Nuprl Lemma : lattice-axioms-from-order
∀[l:LatticeStructure]
lattice-axioms(l)
supposing ∃R:Point(l) ⟶ Point(l) ⟶ ℙ
(((∀[a,b:Point(l)]. least-upper-bound(Point(l);x,y.R[x;y];a;b;a ∨ b))
∧ (∀[a,b:Point(l)]. greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a ∧ b)))
∧ Order(Point(l);x,y.R[x;y]))
Proof
Definitions occuring in Statement :
lattice-axioms: lattice-axioms(l),
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
lattice-structure: LatticeStructure,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
order: Order(T;x,y.R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
lattice-axioms: lattice-axioms(l),
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
all: ∀x:A. B[x],
order: Order(T;x,y.R[x; y]),
refl: Refl(T;x,y.E[x; y]),
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c)
Lemmas referenced :
lattice-point_wf,
exists_wf,
uall_wf,
least-upper-bound_wf,
lattice-join_wf,
greatest-lower-bound_wf,
lattice-meet_wf,
order_wf,
lattice-structure_wf,
glb-com,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lub-com,
glb-assoc,
lub-assoc,
least-upper-bound-unique,
greatest-lower-bound-unique
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
hypothesis,
because_Cache,
sqequalRule,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
independent_pairEquality,
extract_by_obid,
instantiate,
functionEquality,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
productEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination
Latex:
\mforall{}[l:LatticeStructure]
lattice-axioms(l)
supposing \mexists{}R:Point(l) {}\mrightarrow{} Point(l) {}\mrightarrow{} \mBbbP{}
(((\mforall{}[a,b:Point(l)]. least-upper-bound(Point(l);x,y.R[x;y];a;b;a \mvee{} b))
\mwedge{} (\mforall{}[a,b:Point(l)]. greatest-lower-bound(Point(l);x,y.R[x;y];a;b;a \mwedge{} b)))
\mwedge{} Order(Point(l);x,y.R[x;y]))
Date html generated:
2017_10_05-AM-00_30_25
Last ObjectModification:
2017_07_28-AM-09_12_31
Theory : lattices
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