Nuprl Lemma : uabeta_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. (uabeta(G;A;B) ∈ {G ⊢ _:uabeta-type(G;A;B)})
Proof
Definitions occuring in Statement :
uabeta: uabeta(G;A;B),
uabeta-type: uabeta-type(G;A;B),
cubical-universe: c𝕌,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uabeta-type: uabeta-type(G;A;B),
uimplies: b supposing a,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uabeta: uabeta(G;A;B),
let: let,
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
universe-comp-fun: CompFun(A),
csm-comp-structure: (cA)tau,
universe-comp-op: compOp(t),
comp-op-to-comp-fun: cop-to-cfun(cA),
csm-composition: (comp)sigma,
composition-term: comp cA [phi ⊢→ u] a0,
cubical-term-at: u(a),
cc-adjoin-cube: (v;u),
interval-type: 𝕀,
subset-iota: iota,
csm-comp: G o F,
csm-ap-term: (t)s,
cc-fst: p,
csm-ap-type: (AF)s,
csm-ap: (s)x,
compose: f o g,
constant-cubical-type: (X),
path-trans: PathTransport(p),
cc-snd: q,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm+: tau+,
csm-id: 1(X),
csm-adjoin: (s;u),
pi2: snd(t),
pi1: fst(t),
interval-1: 1(𝕀),
same-cubical-term: X ⊢ u=v:A,
universe-decode: decode(t),
label: ...$L... t,
equiv-path: EquivPath(G;A;B;f),
term-to-path: <>(a),
path-eta: path-eta(pth),
cubicalpath-app: pth @ r,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C)
Lemmas referenced :
universe-decode_wf,
csm-ap-term-universe,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cc-fst_wf,
cubical-equiv_wf,
path-type_wf,
csm-ap-type_wf,
csm-ap-term_wf,
cc-snd_wf,
cubical-equiv-p,
cubical-term-eqcd,
equiv-fun_wf,
cubical-app_wf_fun,
csm-universe-decode,
universe-comp-fun_wf,
cubical-universe-p,
istype-cubical-universe-term,
cubical_set_wf,
cubical-type_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube_set_map_wf,
subtype_rel_self,
iff_weakening_equal,
univ-a_wf,
cubical-universe_wf,
path-trans_wf,
cubical-lambda_wf,
cubical-pi_wf,
uabeta_aux_wf,
transprt-const_wf,
csm-comp-structure_wf2,
istype-cubical-term,
univ-trans-equiv_path,
cubical-fun_wf,
csm-cubical-fun,
app-trans-equiv-path,
cubical-term_wf,
csm-trans-equiv-path,
composition-structure_wf,
csm-universe-comp-fun,
csm-univ-trans,
equiv_path_wf,
equiv_path-0,
subtype_rel_universe1,
equiv_path-1,
interval-type_wf,
csm-equiv_path,
csm-cubical-equiv,
app-univ-a,
path-eta_wf,
csm-cubical-universe,
path-type-sub-pathtype,
csm-cubical-lambda,
cc-fst_wf_interval,
cubical-beta,
csm+_wf_interval,
csm-interval-type,
subset-cubical-term,
sub_cubical_set_self,
csm-adjoin-fst-snd,
csm-ap-id-term,
cubical-app_wf,
cubical-pi-p,
univ-trans_wf,
csm-id-adjoin_wf-interval-0,
csm-id-adjoin_wf-interval-1,
subset-cubical-term2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
dependent_functionElimination,
Error :memTop,
natural_numberEquality,
imageElimination,
universeEquality,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
lambdaFormation_alt,
equalityIstype,
applyLambdaEquality,
equalityElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
setElimination,
rename,
cumulativity,
promote_hyp
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. (uabeta(G;A;B) \mmember{} \{G \mvdash{} \_:uabeta-type(G;A;B)\})
Date html generated:
2020_05_20-PM-07_44_21
Last ObjectModification:
2020_05_01-PM-05_55_07
Theory : cubical!type!theory
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