
true_wf,
false_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalRule,
sqequalHypSubstitution,
independent_functionElimination,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
setEquality,
universeEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
unionElimination,
dependent_set_memberEquality,
dependent_functionElimination
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]. \mforall{}[x:A].
do-apply(p-lift(d;f);x) = (f x) supposing \muparrow{}can-apply(p-lift(d;f);x)
Date html generated:
2016_05_15-PM-03_29_25
Last ObjectModification:
2015_12_27-PM-01_09_45
Theory : general
Home
Index