Nuprl Lemma : face-lattice-subset-le
∀T:Type. ∀eq:EqDecider(T). ∀x,y:Point(face-lattice(T;eq)). (x ⊆ y ⇒ x ≤ y)
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: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
all: ∀x:A. B[x],
implies: 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,
squash: ↓T,
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T},
f-subset: xs ⊆ ys
Lemmas referenced :
f-subset_weakening,
deq_wf,
lattice-join_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,
face-lattice_wf,
lattice-point_wf,
f-subset_wf,
deq-fset_wf,
fset-member_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,
productElimination,
independent_functionElimination,
introduction,
imageElimination,
imageMemberEquality,
baseClosed,
applyEquality,
because_Cache,
cumulativity,
instantiate,
productEquality,
universeEquality,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}x,y:Point(face-lattice(T;eq)). (x \msubseteq{} y {}\mRightarrow{} x \mleq{} y)
Date html generated:
2020_05_20-AM-08_52_05
Last ObjectModification:
2016_01_20-AM-00_35_06
Theory : lattices
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