Nuprl Lemma : fset-minimals-antichain
∀[T:Type]
∀eq:EqDecider(T). ∀s:fset(fset(T)). (↑fset-antichain(eq;fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); s)))
Proof
Definitions occuring in Statement :
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-antichain: fset-antichain(eq;ac),
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
f-proper-subset: xs ⊆≠ ys,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert-fset-antichain,
fset-minimals_wf,
fset_wf,
f-proper-subset-dec_wf,
f-proper-subset_wf,
fset-member_wf,
deq-fset_wf,
deq_wf,
assert_witness,
fset-antichain_wf,
member-fset-minimals,
fset-all-iff,
bnot_wf,
assert_wf,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert-f-proper-subset-dec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
productElimination,
independent_isectElimination,
because_Cache,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}s:fset(fset(T)).
(\muparrow{}fset-antichain(eq;fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); s)))
Date html generated:
2016_05_14-PM-03_47_52
Last ObjectModification:
2015_12_26-PM-06_36_29
Theory : finite!sets
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