Nuprl Lemma : assert-fset-minimal
∀[T:Type]. ∀[less:T ⟶ T ⟶ 𝔹]. ∀[s:fset(T)]. ∀[a:T]. uiff(↑fset-minimal(x,y.less[x;y];s;a);fset-all(s;y.¬bless[y;a]))
Proof
Definitions occuring in Statement :
fset-minimal: fset-minimal(x,y.less[x; y];s;a),
fset-all: fset-all(s;x.P[x]),
fset: fset(T),
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fset-all: fset-all(s;x.P[x]),
fset-minimal: fset-minimal(x,y.less[x; y];s;a),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s]
Lemmas referenced :
assert_wf,
squash_wf,
true_wf,
bool_wf,
fset-null_wf,
fset-filter_wf,
equal_wf,
bnot_bnot_elim,
iff_weakening_equal,
assert_witness,
bnot_wf,
fset_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
hyp_replacement,
thin,
equalitySymmetry,
applyEquality,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
because_Cache,
functionExtensionality,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
isect_memberEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[less:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)]. \mforall{}[a:T].
uiff(\muparrow{}fset-minimal(x,y.less[x;y];s;a);fset-all(s;y.\mneg{}\msubb{}less[y;a]))
Date html generated:
2017_04_17-AM-09_23_24
Last ObjectModification:
2017_02_27-PM-05_25_00
Theory : finite!sets
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