Nuprl Lemma : cubical-subset-is-context-subset-canonical
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (I,psi = formal-cube(I), canonical-section(();𝔽;I;⋅;psi) ∈ CubicalSet{j})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-subset: I,psi,
face-presheaf: 𝔽,
trivial-cube-set: (),
formal-cube: formal-cube(I),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
unit: Unit,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
trivial-cube-set: (),
face-presheaf: 𝔽,
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,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
uimplies: b supposing a,
all: ∀x:A. B[x],
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
true: True,
canonical-section: canonical-section(Gamma;A;I;rho;a)
Lemmas referenced :
cubical-subset-is-context-subset,
formal-cube_wf,
context-subset_wf,
squash_wf,
true_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
cubical-term-equal,
canonical-section_wf,
trivial-cube-set_wf,
it_wf,
subtype_rel_self,
I_cube_wf,
cubical-type-at_wf_face-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
context-map_wf,
csm-face-type,
cube-set-restriction_wf,
face-presheaf_wf2,
names-hom_wf,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
face-type-ap-morph,
cube_set_restriction_pair_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
because_Cache,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
sqequalRule,
Error :memTop,
independent_isectElimination,
dependent_functionElimination,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
rename,
hyp_replacement
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[psi:\mBbbF{}(I)]. (I,psi = formal-cube(I), canonical-section(();\mBbbF{};I;\mcdot{};psi))
Date html generated:
2020_05_20-PM-02_45_57
Last ObjectModification:
2020_04_05-PM-02_50_38
Theory : cubical!type!theory
Home
Index