Nuprl Lemma : name-morph-satisfies-fset-join
∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀s:fset(Point(face_lattice(I))).
((\/(s) f) = 1 ⇐⇒ ↓∃a:Point(face_lattice(I)). (a ∈ s ∧ (a f) = 1))
Proof
Definitions occuring in Statement :
name-morph-satisfies: (psi f) = 1,
face_lattice-deq: face_lattice-deq(),
face_lattice: face_lattice(I),
names-hom: I ⟶ J,
lattice-fset-join: \/(s),
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
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,
implies: P ⇒ Q,
guard: {T},
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
iff: P ⇐⇒ Q,
squash: ↓T,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
top: Top,
false: False,
empty-fset: {},
lattice-fset-join: \/(s),
true: True,
not: ¬A,
fset-add: fset-add(eq;x;s),
or: P ∨ Q,
uiff: uiff(P;Q),
fl-join: fl-join(I;x;y),
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
cand: A c∧ B,
decidable: Dec(P)
Lemmas referenced :
fset-induction,
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,
face_lattice-deq_wf,
iff_wf,
name-morph-satisfies_wf,
lattice-fset-join_wf,
decidable__equal_face_lattice,
squash_wf,
exists_wf,
fset-member_wf,
fset_wf,
names-hom_wf,
nat_wf,
sq_stable__iff,
fl-morph_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-1_wf,
sq_stable__equal,
sq_stable__squash,
empty-fset_wf,
member-empty-fset,
reduce_nil_lemma,
true_wf,
fl-morph-0,
lattice-0_wf,
iff_weakening_equal,
face-lattice-0-not-1,
fset-add_wf,
not_wf,
fset-union_wf,
fset-singleton_wf,
or_wf,
iff_functionality_wrt_iff,
lattice-fset-join-union,
member-fset-union,
lattice-fset-join-singleton,
member-fset-singleton,
name-morph-satisfies-join,
fl-join_wf,
subtype_rel_self,
names_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
names-deq_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
deq-implies,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
hyp_replacement,
applyLambdaEquality,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
existsFunctionality,
andLevelFunctionality,
orFunctionality,
addLevel,
impliesFunctionality,
setEquality,
unionEquality,
unionElimination,
dependent_pairFormation,
inlFormation,
inrFormation,
dependent_set_memberEquality
Latex:
\mforall{}I,J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}s:fset(Point(face\_lattice(I))).
((\mbackslash{}/(s) f) = 1 \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mexists{}a:Point(face\_lattice(I)). (a \mmember{} s \mwedge{} (a f) = 1))
Date html generated:
2017_10_05-AM-01_17_49
Last ObjectModification:
2017_03_02-PM-10_33_43
Theory : cubical!type!theory
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