Nuprl Lemma : lattice-fset-meet-is-1
∀[l:BoundedLattice]. ∀[eq:EqDecider(Point(l))]. ∀[s:fset(Point(l))].
uiff(/\(s) = 1 ∈ Point(l);∀x:Point(l). (x ∈ s ⇒ (x = 1 ∈ Point(l))))
Proof
Definitions occuring in Statement :
lattice-fset-meet: /\(s),
bdd-lattice: BoundedLattice,
lattice-1: 1,
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice-fset-meet-is-glb,
fset-member_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,
equal_wf,
lattice-fset-meet_wf,
decidable-equal-deq,
lattice-1_wf,
all_wf,
fset_wf,
deq_wf,
bdd-lattice_wf,
lattice-le_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-1-le-iff,
le-lattice-1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
lambdaFormation,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
axiomEquality,
independent_functionElimination,
setElimination,
rename,
functionEquality,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[eq:EqDecider(Point(l))]. \mforall{}[s:fset(Point(l))].
uiff(/\mbackslash{}(s) = 1;\mforall{}x:Point(l). (x \mmember{} s {}\mRightarrow{} (x = 1)))
Date html generated:
2020_05_20-AM-08_44_15
Last ObjectModification:
2017_07_28-AM-09_14_06
Theory : lattices
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