Nuprl Lemma : cal-filter-decomp
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:fset(T) ⟶ 𝔹]. ∀[s:Point(constrained-antichain-lattice(T;eq;P))]. ∀[Q:{x:fset(T)|
↑P[x]} ⟶ 𝔹].
(s = cal-filter(s;x.Q[x]) ∨ cal-filter(s;x.¬bQ[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-join: a ∨ b,
lattice-point: Point(l),
fset: fset(T),
deq: EqDecider(T),
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
lattice-join: a ∨ b,
and: P ∧ Q,
guard: {T},
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
btrue: tt,
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
record-select: r.x,
lattice-point: Point(l),
top: Top,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
cal-filter: cal-filter(s;x.P[x]),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
fset-ac-le: fset-ac-le(eq;ac1;ac2),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
deq_wf,
set_wf,
bool_wf,
fset-all_wf,
fset-antichain_wf,
rec_select_update_lemma,
fset-constrained-ac-lub-is-lub,
bnot_wf,
assert_wf,
fset_wf,
cal-filter_wf,
cal-point,
equal_functionality_wrt_subtype_rel2,
fset-all-iff,
fset-member_wf,
deq-fset_wf,
fset-filter-subset2,
fset-ac-le_weakening_f-subset,
fset-constrained-ac-lub_wf,
fset-ac-order-constrained,
fset-ac-le_wf,
least-upper-bound-unique,
fset-ac-order,
fset-subtype2,
subtype_rel_sets,
fset-subtype,
strong-subtype-deq-subtype,
strong-subtype-set2,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
istype-assert,
iff_weakening_uiff,
eqtt_to_assert,
member-fset-filter,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert_of_bnot,
assert_witness
Rules used in proof :
universeEquality,
axiomEquality,
functionEquality,
productEquality,
productElimination,
dependent_set_memberEquality,
dependent_functionElimination,
setEquality,
rename,
setElimination,
lambdaFormation,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
because_Cache,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
independent_isectElimination,
lambdaEquality_alt,
universeIsType,
inhabitedIsType,
setIsType,
isectEquality,
isect_memberFormation_alt,
lambdaFormation_alt,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
isect_memberEquality_alt,
isectIsTypeImplies
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{}].
(s = cal-filter(s;x.Q[x]) \mvee{} cal-filter(s;x.\mneg{}\msubb{}Q[x]))
Date html generated:
2020_05_20-AM-08_48_09
Last ObjectModification:
2020_02_04-PM-02_33_32
Theory : lattices
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