Nuprl Lemma : do-apply-p-restrict
∀[A,B:Type]. ∀[f:A ⟶ (B + Top)]. ∀[P:A ⟶ ℙ]. ∀[p:∀x:A. Dec(P[x])]. ∀[x:A].
do-apply(p-restrict(f;p);x) = do-apply(f;x) ∈ B supposing ↑can-apply(p-restrict(f;p);x)
Proof
Definitions occuring in Statement :
p-restrict: p-restrict(f;p),
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],
top: Top,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
p-restrict: p-restrict(f;p),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
squash: ↓T,
true: True,
guard: {T},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
assert_wf,
can-apply_wf,
p-compose_wf,
top_wf,
p-filter_wf,
all_wf,
decidable_wf,
equal_wf,
do-apply_wf,
do-apply-p-filter,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
iff_weakening_equal,
do-apply-compose,
can-apply-compose-iff
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
productElimination,
imageElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
unionEquality,
natural_numberEquality,
independent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} (B + Top)]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[p:\mforall{}x:A. Dec(P[x])]. \mforall{}[x:A].
do-apply(p-restrict(f;p);x) = do-apply(f;x) supposing \muparrow{}can-apply(p-restrict(f;p);x)
Date html generated:
2017_10_01-AM-09_14_16
Last ObjectModification:
2017_07_26-PM-04_49_25
Theory : general
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