Nuprl Lemma : lattice-meet-fset-join-distrib
∀[l:BoundedDistributiveLattice]. ∀[eq:EqDecider(Point(l))]. ∀[s1,s2:fset(Point(l))].
(\/(s1) ∧ \/(s2) = \/(f-union(eq;eq;s1;a.λb.a ∧ b"(s2))) ∈ Point(l))
Proof
Definitions occuring in Statement :
lattice-fset-join: \/(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
fset: fset(T),
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
bdd-lattice: BoundedLattice,
implies: P ⇒ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
nil: [],
it: ⋅,
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
lattice-0: 0,
record-select: r.x,
top: Top,
listp: A List+,
or: P ∨ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
eqof: eqof(d),
fset-member: a ∈ s,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
lattice-meet: a ∧ b,
fset-image: f"(s)
Lemmas referenced :
bdd-distributive-lattice_wf,
deq_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
lattice-point_wf,
fset_wf,
set-equal_wf,
equal-wf-base,
fset-image_wf,
f-union_wf,
set-equal-equiv,
list_wf,
quotient-member-eq,
decidable-equal-deq,
bdd-distributive-lattice-subtype-bdd-lattice,
bdd-lattice_wf,
decidable_wf,
all_wf,
squash_wf,
lattice-fset-join_wf,
iff_weakening_equal,
true_wf,
list_subtype_fset,
list_induction,
lattice-meet-0,
reduce_cons_lemma,
length_wf,
less_than_wf,
cons_wf_listp,
bdd-distributive-lattice-subtype-distributive-lattice,
distributive-lattice-distrib,
uiff_wf,
member-fset-union,
or_wf,
fset-member_wf,
fset-member_witness,
fset-union_wf,
fset-extensionality,
member-fset-image-iff,
decidable__fset-member,
decidable__or,
sq_stable_from_decidable,
member-f-union,
safe-assert-deq,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
assert-deq-member,
l_member_wf,
deq-member_wf,
eqof_wf,
bor_wf,
assert_wf,
deq_member_cons_lemma,
and_wf,
sq_stable__fset-member,
member_wf,
cons_member,
cons_wf,
lattice-fset-join-union,
lattice_properties,
bdd-distributive-lattice-subtype-lattice,
lattice-0_wf,
lattice-fset-join-singleton,
fset-singleton_wf,
member-fset-singleton
Rules used in proof :
axiomEquality,
isect_memberEquality,
independent_isectElimination,
universeEquality,
cumulativity,
lambdaEquality,
instantiate,
applyEquality,
hypothesisEquality,
isectElimination,
extract_by_obid,
productEquality,
hypothesis,
thin,
productElimination,
pertypeElimination,
sqequalRule,
because_Cache,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
natural_numberEquality,
baseClosed,
imageMemberEquality,
dependent_functionElimination,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
imageElimination,
promote_hyp,
rename,
setElimination,
lambdaFormation_alt,
inhabitedIsType,
universeIsType,
lambdaEquality_alt,
isectEquality,
voidEquality,
voidElimination,
applyLambdaEquality,
hyp_replacement,
independent_pairEquality,
addLevel,
independent_pairFormation,
orFunctionality,
inrFormation,
inlFormation,
unionElimination,
dependent_set_memberEquality,
dependent_pairFormation,
equalityElimination,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[l:BoundedDistributiveLattice]. \mforall{}[eq:EqDecider(Point(l))]. \mforall{}[s1,s2:fset(Point(l))].
(\mbackslash{}/(s1) \mwedge{} \mbackslash{}/(s2) = \mbackslash{}/(f-union(eq;eq;s1;a.\mlambda{}b.a \mwedge{} b"(s2))))
Date html generated:
2020_05_20-AM-08_44_30
Last ObjectModification:
2020_02_03-PM-03_10_53
Theory : lattices
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