Nuprl Lemma : empty-cubical-subset-term
∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))].
∀[X,A,B:Top]. (A = B ∈ {I,phi ⊢ _:X}) supposing phi = 0 ∈ Point(face_lattice(I))
Proof
Definitions occuring in Statement :
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
face_lattice: face_lattice(I),
lattice-0: 0,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cubical-term: {X ⊢ _:A},
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
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: functor-ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-type-ap-morph_wf,
cubical-type-at_wf,
all_wf,
iff_weakening_equal,
face-presheaf_wf,
cubical_set_wf,
true_wf,
squash_wf,
bdd-distributive-lattice_wf,
lattice-0_wf,
lattice-join_wf,
lattice-meet_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
face_lattice_wf,
lattice-point_wf,
equal_wf,
top_wf,
names-hom_wf,
nat_wf,
face-lattice-constraints_wf,
fset-contains-none_wf,
fset-all_wf,
names-deq_wf,
union-deq_wf,
fset-antichain_wf,
assert_wf,
names_wf,
fset_wf,
subtype_rel_self,
cubical-subset_wf,
I_cube_wf,
empty-cubical-subset-I_cube
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
functionExtensionality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
voidElimination,
applyEquality,
sqequalRule,
setEquality,
unionEquality,
because_Cache,
hypothesis,
productEquality,
lambdaEquality,
lambdaFormation,
isect_memberEquality,
axiomEquality,
instantiate,
cumulativity,
universeEquality,
independent_isectElimination,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:Point(face\_lattice(I))]. \mforall{}[X,A,B:Top]. (A = B) supposing phi = 0
Date html generated:
2016_05_18-PM-01_58_11
Last ObjectModification:
2016_01_28-PM-01_20_49
Theory : cubical!type!theory
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