Nuprl Lemma : lattice-fset-meet-union
∀[l:BoundedLattice]. ∀[eq:EqDecider(Point(l))]. ∀[s1,s2:fset(Point(l))]. (/\(s1 ⋃ s2) = /\(s1) ∧ /\(s2) ∈ Point(l))
Proof
Definitions occuring in Statement :
lattice-fset-meet: /\(s),
bdd-lattice: BoundedLattice,
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
bdd-lattice: BoundedLattice,
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
false: False,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lattice-fset-meet: /\(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
lattice-1: 1,
record-select: r.x,
lattice-axioms: lattice-axioms(l)
Lemmas referenced :
fset-induction,
lattice-point_wf,
all_wf,
fset_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,
fset-union_wf,
lattice-meet_wf,
sq_stable__all,
sq_stable__equal,
fset-member_wf,
istype-void,
deq_wf,
bdd-lattice_wf,
squash_wf,
true_wf,
istype-universe,
decidable_wf,
empty-fset-union,
empty-fset_wf,
subtype_rel_self,
iff_weakening_equal,
lattice-1_wf,
lattice-1-meet,
fset-add-union,
fset-add_wf,
reduce_cons_lemma,
fset-add-as-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
dependent_functionElimination,
lambdaEquality_alt,
instantiate,
productEquality,
cumulativity,
universeIsType,
independent_isectElimination,
independent_functionElimination,
lambdaFormation_alt,
inhabitedIsType,
axiomEquality,
functionIsTypeImplies,
functionIsType,
equalityIstype,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionEquality,
imageMemberEquality,
baseClosed,
natural_numberEquality,
productElimination,
setElimination,
rename,
hyp_replacement,
applyLambdaEquality,
Error :memTop
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[eq:EqDecider(Point(l))]. \mforall{}[s1,s2:fset(Point(l))].
(/\mbackslash{}(s1 \mcup{} s2) = /\mbackslash{}(s1) \mwedge{} /\mbackslash{}(s2))
Date html generated:
2020_05_20-AM-08_44_03
Last ObjectModification:
2020_01_31-AM-10_28_54
Theory : lattices
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