Nuprl Lemma : free-dl-basis
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:Point(free-dist-lattice(T; eq))].
(x = \/(λs./\(λx.free-dl-inc(x)"(s))"(x)) ∈ Point(free-dist-lattice(T; eq)))
Proof
Definitions occuring in Statement :
free-dl-inc: free-dl-inc(x),
free-dist-lattice: free-dist-lattice(T; eq),
lattice-fset-join: \/(s),
lattice-fset-meet: /\(s),
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
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],
cand: A c∧ B,
guard: {T},
lattice-point: Point(l),
record-select: r.x,
free-dist-lattice: free-dist-lattice(T; eq),
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,
fset-singleton: {x},
cons: [a / b],
sq_stable: SqStable(P),
fset-member: a ∈ s,
assert: ↑b,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
lattice-fset-join: \/(s),
lattice-0: 0,
fset-add: fset-add(eq;x;s),
true: True,
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
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_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,
deq_wf,
free-dl-point,
fset-antichain-singleton,
fset-singleton_wf,
fset_wf,
assert_wf,
fset-antichain_wf,
deq-fset_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
lattice-fset-join-is-lub,
bdd-distributive-lattice-subtype-bdd-lattice,
fset-image_wf,
fset-member_wf,
member-fset-image-iff,
free-dl-le,
fset-all-iff,
bnot_wf,
fset-null_wf,
fset-filter_wf,
deq-f-subset_wf,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
assert_of_bnot,
member-fset-singleton,
not_wf,
assert_witness,
assert-fset-null,
fset-filter-is-empty,
assert-deq-f-subset,
f-subset_weakening,
lattice-fset-join_wf,
decidable__equal_free-dl,
subtype_rel_self,
fset-ac-le-implies2,
sq_stable__fset-member,
set_wf,
fset-induction,
squash_wf,
exists_wf,
sq_stable__all,
sq_stable__squash,
empty-fset_wf,
fset-add_wf,
fset-union_wf,
true_wf,
lattice-fset-join-union,
iff_weakening_equal,
free-dl-join,
member-fset-minimals,
f-proper-subset-dec_wf,
member-fset-union,
member-fset-add,
lattice-fset-join-singleton,
and_wf,
assert-fset-antichain,
deq-implies,
lattice-le-order,
bdd-distributive-lattice-subtype-lattice,
lattice-fset-meet-free-dl-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
voidElimination,
voidEquality,
lambdaFormation,
dependent_set_memberEquality,
setElimination,
rename,
setEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
functionExtensionality,
imageElimination,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
natural_numberEquality,
unionElimination,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:Point(free-dist-lattice(T; eq))].
(x = \mbackslash{}/(\mlambda{}s./\mbackslash{}(\mlambda{}x.free-dl-inc(x)"(s))"(x)))
Date html generated:
2017_10_05-AM-00_36_16
Last ObjectModification:
2017_07_28-AM-09_14_53
Theory : lattices
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