Nuprl Lemma : face_lattice-basis
∀I:fset(ℕ). ∀x:Point(face_lattice(I)).
∃fs:fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )
(x = \/(λpr.irr_face(I;fst(pr);snd(pr))"(fs)) ∈ Point(face_lattice(I)))
Proof
Definitions occuring in Statement :
irr_face: irr_face(I;as;bs),
face_lattice-deq: face_lattice-deq(),
face_lattice: face_lattice(I),
names-deq: NamesDeq,
names: names(I),
lattice-fset-join: \/(s),
lattice-point: Point(l),
fset-image: f"(s),
deq-fset: deq-fset(eq),
fset-disjoint: fset-disjoint(eq;as;bs),
fset: fset(T),
product-deq: product-deq(A;B;a;b),
nat: ℕ,
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
lambda: λx.A[x],
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
pi1: fst(t),
pi2: snd(t),
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q
Lemmas referenced :
face_lattice_components_wf,
equal_wf,
lattice-fset-join_wf,
decidable__equal_face_lattice,
fset-image_wf,
product-deq_wf,
deq-fset_wf,
names-deq_wf,
strong-subtype-deq-subtype,
fset_wf,
names_wf,
assert_wf,
fset-disjoint_wf,
pi1_wf_top,
pi2_wf,
strong-subtype-set2,
face_lattice-deq_wf,
irr_face_wf,
set_wf,
face_lattice_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
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,
lattice-meet_wf,
lattice-join_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
isectElimination,
independent_functionElimination,
productEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
universeEquality
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}x:Point(face\_lattice(I)).
\mexists{}fs:fset(\{p:fset(names(I)) \mtimes{} fset(names(I))| \muparrow{}fset-disjoint(NamesDeq;fst(p);snd(p))\} )
(x = \mbackslash{}/(\mlambda{}pr.irr\_face(I;fst(pr);snd(pr))"(fs)))
Date html generated:
2017_10_05-AM-01_11_39
Last ObjectModification:
2017_03_02-PM-10_27_47
Theory : cubical!type!theory
Home
Index