Nuprl Lemma : cal-filter_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:fset(T) ⟶ 𝔹]. ∀[s:Point(constrained-antichain-lattice(T;eq;P))]. ∀[Q:{x:fset(T)|
↑P[x]} ⟶ 𝔹].
(cal-filter(s;x.Q[x]) ∈ Point(constrained-antichain-lattice(T;eq;P)))
Proof
Definitions occuring in Statement :
cal-filter: cal-filter(s;x.P[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
cal-filter: cal-filter(s;x.P[x]),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
istype: istype(T),
not: ¬A,
false: False,
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_type: SQType(T),
cand: A c∧ B
Lemmas referenced :
cal-point,
istype-void,
istype-assert,
bool_wf,
fset_wf,
fset-antichain_wf,
fset-all_wf,
deq_wf,
istype-universe,
set_wf,
subtype_rel_self,
assert_wf,
fset-subtype2,
deq-fset_wf,
fset-member_wf,
fset-filter_wf,
subtype_rel_dep_function,
subtype_rel_sets,
fset-subtype,
fset-all-iff,
assert-fset-antichain,
iff_weakening_uiff,
equal_wf,
f-proper-subset_wf,
not_wf,
isect_wf,
all_wf,
member-fset-filter2,
subtype_rel_sets_simple,
assert_witness,
bool_subtype_base,
subtype_base_sq,
assert_elim,
strong-subtype-set2,
strong-subtype-deq-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
because_Cache,
setElimination,
rename,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
setIsType,
hypothesisEquality,
universeIsType,
isectIsTypeImplies,
inhabitedIsType,
productIsType,
lambdaEquality_alt,
instantiate,
universeEquality,
lambdaEquality,
functionExtensionality,
productElimination,
cumulativity,
setEquality,
independent_isectElimination,
lambdaFormation,
dependent_set_memberEquality_alt,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation_alt,
equalityIsType1,
hyp_replacement,
applyLambdaEquality,
functionIsTypeImplies,
productEquality,
isect_memberEquality,
natural_numberEquality,
isect_memberFormation,
independent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:fset(T) {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:Point(constrained-antichain-lattice(T;eq;P))].
\mforall{}[Q:\{x:fset(T)| \muparrow{}P[x]\} {}\mrightarrow{} \mBbbB{}].
(cal-filter(s;x.Q[x]) \mmember{} Point(constrained-antichain-lattice(T;eq;P)))
Date html generated:
2019_10_31-AM-07_21_14
Last ObjectModification:
2018_11_10-PM-00_29_52
Theory : lattices
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