Nuprl Lemma : context-subset-subtype
∀[Gamma:j⊢]. ∀[phi1,phi2:{Gamma ⊢ _:𝔽}].
{Gamma, phi1 ⊢ _} ⊆r {Gamma, phi2 ⊢ _}
supposing ∀I:fset(ℕ). ∀rho:Gamma(I).
((phi2(rho) = 1 ∈ Point(face_lattice(I))) ⇒ (phi1(rho) = 1 ∈ Point(face_lattice(I))))
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
face_lattice: face_lattice(I),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
lattice-1: 1,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
context-subset: Gamma, phi,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
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,
so_apply: x[s],
cubical-type: {X ⊢ _},
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
I_cube_pair_redex_lemma,
subtype_rel_sets_simple,
I_cube_wf,
equal_wf,
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
face-type_wf,
subtype_rel_self,
lattice-1_wf,
cubical-type_wf,
context-subset_wf,
fset_wf,
nat_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,
cubical-term_wf,
cubical_set_wf,
subtype_rel_dep_function,
istype-universe,
names-hom_wf,
context-subset-restriction,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel-equal,
squash_wf,
true_wf,
cube-set-restriction-id,
iff_weakening_equal,
nh-comp_wf,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
lambdaFormation_alt,
hypothesisEquality,
instantiate,
isectElimination,
cumulativity,
applyEquality,
because_Cache,
universeIsType,
independent_isectElimination,
setElimination,
rename,
productElimination,
dependent_set_memberEquality_alt,
functionIsType,
equalityIstype,
productEquality,
isectEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality_alt,
functionExtensionality,
universeEquality,
independent_pairFormation,
promote_hyp,
productIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi1,phi2:\{Gamma \mvdash{} \_:\mBbbF{}\}].
\{Gamma, phi1 \mvdash{} \_\} \msubseteq{}r \{Gamma, phi2 \mvdash{} \_\}
supposing \mforall{}I:fset(\mBbbN{}). \mforall{}rho:Gamma(I). ((phi2(rho) = 1) {}\mRightarrow{} (phi1(rho) = 1))
Date html generated:
2020_05_20-PM-02_51_25
Last ObjectModification:
2020_04_06-AM-10_49_13
Theory : cubical!type!theory
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