Nuprl Lemma : fset-ac-lub-is-lub-constrained
∀[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])} ].
least-upper-bound({ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])} ;ac1,ac2.fset-ac-le(eq;ac1;ac2);
ac1;ac2;fset-ac-lub(eq;ac1;ac2))
Proof
Definitions occuring in Statement :
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-antichain: fset-antichain(eq;ac),
fset-all: fset-all(s;x.P[x]),
fset: fset(T),
deq: EqDecider(T),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
and: P ∧ Q,
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
or: P ∨ Q
Lemmas referenced :
member-fset-union,
member-fset-minimals,
empty-fset_wf,
equal_wf,
not_wf,
assert-fset-null,
iff_wf,
all_wf,
assert_of_bnot,
fset-member_wf,
isect_wf,
uall_wf,
iff_weakening_uiff,
fset-all-iff,
deq_wf,
bool_wf,
set_wf,
fset-all_wf,
fset-antichain_wf,
assert_wf,
and_wf,
subtype_rel_sets,
fset-ac-lub_wf,
deq-f-subset_wf,
bnot_wf,
fset-filter_wf,
fset-null_wf,
assert_witness,
fset-ac-le_wf,
fset-ac-le-implies,
iff_weakening_equal,
fset-union-commutes,
true_wf,
squash_wf,
f-subset_wf,
f-subset-union,
fset-ac-le_weakening_f-subset,
fset-minimals-ac-le,
f-proper-subset-dec_wf,
fset-minimals_wf,
deq-fset_wf,
fset_wf,
fset-union_wf,
fset-ac-le_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
because_Cache,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
lambdaFormation,
independent_pairEquality,
cumulativity,
setEquality,
isect_memberEquality,
functionEquality,
addLevel,
impliesFunctionality,
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])\} ].
least-upper-bound(\{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\} ;
ac1,ac2.fset-ac-le(eq;ac1;ac2);ac1;ac2;fset-ac-lub(eq;ac1;ac2))
Date html generated:
2016_05_14-PM-03_48_59
Last ObjectModification:
2016_01_14-PM-10_40_39
Theory : finite!sets
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