Nuprl Lemma : lattice-meet-is-glb
∀l:Lattice. ∀a,b:Point(l). greatest-lower-bound(Point(l);x,y.x ≤ y;a;b;a ∧ b)
Proof
Definitions occuring in Statement :
lattice-le: a ≤ b,
lattice: Lattice,
lattice-meet: a ∧ b,
lattice-point: Point(l),
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
lattice: Lattice,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
cand: A c∧ B,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
lattice-le: a ≤ b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
lattice-le-iff,
lattice-meet_wf,
lattice-join_wf,
lattice_properties,
iff_weakening_uiff,
lattice-le_wf,
equal_wf,
lattice-point_wf,
lattice_wf,
squash_wf,
true_wf,
istype-universe,
lattice-structure_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
independent_isectElimination,
equalityIstype,
independent_functionElimination,
promote_hyp,
dependent_functionElimination,
universeIsType,
inhabitedIsType,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
instantiate,
universeEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed
Latex:
\mforall{}l:Lattice. \mforall{}a,b:Point(l). greatest-lower-bound(Point(l);x,y.x \mleq{} y;a;b;a \mwedge{} b)
Date html generated:
2020_05_20-AM-08_25_34
Last ObjectModification:
2020_01_03-AM-00_38_51
Theory : lattices
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