Nuprl Lemma : lattice-axioms-iff-order
∀l:LatticeStructure
(∃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]))
⇐⇒ lattice-axioms(l))
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]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
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],
rev_implies: P ⇐ Q,
cand: A c∧ B,
lattice: Lattice,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
lattice-le: a ≤ b,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
exists: ∃x:A. B[x]
Lemmas referenced :
lattice-axioms-from-order,
exists_wf,
lattice-point_wf,
uall_wf,
least-upper-bound_wf,
lattice-join_wf,
greatest-lower-bound_wf,
lattice-meet_wf,
order_wf,
lattice-axioms_wf,
lattice-structure_wf,
lattice-le_wf,
lattice-join-is-lub,
lattice-meet-is-glb,
lattice-le-order
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
instantiate,
functionEquality,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
because_Cache,
productEquality,
isect_memberFormation,
introduction,
dependent_functionElimination,
dependent_set_memberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
rename,
dependent_pairEquality
Latex:
\mforall{}l:LatticeStructure
(\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]))
\mLeftarrow{}{}\mRightarrow{} lattice-axioms(l))
Date html generated:
2016_05_18-AM-11_22_33
Last ObjectModification:
2015_12_28-PM-02_03_42
Theory : lattices
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