Nuprl Lemma : discrete-sigma-equiv
∀A:Type. ∀B:A ⟶ Type. ∀X:j⊢. {X ⊢ _:Equiv(Σ discr(A) discrete-family(A;a.B[a]);discr(a:A × B[a]))}
Proof
Definitions occuring in Statement :
discrete-family: discrete-family(A;a.B[a]),
cubical-equiv: Equiv(T;A),
discrete-cubical-type: discr(T),
cubical-sigma: Σ A B,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
discrete-family: discrete-family(A;a.B[a]),
cc-snd: q,
cc-fst: p,
csm-comp: G o F,
csm-adjoin: (s;u),
csm-ap-type: (AF)s,
compose: f o g,
csm-ap: (s)x,
pi2: snd(t),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
discrete-cubical-type: discr(T),
cubical-lam: cubical-lam(X;b),
cubical-app: app(w; u),
cubical-lambda: (λb),
cube-context-adjoin: X.A,
discrete-pair-inv: discrete-pair-inv(X;b),
discrete-pair: discrete-pair(p),
cubical-pair: cubical-pair(u;v),
cubical-snd: p.2,
cubical-fst: p.1,
cubical-term-at: u(a),
cc-adjoin-cube: (v;u),
csm-ap-term: (t)s,
pi1: fst(t),
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
fiber-member: fiber-member(p),
fiber-point: fiber-point(t;c)
Lemmas referenced :
cubical_set_wf,
istype-universe,
cubical-sigma-p,
cubical-sigma_wf,
discrete-cubical-type_wf,
discrete-family_wf,
csm-discrete-cubical-type,
discrete-pair_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cc-snd_wf,
csm-discrete-family,
cubical-term-eqcd,
discrete-pair-inv_wf,
csm-discrete-sigma,
cc-fst_wf,
path-type_wf,
squash_wf,
true_wf,
istype-cubical-term,
cubical-type_wf,
csm-ap-type_wf,
cubical-refl_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical_type_at_pair_lemma,
I_cube_pair_redex_lemma,
cubical-fiber_wf,
csm-ap-term_wf,
cubical-fun_wf,
cubical-lam_wf,
csm-cubical-fun,
csm-cubical-fiber,
equal-fiber-discrete,
fiber-point_wf,
csm-fiber-point,
fiber-discrete-equal,
fiber-member_wf,
equal_wf,
discrete-pair-injection,
cubical-term_wf,
cubical-term-at_wf,
discrete-pair-inv-property,
cubical-fst-pair,
equiv-witness_wf,
cubical-lambda_wf,
contractible-type_wf,
contr-witness_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
universeIsType,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
functionIsType,
hypothesisEquality,
inhabitedIsType,
sqequalHypSubstitution,
isectElimination,
universeEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
Error :memTop,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
hyp_replacement,
cumulativity,
productEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
productElimination,
dependent_pairEquality_alt,
equalityIstype,
independent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type. \mforall{}X:j\mvdash{}. \{X \mvdash{} \_:Equiv(\mSigma{} discr(A) discrete-family(A;a.B[a]);discr(a:A \mtimes{} B[a]))\}
Date html generated:
2020_05_20-PM-03_41_58
Last ObjectModification:
2020_04_20-PM-06_17_22
Theory : cubical!type!theory
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