Nuprl Lemma : lattice-join-is-lub
∀l:Lattice. ∀a,b:Point(l). least-upper-bound(Point(l);x,y.x ≤ y;a;b;a ∨ b)
Proof
Definitions occuring in Statement :
lattice-le: a ≤ b,
lattice: Lattice,
lattice-join: a ∨ b,
lattice-point: Point(l),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
and: P ∧ Q,
lattice-le: a ≤ b,
uall: ∀[x:A]. B[x],
member: t ∈ T,
cand: A c∧ B,
implies: P ⇒ Q,
prop: ℙ,
lattice: Lattice,
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice_properties,
equal_wf,
lattice-point_wf,
lattice-meet_wf,
lattice_wf,
lattice-join_wf,
lattice-le-iff,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
equalitySymmetry,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
sqequalRule,
setElimination,
rename,
because_Cache,
hyp_replacement,
applyLambdaEquality,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}l:Lattice. \mforall{}a,b:Point(l). least-upper-bound(Point(l);x,y.x \mleq{} y;a;b;a \mvee{} b)
Date html generated:
2017_10_05-AM-00_31_02
Last ObjectModification:
2017_07_28-AM-09_12_53
Theory : lattices
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