Nuprl Lemma : free-dlwc-basis
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))]. ∀[x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
(x = \/(λs./\(λx.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)) ∈ 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-join: \/(s),
lattice-fset-meet: /\(s),
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset: fset(T),
deq: EqDecider(T),
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
top: Top,
bdd-distributive-lattice: BoundedDistributiveLattice,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
cand: A c∧ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
false: False,
not: ¬A,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
guard: {T},
cons: [a / b],
fset-singleton: {x},
sq_stable: SqStable(P),
btrue: tt,
eq_atom: x =a y,
record-update: r[x := v],
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
record-select: r.x,
lattice-0: 0,
lattice-fset-join: \/(s),
bfalse: ff,
it: ⋅,
nil: [],
empty-fset: {},
list_ind: list_ind,
reduce: reduce(f;k;as),
deq-member: x ∈b L,
ifthenelse: if b then t else f fi ,
assert: ↑b,
fset-member: a ∈ s,
fset-add: fset-add(eq;x;s),
true: True,
fset-constrained-ac-lub: lub(P;ac1;ac2),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
or: P ∨ Q,
decidable: Dec(P),
f-proper-subset: xs ⊆≠ ys,
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
lattice-point_wf,
free-dist-lattice-with-constraints_wf,
fset_wf,
deq_wf,
istype-universe,
strong-subtype-set2,
fset-contains-none_wf,
fset-all_wf,
fset-antichain_wf,
assert_wf,
strong-subtype-deq-subtype,
deq-fset_wf,
free-dlwc-point,
lattice-join_wf,
lattice-meet_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
fset-image_wf,
fset-subtype2,
assert_witness,
equal_wf,
and_wf,
member-fset-singleton,
fset-member_wf,
isect_wf,
uall_wf,
iff_weakening_uiff,
fset-all-iff,
fset-antichain-singleton,
fset-singleton_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-fset-join-is-lub,
member-fset-image-iff,
free-dlwc-le,
not_wf,
assert_of_bnot,
f-subset_wf,
iff_wf,
all_wf,
bool_wf,
deq-f-subset_wf,
fset-filter_wf,
fset-null_wf,
bnot_wf,
assert-fset-null,
assert-deq-f-subset,
fset-filter-is-empty,
f-subset_weakening,
decidable__equal-free-dist-lattice-with-constraints-point,
lattice-fset-join_wf,
fset-ac-le-implies2,
sq_stable__fset-member,
set_wf,
fset-add_wf,
empty-fset_wf,
sq_stable__squash,
sq_stable__all,
exists_wf,
squash_wf,
fset-induction,
fset-union_wf,
iff_weakening_equal,
lattice-fset-join-union,
true_wf,
free-dlwc-join,
member-fset-union,
f-proper-subset-dec_wf,
member-fset-minimals,
member-fset-add,
lattice-fset-join-singleton,
assert-fset-antichain,
deq-implies,
bdd-distributive-lattice-subtype-lattice,
lattice-le-order,
istype-void,
lattice-fset-meet-free-dlwc-inc,
sq_stable__assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
instantiate,
universeEquality,
independent_isectElimination,
functionExtensionality,
lambdaEquality,
productEquality,
setEquality,
cumulativity,
productElimination,
rename,
setElimination,
voidEquality,
voidElimination,
isect_memberEquality,
equalityTransitivity,
dependent_functionElimination,
levelHypothesis,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
addLevel,
isect_memberFormation,
independent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
lambdaFormation,
imageElimination,
functionEquality,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
unionElimination,
inlFormation,
inrFormation,
lambdaFormation_alt,
equalityIsType1,
setIsType
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))].
\mforall{}[x:Point(free-dist-lattice-with-constraints(T;eq;x.Cs[x]))].
(x = \mbackslash{}/(\mlambda{}s./\mbackslash{}(\mlambda{}x.free-dlwc-inc(eq;a.Cs[a];x)"(s))"(x)))
Date html generated:
2020_05_20-AM-08_49_49
Last ObjectModification:
2018_11_08-PM-06_01_41
Theory : lattices
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