Nuprl Lemma : member-fset-minimals
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[less:T ⟶ T ⟶ 𝔹]. ∀[s:fset(T)]. ∀[a:T].
uiff(a ∈ fset-minimals(x,y.less[x;y]; s);a ∈ s ∧ fset-all(s;y.¬bless[y;a]))
Proof
Definitions occuring in Statement :
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-all: fset-all(s;x.P[x]),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
bnot: ¬bb,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fset-minimals: fset-minimals(x,y.less[x; y]; s),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
fset-all: fset-all(s;x.P[x]),
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset-member_witness,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
and_wf,
fset-member_wf,
fset-all_wf,
assert-fset-minimal,
assert_wf,
fset-minimal_wf,
uiff_wf,
iff_weakening_uiff,
guard_wf,
member-fset-filter,
fset-minimals_wf,
fset_wf,
bool_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
independent_pairFormation,
isect_memberFormation,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
sqequalRule,
independent_pairEquality,
lemma_by_obid,
isectElimination,
because_Cache,
independent_functionElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
addLevel,
independent_isectElimination,
cumulativity,
functionEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[less:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)]. \mforall{}[a:T].
uiff(a \mmember{} fset-minimals(x,y.less[x;y]; s);a \mmember{} s \mwedge{} fset-all(s;y.\mneg{}\msubb{}less[y;a]))
Date html generated:
2016_05_14-PM-03_47_37
Last ObjectModification:
2015_12_26-PM-06_36_34
Theory : finite!sets
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