Nuprl Lemma : face-term-implies-subset
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. sub_cubical_set{j:l}(Gamma, phi; Gamma, psi) supposing Gamma ⊢ (phi ⇒ psi)
Proof
Definitions occuring in Statement :
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
sub_cubical_set: Y ⊆ X,
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
context-subset: Gamma, phi,
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
so_apply: x[s],
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,
implies: P ⇒ Q,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
implies-sub_cubical_set,
context-subset_wf,
I_cube_pair_redex_lemma,
subtype_rel_sets_simple,
I_cube_wf,
equal_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,
cubical-term-at_wf,
face-type_wf,
subtype_rel_self,
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,
face-term-implies_wf,
cubical-term_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
lambdaFormation_alt,
sqequalRule,
dependent_functionElimination,
Error :memTop,
instantiate,
cumulativity,
lambdaEquality_alt,
applyEquality,
productEquality,
isectEquality,
because_Cache,
universeIsType,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
equalityIstype,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
sub\_cubical\_set\{j:l\}(Gamma, phi; Gamma, psi) supposing Gamma \mvdash{} (phi {}\mRightarrow{} psi)
Date html generated:
2020_05_20-PM-02_52_11
Last ObjectModification:
2020_04_04-PM-05_06_51
Theory : cubical!type!theory
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