Nuprl Lemma : face-lattice-le
∀T:Type. ∀eq:EqDecider(T). ∀x,y:Point(face-lattice(T;eq)).
(x ≤ y ⇐⇒ ∀s:fset(T + T). (s ∈ x ⇒ (↓∃t:fset(T + T). (t ∈ y ∧ t ⊆ s))))
Proof
Definitions occuring in Statement :
face-lattice: face-lattice(T;eq),
lattice-le: 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],
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
sq_stable: SqStable(P),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
cand: A c∧ B,
false: False,
fset-member: a ∈ s,
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
bfalse: ff
Lemmas referenced :
face-lattice-le-1,
fl-point-sq,
istype-void,
fset-ac-le-implies2,
union-deq_wf,
fset-member_wf,
fset_wf,
deq-fset_wf,
fset-ac-le_wf,
squash_wf,
f-subset_wf,
lattice-le_wf,
face-lattice_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
deq_wf,
istype-universe,
fset-all-iff,
iff_weakening_uiff,
fset-all_wf,
bnot_wf,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
isect_wf,
assert_wf,
assert_witness,
sq_stable__assert,
assert_of_bnot,
equal-wf-T-base,
assert-fset-null,
istype-assert,
member-fset-filter,
assert-deq-f-subset
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
isect_memberEquality_alt,
voidElimination,
productElimination,
independent_pairFormation,
unionEquality,
setElimination,
rename,
because_Cache,
independent_functionElimination,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
universeIsType,
functionIsType,
productEquality,
applyEquality,
promote_hyp,
inhabitedIsType,
instantiate,
lambdaEquality_alt,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeEquality,
isect_memberFormation_alt,
isectIsTypeImplies,
equalityIsType3,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}x,y:Point(face-lattice(T;eq)).
(x \mleq{} y \mLeftarrow{}{}\mRightarrow{} \mforall{}s:fset(T + T). (s \mmember{} x {}\mRightarrow{} (\mdownarrow{}\mexists{}t:fset(T + T). (t \mmember{} y \mwedge{} t \msubseteq{} s))))
Date html generated:
2019_10_31-AM-07_22_24
Last ObjectModification:
2018_11_10-PM-00_15_49
Theory : lattices
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