Nuprl Lemma : fpf-join-list-domain
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]. ∀L:a:A fp-> B[a] List. ∀x:A. ((x ∈ fpf-domain(⊕(L))) ⇐⇒ (∃f∈L. (x ∈ fpf-domain(f))))
Proof
Definitions occuring in Statement :
fpf-join-list: ⊕(L),
fpf-domain: fpf-domain(f),
fpf: a:A fp-> B[a],
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
rev_implies: P ⇐ Q,
prop: ℙ,
guard: {T}
Lemmas referenced :
fpf-join-list-dom,
member-fpf-domain,
fpf-join-list_wf,
top_wf,
subtype_rel_list,
fpf_wf,
subtype-fpf2,
l_exists_functionality,
l_member_wf,
assert_wf,
fpf-dom_wf,
fpf-domain_wf,
subtype_rel_set,
set_wf,
l_exists_wf,
list_wf,
deq_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
productElimination,
independent_pairFormation,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
setElimination,
rename,
cumulativity,
setEquality,
promote_hyp,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}L:a:A fp-> B[a] List. \mforall{}x:A. ((x \mmember{} fpf-domain(\moplus{}(L))) \mLeftarrow{}{}\mRightarrow{} (\mexists{}f\mmember{}L. (x \mmember{} fpf-domain(f))))
Date html generated:
2018_05_21-PM-09_22_47
Last ObjectModification:
2018_02_09-AM-10_18_54
Theory : finite!partial!functions
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