Nuprl Lemma : lattice-fset-join-is-lub
∀[l:BoundedLattice]. ∀[eq:EqDecider(Point(l))].
((∀[s:fset(Point(l))]. ∀[x:Point(l)]. x ≤ \/(s) supposing x ∈ s)
∧ (∀[s:fset(Point(l))]. ∀[u:Point(l)]. ((∀x:Point(l). (x ∈ s ⇒ x ≤ u)) ⇒ \/(s) ≤ u)))
Proof
Definitions occuring in Statement :
lattice-fset-join: \/(s),
bdd-lattice: BoundedLattice,
lattice-le: a ≤ b,
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
lattice-le: a ≤ b,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bdd-lattice: BoundedLattice,
sq_stable: SqStable(P),
top: Top,
false: False,
guard: {T},
fset-add: fset-add(eq;x;s),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
empty-fset: {},
lattice-fset-join: \/(s)
Lemmas referenced :
fset_wf,
all_wf,
fset-member_wf,
lattice-le_wf,
deq_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
bdd-lattice_wf,
fset-induction,
uall_wf,
isect_wf,
lattice-fset-join_wf,
decidable-equal-deq,
sq_stable__uall,
squash_wf,
mem_empty_lemma,
empty-fset_wf,
fset-add_wf,
not_wf,
sq_stable__equal,
lattice-meet_wf,
true_wf,
lattice-fset-join-union,
fset-singleton_wf,
iff_weakening_equal,
lattice-join_wf,
lattice-fset-join-singleton,
member-fset-add,
lattice-le_transitivity,
bdd-lattice-subtype-lattice,
lattice-join-is-lub,
lattice-le_weakening,
sq_stable__all,
reduce_nil_lemma,
lattice-0-le,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
isectElimination,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
lambdaEquality,
functionEquality,
productElimination,
independent_pairEquality,
applyEquality,
instantiate,
productEquality,
cumulativity,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
voidElimination,
voidEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
hyp_replacement,
applyLambdaEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[eq:EqDecider(Point(l))].
((\mforall{}[s:fset(Point(l))]. \mforall{}[x:Point(l)]. x \mleq{} \mbackslash{}/(s) supposing x \mmember{} s)
\mwedge{} (\mforall{}[s:fset(Point(l))]. \mforall{}[u:Point(l)]. ((\mforall{}x:Point(l). (x \mmember{} s {}\mRightarrow{} x \mleq{} u)) {}\mRightarrow{} \mbackslash{}/(s) \mleq{} u)))
Date html generated:
2020_05_20-AM-08_43_51
Last ObjectModification:
2017_07_28-AM-09_13_56
Theory : lattices
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