Nuprl Lemma : context-map-lemma1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[i:ℕ]. (<s(rho)> ∈ I+i,s(phi(rho)) j⟶ Gamma, phi)
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
face-presheaf: 𝔽,
context-map: <rho>,
cube_set_map: A ⟶ B,
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-s: s,
add-name: I+i,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
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,
guard: {T},
formal-cube: formal-cube(I),
cubical-term-at: u(a),
context-map: <rho>,
csm-ap: (s)x,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
names-hom: I ⟶ J,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
istype-nat,
I_cube_wf,
fset_wf,
nat_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
context-subset-map,
formal-cube_wf1,
add-name_wf,
context-map_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
cube_set_map_wf,
squash_wf,
true_wf,
cubical-subset-is-context-subset,
face-presheaf_wf2,
cubical-term-at_wf,
subtype_rel_self,
context-subset_wf,
cubical-term-equal,
csm-ap-term_wf,
csm-face-type,
I_cube_pair_redex_lemma,
csm-ap-term-at,
cubical-type-at_wf_face-type,
names-hom_wf,
equal_wf,
istype-universe,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cube-set-restriction-comp,
subtype_rel-equal,
iff_weakening_equal,
cubical-term-at-morph,
nh-comp_wf,
face-type-ap-morph,
cube_set_restriction_pair_lemma,
fl-morph_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,
lattice-meet_wf,
lattice-join_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isect_memberEquality_alt,
isectElimination,
thin,
hypothesisEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
Error :memTop,
functionExtensionality,
natural_numberEquality,
universeEquality,
productElimination,
independent_functionElimination,
productEquality,
cumulativity,
isectEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)]. \mforall{}[i:\mBbbN{}].
(<s(rho)> \mmember{} I+i,s(phi(rho)) j{}\mrightarrow{} Gamma, phi)
Date html generated:
2020_05_20-PM-04_07_32
Last ObjectModification:
2020_04_17-PM-01_09_40
Theory : cubical!type!theory
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