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
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