Nuprl Lemma : fset-constrained-ac-lub_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:fset(T) ⟶ 𝔹]. ∀[ac1,ac2:{ac:fset(fset(T))| 
                                                             (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P a)} ].
  (lub(P;ac1;ac2) ∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P a)} )
Proof
Definitions occuring in Statement : 
fset-constrained-ac-lub: lub(P;ac1;ac2)
, 
fset-antichain: fset-antichain(eq;ac)
, 
fset-all: fset-all(s;x.P[x])
, 
fset: fset(T)
, 
deq: EqDecider(T)
, 
assert: ↑b
, 
bool: 𝔹
, 
uall: ∀[x:A]. B[x]
, 
and: P ∧ Q
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
fset-constrained-ac-lub: lub(P;ac1;ac2)
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
prop: ℙ
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
uiff: uiff(P;Q)
, 
uimplies: b supposing a
, 
fset-ac-lub: fset-ac-lub(eq;ac1;ac2)
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
iff: P 
⇐⇒ Q
, 
or: P ∨ Q
, 
guard: {T}
Lemmas referenced : 
member-fset-union, 
fset-union_wf, 
f-proper-subset-dec_wf, 
member-fset-minimals, 
fset-member_wf, 
assert_witness, 
deq-fset_wf, 
fset-all-iff, 
equal_wf, 
decidable__assert, 
sq_stable_from_decidable, 
fset-ac-lub_wf, 
deq_wf, 
bool_wf, 
set_wf, 
fset_wf, 
fset-all_wf, 
fset-antichain_wf, 
assert_wf, 
and_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
setElimination, 
thin, 
rename, 
sqequalRule, 
sqequalHypSubstitution, 
productElimination, 
dependent_set_memberEquality, 
because_Cache, 
independent_pairFormation, 
hypothesis, 
lemma_by_obid, 
isectElimination, 
hypothesisEquality, 
lambdaEquality, 
applyEquality, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
isect_memberEquality, 
functionEquality, 
universeEquality, 
cumulativity, 
lambdaFormation, 
independent_functionElimination, 
dependent_functionElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
independent_isectElimination, 
unionElimination
Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[P:fset(T)  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[ac1,ac2:\{ac:fset(fset(T))| 
                                                                                                                          (\muparrow{}fset-antichain(eq;ac))
                                                                                                                          \mwedge{}  fset-all(ac;a.P  a)\}  ].
    (lub(P;ac1;ac2)  \mmember{}  \{ac:fset(fset(T))|  (\muparrow{}fset-antichain(eq;ac))  \mwedge{}  fset-all(ac;a.P  a)\}  )
Date html generated:
2016_05_14-PM-03_49_07
Last ObjectModification:
2016_01_14-PM-10_39_31
Theory : finite!sets
Home
Index