Nuprl Lemma : fl-filter_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Q:{x:fset(T + T)|
↑fset-contains-none(union-deq(T;T;eq;eq);x;x.face-lattice-constraints(x))} ⟶ 𝔹].
∀[s:Point(face-lattice(T;eq))].
(fl-filter(s;x.Q[x]) ∈ Point(face-lattice(T;eq)))
Proof
Definitions occuring in Statement :
fl-filter: fl-filter(s;x.Q[x]),
face-lattice: face-lattice(T;eq),
face-lattice-constraints: face-lattice-constraints(x),
lattice-point: Point(l),
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
fset: fset(T),
union-deq: union-deq(A;B;a;b),
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],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fl-filter: fl-filter(s;x.Q[x]),
so_lambda: λ2x.t[x],
so_apply: x[s],
lattice-point: Point(l),
record-select: r.x,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
all: ∀x:A. B[x],
prop: ℙ,
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
uimplies: b supposing a
Lemmas referenced :
cal-filter_wf,
union-deq_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
fset_wf,
assert_wf,
lattice-point_wf,
face-lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
bool_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
because_Cache,
hypothesis,
lambdaEquality,
lambdaFormation,
applyEquality,
setElimination,
rename,
functionExtensionality,
setEquality,
dependent_set_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
productEquality,
independent_isectElimination,
isect_memberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)].
\mforall{}[Q:\{x:fset(T + T)| \muparrow{}fset-contains-none(union-deq(T;T;eq;eq);x;x.face-lattice-constraints(x))\}
{}\mrightarrow{} \mBbbB{}]. \mforall{}[s:Point(face-lattice(T;eq))].
(fl-filter(s;x.Q[x]) \mmember{} Point(face-lattice(T;eq)))
Date html generated:
2020_05_20-AM-08_52_12
Last ObjectModification:
2018_05_20-PM-10_12_29
Theory : lattices
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