Nuprl Lemma : free-dlwc-le
∀[T:Type]
∀eq:EqDecider(T). ∀cs:T ⟶ fset(fset(T)). ∀x,y:Point(free-dist-lattice-with-constraints(T;eq;x.cs[x])).
(x ≤ y ⇐⇒ fset-ac-le(eq;x;y))
Proof
Definitions occuring in Statement :
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
lattice-le: a ≤ b,
lattice-point: Point(l),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
lattice-le: a ≤ b,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
prop: ℙ,
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
iff: P ⇐⇒ Q,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
order: Order(T;x,y.R[x; y]),
refl: Refl(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
free-dlwc-point,
lattice-point_wf,
free-dist-lattice-with-constraints_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,
fset_wf,
deq_wf,
fset-ac-order-constrained,
fset-contains-none_wf,
fset-constrained-ac-glb-is-glb,
fset-contains-none-closed-downward,
assert_wf,
f-subset_wf,
fset-ac-le_wf,
squash_wf,
true_wf,
fset-antichain_wf,
fset-all_wf,
subtype_rel_self,
iff_weakening_equal,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
fset-constrained-ac-glb_wf,
free-dlwc-meet
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
hypothesis,
because_Cache,
hypothesisEquality,
lambdaEquality,
applyEquality,
instantiate,
productEquality,
cumulativity,
independent_isectElimination,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
productElimination,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}cs:T {}\mrightarrow{} fset(fset(T)).
\mforall{}x,y:Point(free-dist-lattice-with-constraints(T;eq;x.cs[x])).
(x \mleq{} y \mLeftarrow{}{}\mRightarrow{} fset-ac-le(eq;x;y))
Date html generated:
2020_05_20-AM-08_48_35
Last ObjectModification:
2018_05_20-PM-10_12_18
Theory : lattices
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