Nuprl Lemma : lattice-meet-join-images-distrib
∀L:BoundedDistributiveLattice. ∀eqL:EqDecider(Point(L)). ∀as,bs:fset(fset(Point(L))).
  (\/(λls./\(ls)"(as)) ∧ \/(λls./\(ls)"(bs))
  = \/(λls./\(ls)"(f-union(deq-fset(eqL);deq-fset(eqL);as;as.λbs.as ⋃ bs"(bs))))
  ∈ Point(L))
Proof
Definitions occuring in Statement : 
lattice-fset-join: \/(s)
, 
lattice-fset-meet: /\(s)
, 
bdd-distributive-lattice: BoundedDistributiveLattice
, 
lattice-meet: a ∧ b
, 
lattice-point: Point(l)
, 
fset-image: f"(s)
, 
deq-fset: deq-fset(eq)
, 
f-union: f-union(domeq;rngeq;s;x.g[x])
, 
fset-union: x ⋃ y
, 
fset: fset(T)
, 
deq: EqDecider(T)
, 
all: ∀x:A. B[x]
, 
lambda: λx.A[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
squash: ↓T
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
implies: P 
⇒ Q
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
true: True
, 
uimplies: b supposing a
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
bdd-distributive-lattice: BoundedDistributiveLattice
, 
uiff: uiff(P;Q)
, 
sq_stable: SqStable(P)
, 
exists: ∃x:A. B[x]
, 
cand: A c∧ B
, 
fset-union: x ⋃ y
, 
l-union: as ⋃ bs
, 
reduce: reduce(f;k;as)
, 
list_ind: list_ind, 
lattice-fset-meet: /\(s)
Lemmas referenced : 
lattice-meet-fset-join-distrib, 
equal_wf, 
squash_wf, 
true_wf, 
fset-image_wf, 
deq-fset_wf, 
lattice-fset-meet_wf, 
bdd-distributive-lattice-subtype-bdd-lattice, 
decidable-equal-deq, 
lattice-fset-join_wf, 
f-union_wf, 
fset-union_wf, 
iff_weakening_equal, 
fset_wf, 
lattice-point_wf, 
subtype_rel_set, 
bounded-lattice-structure_wf, 
lattice-structure_wf, 
lattice-axioms_wf, 
bounded-lattice-structure-subtype, 
bounded-lattice-axioms_wf, 
uall_wf, 
lattice-meet_wf, 
lattice-join_wf, 
deq_wf, 
bdd-distributive-lattice_wf, 
fset-extensionality, 
fset-member_witness, 
fset-member_wf, 
sq_stable__fset-member, 
member-fset-image-iff, 
member-f-union, 
lattice-fset-meet-union
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
lambdaEquality, 
imageElimination, 
equalityTransitivity, 
hypothesis, 
equalitySymmetry, 
universeEquality, 
because_Cache, 
sqequalRule, 
independent_functionElimination, 
dependent_functionElimination, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
independent_isectElimination, 
productElimination, 
instantiate, 
productEquality, 
cumulativity, 
isect_memberFormation, 
independent_pairFormation, 
independent_pairEquality, 
isect_memberEquality, 
hyp_replacement, 
applyLambdaEquality, 
dependent_pairFormation
Latex:
\mforall{}L:BoundedDistributiveLattice.  \mforall{}eqL:EqDecider(Point(L)).  \mforall{}as,bs:fset(fset(Point(L))).
    (\mbackslash{}/(\mlambda{}ls./\mbackslash{}(ls)"(as))  \mwedge{}  \mbackslash{}/(\mlambda{}ls./\mbackslash{}(ls)"(bs))
    =  \mbackslash{}/(\mlambda{}ls./\mbackslash{}(ls)"(f-union(deq-fset(eqL);deq-fset(eqL);as;as.\mlambda{}bs.as  \mcup{}  bs"(bs)))))
Date html generated:
2020_05_20-AM-08_46_00
Last ObjectModification:
2017_07_28-AM-09_14_41
Theory : lattices
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