Nuprl Lemma : can-apply-p-lift
∀[A,B:Type]. ∀[P:A ⟶ ℙ]. ∀d:x:A ⟶ Dec(P[x]). ∀f:{x:A| P[x]} ⟶ B. ∀x:A. (↑can-apply(p-lift(d;f);x) ⇐⇒ P[x])
Proof
Definitions occuring in Statement :
p-lift: p-lift(d;f),
can-apply: can-apply(f;x),
assert: ↑b,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
p-lift: p-lift(d;f),
can-apply: can-apply(f;x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
true: True,
bfalse: ff,
false: False,
not: ¬A
Lemmas referenced :
decidable_wf,
true_wf,
false_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
functionEquality,
setEquality,
cut,
applyEquality,
hypothesis,
thin,
lambdaEquality,
sqequalHypSubstitution,
universeEquality,
lemma_by_obid,
isectElimination,
cumulativity,
unionElimination,
independent_pairFormation,
natural_numberEquality,
voidElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
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. (\muparrow{}can-apply(p-lift(d;f);x) \mLeftarrow{}{}\mRightarrow{} P[x])
Date html generated:
2016_05_15-PM-03_29_20
Last ObjectModification:
2015_12_27-PM-01_09_31
Theory : general
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