Nuprl Lemma : context-adjoin-subset0
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀T:{H ⊢ _}. sub_cubical_set{k:l}(H.T, (phi)p; H, phi.T)
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
sub_cubical_set: Y ⊆ X,
member: t ∈ T,
cubical-term: {X ⊢ _:A},
cc-fst: p,
csm-ap-term: (t)s,
cubical-term-at: u(a),
csm-ap: (s)x,
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,
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
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,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
csm-id: 1(X),
cat-arrow: cat-arrow(C),
op-cat: op-cat(C),
cat-ob: cat-ob(C),
cube-cat: CubeCat,
spreadn: spread4,
cube-context-adjoin: X.A,
type-cat: TypeCat,
pi2: snd(t),
context-subset: Gamma, phi,
functor-arrow: arrow(F),
cat-comp: cat-comp(C),
compose: f o g,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
pi1_wf_top,
I_cube_wf,
istype-cubical-type-at,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
cubical-type_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
cubical-term-at_wf,
subtype_rel_self,
cubical-type-at_wf,
lattice-1_wf,
squash_wf,
true_wf,
istype-universe,
face-term-at-restriction-eq-1,
iff_weakening_equal,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
names-hom_wf,
pi2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
universeIsType,
independent_isectElimination,
productElimination,
independent_pairEquality,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
productIsType,
dependent_set_memberEquality_alt,
dependent_functionElimination,
dependent_pairEquality_alt,
equalityIstype,
inhabitedIsType,
setIsType,
functionExtensionality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
setEquality,
functionIsType
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}T:\{H \mvdash{} \_\}. sub\_cubical\_set\{k:l\}(H.T, (phi)p; H, phi.T)
Date html generated:
2020_05_20-PM-03_04_43
Last ObjectModification:
2020_04_13-PM-05_43_31
Theory : cubical!type!theory
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