Nuprl Lemma : free-dlwc-satisfies-constraints
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))].
((∀x:T. ∀c:fset(T). (c ∈ Cs[x] ⇒ x ∈ c))
⇒ (∀x:T. ∀c:fset(T).
(c ∈ Cs[x]
⇒ (/\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(c)) = 0 ∈ Point(free-dist-lattice-with-constraints(T;eq;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-0: 0,
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
bdd-distributive-lattice: BoundedDistributiveLattice,
lattice-0: 0,
record-select: r.x,
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
empty-fset: {},
nil: [],
it: ⋅,
cand: A c∧ B,
assert: ↑b,
fset-antichain: fset-antichain(eq;ac),
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
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,
true: True,
fset-all: fset-all(s;x.P[x]),
uiff: uiff(P;Q),
false: False,
not: ¬A,
guard: {T},
f-subset: xs ⊆ ys,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
squash: ↓T,
free-dlwc-inc: free-dlwc-inc(eq;a.Cs[a];x),
bool: 𝔹,
unit: Unit,
sq_type: SQType(T),
bnot: ¬bb,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P)
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,
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,
lattice-fset-meet_wf,
decidable__equal-free-dist-lattice-with-constraints-point,
isect_wf,
fset-member_wf,
lattice-le_wf,
all_wf,
deq_wf,
fset-extensionality,
empty-fset_wf,
mem_empty_lemma,
fset-member_witness,
false_wf,
fset-all-iff,
assert-fset-contains-none,
f-subset_wf,
decidable__assert,
fset-singleton_wf,
fset-antichain-singleton,
iff_weakening_uiff,
member-fset-singleton,
assert_witness,
member-fset-image-iff,
subtype_rel-equal,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
bool_wf,
iff_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
assert-fset-null,
fset-filter-is-empty,
assert-deq-f-subset,
exists_wf,
free-dlwc-le,
fset-ac-le-implies2,
sq_stable__fset-member,
f-singleton-subset,
ifthenelse_wf,
equal-wf-base-T,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
cumulativity,
hypothesisEquality,
applyEquality,
setEquality,
productEquality,
lambdaEquality,
functionExtensionality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
because_Cache,
instantiate,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
functionEquality,
axiomEquality,
independent_pairEquality,
unionElimination,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
imageMemberEquality,
baseClosed,
equalityElimination,
promote_hyp,
imageElimination,
addLevel,
levelHypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))].
((\mforall{}x:T. \mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} x \mmember{} c))
{}\mRightarrow{} (\mforall{}x:T. \mforall{}c:fset(T). (c \mmember{} Cs[x] {}\mRightarrow{} (/\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(c)) = 0))))
Date html generated:
2020_05_20-AM-08_49_59
Last ObjectModification:
2017_07_28-AM-09_15_38
Theory : lattices
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