Nuprl Lemma : assert-ac-covers
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[ac:fset(fset(T))]. ∀[x:fset(T)].
uiff(↑ac-covers(eq;ac;x);↓∃y:fset(T). (y ∈ ac ∧ y ⊆ x))
Proof
Definitions occuring in Statement :
ac-covers: ac-covers(eq;ac;x),
deq-fset: deq-fset(eq),
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
ac-covers: ac-covers(eq;ac;x),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
false: False,
exists: ∃x:A. B[x],
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
assert_wf,
ac-covers_wf,
assert_witness,
squash_wf,
exists_wf,
fset_wf,
fset-member_wf,
deq-fset_wf,
f-subset_wf,
deq_wf,
assert_of_bnot,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
assert-fset-null,
equal-wf-T-base,
fset-filter-is-empty,
not_wf,
decidable__squash_exists_fset,
decidable__f-subset,
assert-deq-f-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
independent_functionElimination,
lambdaEquality,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
applyEquality,
setElimination,
rename,
setEquality,
functionEquality,
functionExtensionality,
independent_isectElimination,
lambdaFormation,
promote_hyp,
dependent_functionElimination,
unionElimination,
voidElimination,
dependent_pairFormation,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[ac:fset(fset(T))]. \mforall{}[x:fset(T)].
uiff(\muparrow{}ac-covers(eq;ac;x);\mdownarrow{}\mexists{}y:fset(T). (y \mmember{} ac \mwedge{} y \msubseteq{} x))
Date html generated:
2017_04_17-AM-09_20_32
Last ObjectModification:
2017_02_27-PM-05_23_38
Theory : finite!sets
Home
Index