Nuprl Lemma : lattice-fset-meet-is-glb
∀[l:BoundedLattice]. ∀[eq:EqDecider(Point(l))].
((∀[s:fset(Point(l))]. ∀[x:Point(l)]. /\(s) ≤ x supposing x ∈ s)
∧ (∀[s:fset(Point(l))]. ∀[v:Point(l)]. ((∀x:Point(l). (x ∈ s
⇒ v ≤ x))
⇒ v ≤ /\(s))))
Proof
Definitions occuring in Statement :
lattice-fset-meet: /\(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
,
empty-fset: {}
,
lattice-fset-meet: /\(s)
,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c)
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-meet_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-meet-union,
fset-singleton_wf,
iff_weakening_equal,
lattice-fset-meet-singleton,
member-fset-add,
lattice-le_transitivity,
bdd-lattice-subtype-lattice,
lattice-meet-le,
lattice-le_weakening,
sq_stable__all,
reduce_nil_lemma,
le-lattice-1,
lattice-meet-is-glb,
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,
Error :applyLambdaEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[eq:EqDecider(Point(l))].
((\mforall{}[s:fset(Point(l))]. \mforall{}[x:Point(l)]. /\mbackslash{}(s) \mleq{} x supposing x \mmember{} s)
\mwedge{} (\mforall{}[s:fset(Point(l))]. \mforall{}[v:Point(l)]. ((\mforall{}x:Point(l). (x \mmember{} s {}\mRightarrow{} v \mleq{} x)) {}\mRightarrow{} v \mleq{} /\mbackslash{}(s))))
Date html generated:
2016_10_26-PM-00_55_39
Last ObjectModification:
2016_07_12-AM-08_58_02
Theory : lattices
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