Nuprl Lemma : fl-point
∀[T:Type]. ∀[eq:EqDecider(T)].
Point(face-lattice(T;eq)) ≡ {ac:fset(fset(T + T))|
(↑fset-antichain(union-deq(T;T;eq;eq);ac))
∧ (∀a:fset(T + T). (a ∈ ac ⇒ (∀z:T. (¬(inl z ∈ a ∧ inr z ∈ a)))))}
Proof
Definitions occuring in Statement :
face-lattice: face-lattice(T;eq),
lattice-point: Point(l),
fset-antichain: fset-antichain(eq;ac),
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
assert: ↑b,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
face-lattice: face-lattice(T;eq),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
face-lattice-constraints: face-lattice-constraints(x),
f-subset: xs ⊆ ys,
or: P ∨ Q,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
free-dlwc-point,
fset-member_wf,
union-deq_wf,
fset_wf,
deq-fset_wf,
assert_wf,
fset-antichain_wf,
all_wf,
not_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
deq_wf,
fset-all-iff,
assert-fset-contains-none,
fset-pair_wf,
member-fset-singleton,
fset-member_witness,
member-fset-pair,
squash_wf,
true_wf,
iff_weakening_equal,
f-subset_wf,
assert_witness,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
independent_pairFormation,
lambdaEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
productElimination,
lambdaFormation,
independent_functionElimination,
productEquality,
unionEquality,
cumulativity,
inlEquality,
inrEquality,
because_Cache,
functionEquality,
setEquality,
independent_pairEquality,
axiomEquality,
universeEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inrFormation,
hyp_replacement,
applyLambdaEquality,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)].
Point(face-lattice(T;eq)) \mequiv{} \{ac:fset(fset(T + T))|
(\muparrow{}fset-antichain(union-deq(T;T;eq;eq);ac))
\mwedge{} (\mforall{}a:fset(T + T). (a \mmember{} ac {}\mRightarrow{} (\mforall{}z:T. (\mneg{}(inl z \mmember{} a \mwedge{} inr z \mmember{} a)))))\}
Date html generated:
2020_05_20-AM-08_51_08
Last ObjectModification:
2017_07_28-AM-09_15_50
Theory : lattices
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