Nuprl Lemma : fpf-join-list-ap2
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀L:a:A fp-> B[a] List. ∀x:A. ((x ∈ fpf-domain(⊕(L))) ⇒ (∃f∈L. (↑x ∈ dom(f)) ∧ (⊕(L)(x) = f(x) ∈ B[x])))
Proof
Definitions occuring in Statement :
fpf-join-list: ⊕(L),
fpf-ap: f(x),
fpf-domain: fpf-domain(f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fpf-join-list-ap,
list_wf,
fpf_wf,
deq_wf,
l_member_wf,
fpf-domain_wf,
fpf-join-list_wf,
top_wf,
subtype_rel_list,
subtype-fpf2,
member-fpf-domain
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
independent_isectElimination,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_functionElimination
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))) {}\mRightarrow{} (\mexists{}f\mmember{}L. (\muparrow{}x \mmember{} dom(f)) \mwedge{} (\moplus{}(L)(x) = f(x))))
Date html generated:
2018_05_21-PM-09_22_59
Last ObjectModification:
2018_02_09-AM-10_19_02
Theory : finite!partial!functions
Home
Index