Nuprl Lemma : fpf-union-join-dom
∀[A:Type]
∀eq:EqDecider(A). ∀f,g:a:A fp-> Top. ∀x:A. ∀R:Top.
(↑x ∈ dom(fpf-union-join(eq;R;f;g)) ⇐⇒ (↑x ∈ dom(f)) ∨ (↑x ∈ dom(g)))
Proof
Definitions occuring in Statement :
fpf-union-join: fpf-union-join(eq;R;f;g),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
fpf: a:A fp-> B[a],
fpf-dom: x ∈ dom(f),
fpf-union-join: fpf-union-join(eq;R;f;g),
pi1: fst(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
false: False
Lemmas referenced :
top_wf,
fpf_wf,
deq_wf,
l_member_wf,
or_wf,
and_wf,
assert_wf,
bnot_wf,
deq-member_wf,
member_filter,
filter_wf5,
iff_wf,
member_append,
append_wf,
assert-deq-member,
decidable__assert,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
hypothesisEquality,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
universeEquality,
productElimination,
independent_pairFormation,
unionElimination,
inlFormation,
inrFormation,
addLevel,
independent_functionElimination,
orFunctionality,
dependent_functionElimination,
applyEquality,
cumulativity,
because_Cache,
setElimination,
rename,
setEquality,
impliesFunctionality,
orLevelFunctionality,
promote_hyp,
voidElimination,
andLevelFunctionality
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A). \mforall{}f,g:a:A fp-> Top. \mforall{}x:A. \mforall{}R:Top.
(\muparrow{}x \mmember{} dom(fpf-union-join(eq;R;f;g)) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}x \mmember{} dom(f)) \mvee{} (\muparrow{}x \mmember{} dom(g)))
Date html generated:
2018_05_21-PM-09_23_23
Last ObjectModification:
2018_02_09-AM-10_19_17
Theory : finite!partial!functions
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