Nuprl Lemma : filter-fset-minimals
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[less:T ⟶ T ⟶ 𝔹]. ∀[s:fset(T)]. ∀[P:T ⟶ 𝔹].
{a ∈ fset-minimals(x,y.less[x;y]; s) | P[a]} = fset-minimals(x,y.less[x;y]; {a ∈ s | P[a]}) ∈ fset(T)
supposing ∀a,y:T. ((↑P[a]) ⇒ (↑less[y;a]) ⇒ (↑P[y]))
Proof
Definitions occuring in Statement :
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
cand: A c∧ B,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
false: False,
btrue: tt,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
not: ¬A
Lemmas referenced :
fset-extensionality,
fset-filter_wf,
istype-universe,
fset-minimals_wf,
member-fset-filter,
fset-member_witness,
fset-member_wf,
fset-all_wf,
bnot_wf,
assert_wf,
iff_weakening_uiff,
member-fset-minimals,
fset-all-iff,
uall_wf,
isect_wf,
assert_witness,
bool_wf,
fset_wf,
deq_wf,
iff_transitivity,
assert_of_bnot,
assert_elim,
bfalse_wf,
and_wf,
equal_wf,
btrue_neq_bfalse
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
hypothesis,
because_Cache,
Error :inhabitedIsType,
productElimination,
independent_isectElimination,
Error :isect_memberFormation_alt,
independent_pairFormation,
independent_functionElimination,
Error :universeIsType,
independent_pairEquality,
Error :isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
productEquality,
Error :productIsType,
promote_hyp,
Error :functionIsType,
universeEquality,
axiomEquality,
Error :lambdaFormation_alt,
voidElimination,
setEquality,
rename,
setElimination,
dependent_set_memberEquality,
levelHypothesis,
addLevel,
lambdaFormation,
isect_memberFormation,
lambdaEquality,
lemma_by_obid,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[less:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
\{a \mmember{} fset-minimals(x,y.less[x;y]; s) | P[a]\} = fset-minimals(x,y.less[x;y]; \{a \mmember{} s | P[a]\})
supposing \mforall{}a,y:T. ((\muparrow{}P[a]) {}\mRightarrow{} (\muparrow{}less[y;a]) {}\mRightarrow{} (\muparrow{}P[y]))
Date html generated:
2019_06_20-PM-02_00_20
Last ObjectModification:
2018_10_06-PM-11_55_40
Theory : finite!sets
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