Nuprl Lemma : context-subset-map
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[Z:j⊢]. ∀[s:Z j⟶ Gamma]. (s ∈ Z, (phi)s j⟶ Gamma, phi)
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
type-cat: TypeCat,
cube-cat: CubeCat,
op-cat: op-cat(C),
nat-trans: nat-trans(C;D;F;G),
spreadn: spread4,
all: ∀x:A. B[x],
functor-arrow: arrow(F),
functor-ob: ob(F),
context-subset: Gamma, phi,
compose: f o g,
pi1: fst(t),
pi2: snd(t),
I_cube: A(I),
subtype_rel: A ⊆r B,
csm-ap: (s)x,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
cube-set-restriction: f(s),
squash: ↓T,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-ob: cat-ob(C),
cat-arrow: cat-arrow(C),
cubical_set: CubicalSet,
ps_context: __⊢,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_ob_pair_lemma,
cube_set_map_wf,
cubical-term_wf,
cubical_set_wf,
subtype_rel_self,
I_cube_wf,
csm-ap-term-at,
cubical-term-at_wf,
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,
lattice-1_wf,
fset_wf,
nat_wf,
names-hom_wf,
csm-ap-restriction,
squash_wf,
true_wf,
istype-universe,
face-term-at-restriction-eq-1,
cat-ob_wf,
op-cat_wf,
cube-cat_wf,
cat-arrow_wf,
type-cat_wf,
functor-ob_wf,
small-category-cumulativity-2,
iff_weakening_equal,
cube-set-restriction_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
universeIsType,
instantiate,
inhabitedIsType,
functionExtensionality,
applyEquality,
equalityIstype,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
independent_isectElimination,
setEquality,
lambdaFormation_alt,
imageElimination,
universeEquality,
functionEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
functionIsType,
setIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[Z:j\mvdash{}]. \mforall{}[s:Z j{}\mrightarrow{} Gamma]. (s \mmember{} Z, (phi)s j{}\mrightarrow{} Gamma, phi)
Date html generated:
2020_05_20-PM-02_45_08
Last ObjectModification:
2020_04_04-PM-04_59_19
Theory : cubical!type!theory
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