Nuprl Lemma : can-apply-p-first
∀[A,B:Type]. ∀L:(A ⟶ (B + Top)) List. ∀x:A. (↑can-apply(p-first(L);x) ⇐⇒ (∃f∈L. ↑can-apply(f;x)))
Proof
Definitions occuring in Statement :
p-first: p-first(L),
can-apply: can-apply(f;x),
l_exists: (∃x∈L. P[x]),
list: T List,
assert: ↑b,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
top: Top,
prop: ℙ,
implies: P ⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
or: P ∨ Q,
squash: ↓T,
true: True,
guard: {T},
uiff: uiff(P;Q)
Lemmas referenced :
p-first-singleton,
p-first-append,
true_wf,
squash_wf,
append_wf,
l_exists_append,
l_exists_cons,
p-conditional-to-p-first,
assert_functionality_wrt_uiff,
or_wf,
p-conditional_wf,
nil_wf,
cons_wf,
p-conditional-domain,
list_ind_nil_lemma,
list_ind_cons_lemma,
l_exists_wf_nil,
l_exists_nil,
false_wf,
p_first_nil_lemma,
list_wf,
l_member_wf,
l_exists_wf,
subtype_rel_union,
subtype_rel_dep_function,
subtype_rel_list,
p-first_wf,
can-apply_wf,
assert_wf,
iff_wf,
all_wf,
top_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
hypothesisEquality,
unionEquality,
hypothesis,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
applyEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
introduction,
independent_pairFormation,
productElimination,
independent_pairEquality,
addLevel,
allFunctionality,
impliesFunctionality,
imageElimination,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
orFunctionality,
equalityTransitivity,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[A,B:Type].
\mforall{}L:(A {}\mrightarrow{} (B + Top)) List. \mforall{}x:A. (\muparrow{}can-apply(p-first(L);x) \mLeftarrow{}{}\mRightarrow{} (\mexists{}f\mmember{}L. \muparrow{}can-apply(f;x)))
Date html generated:
2016_05_15-PM-03_45_17
Last ObjectModification:
2016_01_16-AM-10_56_40
Theory : general
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