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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