Nuprl Lemma : fpf-domain-union-join
∀[A:Type]
  ∀f:a:A fp-> Top List. ∀g:a:A fp-> Top. ∀eq:EqDecider(A). ∀x:A. ∀R:Top.
    ((x ∈ fpf-domain(fpf-union-join(eq;R;f;g))) 
⇐⇒ (x ∈ fpf-domain(f)) ∨ (x ∈ fpf-domain(g)))
Proof
Definitions occuring in Statement : 
fpf-union-join: fpf-union-join(eq;R;f;g)
, 
fpf-domain: fpf-domain(f)
, 
fpf: a:A fp-> B[a]
, 
l_member: (x ∈ l)
, 
list: T List
, 
deq: EqDecider(T)
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
fpf: a:A fp-> B[a]
, 
fpf-domain: fpf-domain(f)
, 
fpf-union-join: fpf-union-join(eq;R;f;g)
, 
pi1: fst(t)
, 
fpf-dom: x ∈ dom(f)
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
or: P ∨ Q
, 
member: t ∈ T
, 
prop: ℙ
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
not: ¬A
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
false: False
Lemmas referenced : 
l_member_wf, 
or_wf, 
and_wf, 
assert_wf, 
bnot_wf, 
deq-member_wf, 
decidable__assert, 
assert-deq-member, 
member_filter, 
filter_wf5, 
iff_wf, 
member_append, 
append_wf, 
top_wf, 
deq_wf, 
fpf_wf, 
list_wf, 
not_wf, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
sqequalHypSubstitution, 
productElimination, 
thin, 
sqequalRule, 
cut, 
independent_pairFormation, 
unionElimination, 
inlFormation, 
hypothesis, 
lemma_by_obid, 
isectElimination, 
hypothesisEquality, 
inrFormation, 
dependent_functionElimination, 
independent_functionElimination, 
promote_hyp, 
addLevel, 
orFunctionality, 
lambdaEquality, 
applyEquality, 
cumulativity, 
because_Cache, 
setElimination, 
rename, 
setEquality, 
impliesFunctionality, 
universeEquality, 
voidElimination, 
andLevelFunctionality
Latex:
\mforall{}[A:Type]
    \mforall{}f:a:A  fp->  Top  List.  \mforall{}g:a:A  fp->  Top.  \mforall{}eq:EqDecider(A).  \mforall{}x:A.  \mforall{}R:Top.
        ((x  \mmember{}  fpf-domain(fpf-union-join(eq;R;f;g)))  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  fpf-domain(f))  \mvee{}  (x  \mmember{}  fpf-domain(g)))
Date html generated:
2018_05_21-PM-09_23_26
Last ObjectModification:
2018_02_09-AM-10_19_18
Theory : finite!partial!functions
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