Nuprl Lemma : lattice-fset-meet-free-dlwc-inc
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[s:fset(T)].
/\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s)) = {s} ∈ Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))
supposing ↑fset-contains-none(eq;s;x.Cs[x])
Proof
Definitions occuring in Statement :
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
lattice-fset-meet: /\(s),
lattice-point: Point(l),
fset-image: f"(s),
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
deq-fset: deq-fset(eq),
fset-singleton: {x},
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
uiff: uiff(P;Q),
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
all: ∀x:A. B[x],
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
fset-singleton: {x},
fset-filter: {x ∈ s | P[x]},
bool: 𝔹,
unit: Unit,
it: ⋅,
fset-null: fset-null(s),
assert: ↑b,
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
f-subset: xs ⊆ ys,
sq_stable: SqStable(P),
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
free-dlwc-point,
deq-fset_wf,
fset_wf,
strong-subtype-deq-subtype,
assert_wf,
fset-antichain_wf,
fset-all_wf,
fset-contains-none_wf,
strong-subtype-set2,
fset-singleton_wf,
fset-antichain-singleton,
fset-all-iff,
member-fset-singleton,
assert_witness,
fset-member_wf,
lattice-fset-meet-is-glb,
free-dist-lattice-with-constraints_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
fset-image_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,
free-dlwc-inc_wf,
member-fset-image-iff,
deq_wf,
free-dlwc-le,
bnot_wf,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
assert_of_bnot,
not_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
filter_cons_lemma,
filter_nil_lemma,
equal-wf-T-base,
uiff_transitivity,
assert-deq-f-subset,
iff_transitivity,
iff_weakening_uiff,
null_cons_lemma,
false_wf,
f-singleton-subset,
assert-fset-null,
f-union_wf,
fset-filter-is-empty,
exists_wf,
member-f-union,
fset-member_witness,
and_wf,
lattice-fset-meet_wf,
decidable__equal-free-dist-lattice-with-constraints-point,
fset-ac-le-implies2,
ifthenelse_wf,
empty-fset_wf,
sq_stable__fset-member,
mem_empty_lemma,
lattice-le-order,
bdd-distributive-lattice-subtype-lattice
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
cumulativity,
hypothesisEquality,
applyEquality,
setEquality,
productEquality,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
because_Cache,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
equalityTransitivity,
instantiate,
universeEquality,
lambdaFormation,
axiomEquality,
functionEquality,
dependent_functionElimination,
setElimination,
rename,
imageElimination,
unionElimination,
baseClosed,
equalityElimination,
impliesFunctionality,
dependent_pairFormation,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))]. \mforall{}[s:fset(T)].
/\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s)) = \{s\} supposing \muparrow{}fset-contains-none(eq;s;x.Cs[x])
Date html generated:
2017_10_05-AM-00_37_33
Last ObjectModification:
2017_07_28-AM-09_15_32
Theory : lattices
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