Nuprl Lemma : implies-le-face-lattice-join
∀T:Type. ∀eq:EqDecider(T). ∀x,y,z:Point(face-lattice(T;eq)).
((∀s:fset(T + T). (s ∈ z ⇒ ((↓∃t:fset(T + T). (t ∈ x ∧ t ⊆ s)) ∨ (↓∃t:fset(T + T). (t ∈ y ∧ t ⊆ s))))) ⇒ z ≤ x ∨ y)
Proof
Definitions occuring in Statement :
face-lattice: face-lattice(T;eq),
lattice-le: a ≤ b,
lattice-join: a ∨ b,
lattice-point: Point(l),
deq-fset: deq-fset(eq),
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
squash: ↓T,
exists: ∃x:A. B[x],
or: P ∨ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
face-lattice: face-lattice(T;eq),
fset-constrained-ac-lub: lub(P;ac1;ac2),
cand: A c∧ B
Lemmas referenced :
subtype_rel_sets,
fset-ac-lub-covers,
free-dlwc-join,
deq_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
lattice-point_wf,
f-subset_wf,
exists_wf,
squash_wf,
or_wf,
all_wf,
deq-fset_wf,
fset-member_wf,
ac-covers_wf,
assert-ac-covers,
face-lattice_wf,
lattice-join_wf,
face-lattice-le,
face-lattice-constraints_wf,
fset-contains-none_wf,
fset-all_wf,
union-deq_wf,
fset-antichain_wf,
assert_wf,
and_wf,
fset_wf,
fl-point-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
setEquality,
unionEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
productElimination,
independent_functionElimination,
introduction,
independent_pairFormation,
independent_isectElimination,
addLevel,
allFunctionality,
impliesFunctionality,
orFunctionality,
levelHypothesis,
promote_hyp,
allLevelFunctionality,
impliesLevelFunctionality,
orLevelFunctionality,
imageElimination,
imageMemberEquality,
baseClosed,
cumulativity,
functionEquality,
productEquality,
instantiate,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}x,y,z:Point(face-lattice(T;eq)).
((\mforall{}s:fset(T + T)
(s \mmember{} z {}\mRightarrow{} ((\mdownarrow{}\mexists{}t:fset(T + T). (t \mmember{} x \mwedge{} t \msubseteq{} s)) \mvee{} (\mdownarrow{}\mexists{}t:fset(T + T). (t \mmember{} y \mwedge{} t \msubseteq{} s)))))
{}\mRightarrow{} z \mleq{} x \mvee{} y)
Date html generated:
2020_05_20-AM-08_52_39
Last ObjectModification:
2016_01_20-PM-10_31_03
Theory : lattices
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