Nuprl Lemma : fset-ac-lub-covers
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x,y:{ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ]. ∀[a:fset(T)].
(((↑ac-covers(eq;x;a)) ∨ (↑ac-covers(eq;y;a))) ⇒ (↑ac-covers(eq;fset-ac-lub(eq;x;y);a)))
Proof
Definitions occuring in Statement :
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
ac-covers: ac-covers(eq;ac;x),
fset-antichain: fset-antichain(eq;ac),
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
or: P ∨ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
cand: A c∧ B,
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
guard: {T}
Lemmas referenced :
deq_wf,
set_wf,
assert_witness,
ac-covers_wf,
or_wf,
f-subset_transitivity,
member-fset-union,
fset-antichain_wf,
assert_wf,
fset-ac-lub_wf,
f-subset_wf,
fset-member_wf,
and_wf,
exists_wf,
squash_wf,
assert-ac-covers,
f-proper-subset-dec_wf,
fset-minimals_wf,
fset-ac-le-iff,
deq-fset_wf,
fset_wf,
fset-union_wf,
fset-minimals-ac-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
productElimination,
independent_functionElimination,
addLevel,
orFunctionality,
independent_isectElimination,
levelHypothesis,
promote_hyp,
applyEquality,
setEquality,
unionElimination,
imageElimination,
inlFormation,
dependent_pairFormation,
independent_pairFormation,
productEquality,
imageMemberEquality,
baseClosed,
inrFormation,
orLevelFunctionality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x,y:\{ac:fset(fset(T))| \muparrow{}fset-antichain(eq;ac)\} ]. \mforall{}[a:fset(T)].
(((\muparrow{}ac-covers(eq;x;a)) \mvee{} (\muparrow{}ac-covers(eq;y;a))) {}\mRightarrow{} (\muparrow{}ac-covers(eq;fset-ac-lub(eq;x;y);a)))
Date html generated:
2016_05_14-PM-03_48_51
Last ObjectModification:
2016_01_20-PM-09_08_23
Theory : finite!sets
Home
Index