Nuprl Lemma : context-iterated-subset
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
(sub_cubical_set{j:l}(Gamma, (phi ∧ psi); Gamma, psi, phi)
∧ sub_cubical_set{j:l}(Gamma, psi, phi; Gamma, (phi ∧ psi)))
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
context-subset: Gamma, phi,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
sub_cubical_set: Y ⊆ X
Lemmas referenced :
implies-sub_cubical_set,
context-subset_wf,
face-and_wf,
subset-cubical-term,
context-subset-is-subset,
face-type_wf,
I_cube_pair_redex_lemma,
face-and-eq-1,
cubical-term-at_wf,
subtype_rel_self,
I_cube_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,
cube_set_restriction_pair_lemma,
squash_wf,
true_wf,
istype-universe,
face-term-at-restriction-eq-1,
iff_weakening_equal,
cube-set-restriction_wf,
names-hom_wf,
cubical-term_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
instantiate,
because_Cache,
sqequalRule,
lambdaFormation_alt,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
setIsType,
universeIsType,
productEquality,
cumulativity,
isectEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
independent_pairEquality,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
(sub\_cubical\_set\{j:l\}(Gamma, (phi \mwedge{} psi); Gamma, psi, phi)
\mwedge{} sub\_cubical\_set\{j:l\}(Gamma, psi, phi; Gamma, (phi \mwedge{} psi)))
Date html generated:
2020_05_20-PM-02_55_59
Last ObjectModification:
2020_04_04-PM-05_10_22
Theory : cubical!type!theory
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