Nuprl Lemma : face_lattice_components_wf
∀I:fset(ℕ). ∀x:Point(face_lattice(I)).
(face_lattice_components(I;x) ∈ {fs:fset({p:fset(names(I)) × fset(names(I))| ↑fset-disjoint(NamesDeq;fst(p);snd(p))} )\000C|
x = \/(λpr.irr_face(I;fst(pr);snd(pr))"(fs)) ∈ Point(face_lattice(I))} )
Proof
Definitions occuring in Statement :
face_lattice_components: face_lattice_components(I;x),
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],
member: t ∈ T,
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],
face_lattice: face_lattice(I),
uall: ∀[x:A]. B[x],
member: t ∈ T,
face_lattice-deq: face_lattice-deq(),
union-deq: union-deq(A;B;a;b),
face-lattice0: (x=0),
fl0: (x=0),
face-lattice1: (x=1),
fl1: (x=1),
top: Top,
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,
pi1: fst(t),
pi2: snd(t),
outl: outl(x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
outr: outr(x),
isr: isr(x),
uiff: uiff(P;Q),
not: ¬A,
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
true: True,
btrue: tt,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
face-lattice-constraints: face-lattice-constraints(x),
names: names(I),
nat: ℕ,
sq_type: SQType(T),
f-subset: xs ⊆ ys,
or: P ∨ Q,
face_lattice_components: face_lattice_components(I;x),
compose: f o g,
irr_face: irr_face(I;as;bs),
cand: A c∧ B,
respects-equality: respects-equality(S;T),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P)
Lemmas referenced :
face-lattice-basis,
names_wf,
names-deq_wf,
face_lattice-deq_wf,
fl0_wf,
fl1_wf,
fl-point-sq,
istype-void,
lattice-point_wf,
face_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,
fset_wf,
nat_wf,
fset-image_wf,
assert_wf,
fset-contains-none_wf,
union-deq_wf,
face-lattice-constraints_wf,
fset-disjoint_wf,
deq-fset_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
product-deq_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
pi2_wf,
fset-mapfilter_wf,
istype-assert,
isl_wf,
isr_wf,
assert-fset-disjoint,
btrue_wf,
bfalse_wf,
fset-member_wf,
member-fset-mapfilter,
assert-fset-contains-none,
squash_wf,
true_wf,
deq_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
fset-pair_wf,
set_subtype_base,
int-deq_wf,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
istype-nat,
int_subtype_base,
subtype_base_sq,
subtype_rel_universe1,
member-fset-singleton,
fset-singleton_wf,
fset-member_witness,
member-fset-pair,
fset-subtype2,
subtype_rel_fset,
fset-all-iff,
lattice-fset-join_wf,
decidable__equal_face_lattice,
irr_face_wf,
all_wf,
decidable_wf,
bdd-lattice_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-fset-meet_wf,
fset-image-compose,
fset-union_wf,
iff_weakening_uiff,
exists_wf,
member-fset-image-iff,
equal-wf-T-base,
fset-extensionality,
member-fset-union,
equal-wf-base-T,
outl_wf,
union_subtype_base,
respects-equality-face-lattice-point-2,
fset-find_wf,
fl-eq_wf,
assert-fl-eq,
fset-singletons-equal,
sq_stable__fset-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
unionIsType,
universeIsType,
because_Cache,
isect_memberEquality_alt,
voidElimination,
setElimination,
rename,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
setEquality,
unionEquality,
productIsType,
dependent_set_memberEquality_alt,
independent_pairEquality,
setIsType,
independent_functionElimination,
imageElimination,
dependent_functionElimination,
inlEquality_alt,
hyp_replacement,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
independent_pairFormation,
equalityIstype,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
inrEquality_alt,
isect_memberFormation_alt,
functionIsType,
functionExtensionality,
promote_hyp,
inlFormation_alt,
inrFormation_alt,
isectIsTypeImplies,
dependent_pairFormation_alt
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}x:Point(face\_lattice(I)).
(face\_lattice\_components(I;x) \mmember{} \{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:
2019_11_04-PM-05_33_14
Last ObjectModification:
2018_12_13-PM-00_50_29
Theory : cubical!type!theory
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