Nuprl Lemma : csm-univ-trans
∀[G:j⊢]. ∀[T:{G.𝕀 ⊢ _:c𝕌}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((univ-trans(G;T))s = univ-trans(H;(T)s+) ∈ {H ⊢ _:(((decode(T))s+)[0(𝕀)] ⟶ ((decode(T))s+)[1(𝕀)])})
Proof
Definitions occuring in Statement :
univ-trans: univ-trans(G;T),
universe-decode: decode(t),
cubical-universe: c𝕌,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm+: tau+,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
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,
univ-trans: univ-trans(G;T),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
universe-comp-op: compOp(t),
comp-op-to-comp-fun: cop-to-cfun(cA),
csm-composition: (comp)sigma,
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm+: tau+,
csm-comp: G o F,
csm-ap: (s)x,
cubical-term-at: u(a),
csm-ap-term: (t)s,
compose: f o g,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-adjoin: (s;u)
Lemmas referenced :
csm-transprt-fun,
universe-decode_wf,
cube-context-adjoin_wf,
interval-type_wf,
comp-op-to-comp-fun_wf2,
cubical_set_cumulativity-i-j,
universe-comp-op_wf,
cube_set_map_wf,
istype-cubical-universe-term,
cubical_set_wf,
csm-universe-decode
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
sqequalRule,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[T:\{G.\mBbbI{} \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G]. ((univ-trans(G;T))s = univ-trans(H;(T)s+))
Date html generated:
2020_05_20-PM-07_32_14
Last ObjectModification:
2020_04_29-PM-11_16_01
Theory : cubical!type!theory
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