Nuprl Lemma : csm-cubical-equiv
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[A,E:{H ⊢ _}]. ((Equiv(A;E))tau = Equiv((A)tau;(E)tau) ∈ {K ⊢ _})
Proof
Definitions occuring in Statement :
cubical-equiv: Equiv(T;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],
cubical-equiv: Equiv(T;A),
member: t ∈ T,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
is-cubical-equiv: IsEquiv(T;A;w),
csm+: tau+,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
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),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
cc-fst: p,
csm-ap: (s)x,
cc-snd: q,
csm-comp: G o F,
csm-adjoin: (s;u),
cubical-fun: (A ⟶ B),
csm-ap-term: (t)s
Lemmas referenced :
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-cubical-sigma,
cubical-fun_wf,
is-cubical-equiv_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-ap-type_wf,
cc-fst_wf,
cc-snd_wf-cubical-fun,
cubical-sigma_wf,
subtype_rel_self,
iff_weakening_equal,
csm-cubical-fun,
csm-ap-term_wf,
cc-snd_wf,
cubical-term_wf,
cubical-fun-p,
csm-cubical-pi,
csm+_wf,
contractible-type_wf,
cubical-fiber_wf,
subset-cubical-type,
sub_cubical_set_self,
cubical-pi_wf,
subtype_rel-equal,
csm-contractible-type,
csm-adjoin_wf,
csm-comp_wf,
p-csm+-type,
csm-cubical-fiber
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
because_Cache,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
instantiate,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
setElimination,
rename,
Error :memTop
Latex:
\mforall{}[H,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} H]. \mforall{}[A,E:\{H \mvdash{} \_\}]. ((Equiv(A;E))tau = Equiv((A)tau;(E)tau))
Date html generated:
2020_05_20-PM-03_26_22
Last ObjectModification:
2020_04_07-PM-06_41_37
Theory : cubical!type!theory
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