Nuprl Lemma : closed-cubical-universe-cumulativity
closed-cubical-term(cc𝕌) ⊆r closed-cubical-term(cc𝕌')
Proof
Definitions occuring in Statement :
closed-cubical-universe: cc𝕌,
closed-cubical-term: closed-cubical-term(T),
subtype_rel: A ⊆r B
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
member: t ∈ T,
closed-cubical-universe: cc𝕌,
closed-cubical-term: closed-cubical-term(T),
pi1: fst(t),
pi2: snd(t),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
formal-cube: formal-cube(I)
Lemmas referenced :
fibrant-type-cumulativity,
formal-cube_wf1,
fset_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
fibrant-type_wf,
subtype_rel_self,
iff_weakening_equal,
names-hom_wf,
csm-fibrant-type_wf,
context-map_wf,
I_cube_wf,
closed-cubical-term_wf,
closed-cubical-universe_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality_alt,
sqequalHypSubstitution,
sqequalRule,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
lambdaFormation_alt,
instantiate,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
because_Cache,
inhabitedIsType,
functionIsType,
equalityIstype
Latex:
closed-cubical-term(cc\mBbbU{}) \msubseteq{}r closed-cubical-term(cc\mBbbU{}')
Date html generated:
2020_05_20-PM-07_06_01
Last ObjectModification:
2020_04_25-AM-11_33_47
Theory : cubical!type!theory
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