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:
2017_10_05-AM-00_35_27
Last ObjectModification:
2017_07_28-AM-09_14_41
Theory : lattices
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