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
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