Nuprl Lemma : csm-universe-encode
∀[G:j⊢]. ∀[T:{G ⊢ _}]. ∀[cT:G ⊢ CompOp(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G]. ((encode(T;cT))s = encode((T)s;(cT)s) ∈ {H ⊢ _:c𝕌})
Proof
Definitions occuring in Statement :
universe-encode: encode(T;cT),
cubical-universe: c𝕌,
csm-composition: (comp)sigma,
composition-op: Gamma ⊢ CompOp(A),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
cubical-term: {X ⊢ _:A},
universe-encode: encode(T;cT),
csm-ap-term: (t)s,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
csm-composition: (comp)sigma,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g
Lemmas referenced :
cubical-term-equal,
cubical-universe_wf,
cube_set_map_wf,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
csm-ap-term_wf,
universe-encode_wf,
csm-cubical-universe,
csm-ap-type_wf,
csm-composition_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-universe-at,
formal-cube_wf1,
equal_wf,
squash_wf,
true_wf,
istype-universe,
context-map_wf,
csm-ap_wf,
csm-ap-comp-type,
subtype_rel_self,
iff_weakening_equal,
csm-comp-context-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
universeIsType,
inhabitedIsType,
applyEquality,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
setElimination,
rename,
functionExtensionality,
dependent_pairEquality_alt,
imageElimination,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
because_Cache
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[T:\{G \mvdash{} \_\}]. \mforall{}[cT:G \mvdash{} CompOp(T)]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((encode(T;cT))s = encode((T)s;(cT)s))
Date html generated:
2020_05_20-PM-07_10_18
Last ObjectModification:
2020_04_25-PM-08_24_58
Theory : cubical!type!theory
Home
Index