Nuprl Lemma : lattice-fset-join-le
∀[l:BoundedLattice]. ∀[eq:EqDecider(Point(l))]. ∀[s:fset(Point(l))]. ∀[x:Point(l)].
(\/(s) ≤ x ⇐⇒ ∀p:Point(l). (p ∈ s ⇒ p ≤ x))
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),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
lattice-le: a ≤ b,
guard: {T}
Lemmas referenced :
bdd-lattice_wf,
deq_wf,
fset_wf,
all_wf,
lattice-le_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-axioms_wf,
and_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
lattice-point_wf,
fset-member_wf,
decidable-equal-deq,
lattice-fset-join_wf,
bdd-lattice-subtype-lattice,
lattice-le_transitivity,
lattice-fset-join-is-lub
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
introduction,
independent_pairFormation,
lambdaFormation,
applyEquality,
sqequalRule,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
instantiate,
lambdaEquality,
cumulativity,
functionEquality,
independent_pairEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[eq:EqDecider(Point(l))]. \mforall{}[s:fset(Point(l))]. \mforall{}[x:Point(l)].
(\mbackslash{}/(s) \mleq{} x \mLeftarrow{}{}\mRightarrow{} \mforall{}p:Point(l). (p \mmember{} s {}\mRightarrow{} p \mleq{} x))
Date html generated:
2020_05_20-AM-08_43_54
Last ObjectModification:
2016_01_19-PM-03_34_26
Theory : lattices
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