Nuprl Lemma : discrete-pi-equiv
∀A:Type. ∀B:A ⟶ Type. ∀X:j⊢. {X ⊢ _:Equiv(Πdiscr(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-pi: ΠA B,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
so_apply: x[s],
all: ∀x:A. B[x],
function: 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,
is-cubical-equiv: IsEquiv(T;A;w),
squash: ↓T,
prop: ℙ,
true: True,
discrete-cubical-type: discr(T),
cubical-lam: cubical-lam(X;b),
cubical-app: app(w; u),
cube-context-adjoin: X.A,
discrete-function-inv: discrete-function-inv(X; b),
discrete-function: discrete-function(f),
cubical-lambda: (λb),
cc-adjoin-cube: (v;u),
cubical-term-at: u(a),
csm-ap-term: (t)s,
pi1: fst(t),
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
cubical-pi: ΠA B,
cubical-term: {X ⊢ _:A},
fset: fset(T),
quotient: x,y:A//B[x; y],
I_cube: A(I),
functor-ob: ob(F),
cubical-type-at: A(a),
names-hom: I ⟶ J,
cubical-type-ap-morph: (u a f),
cubical-type: {X ⊢ _},
fiber-member: fiber-member(p),
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
fiber-point: fiber-point(t;c)
Lemmas referenced :
cubical_set_wf,
istype-universe,
cubical-pi-p,
cubical-pi_wf,
discrete-cubical-type_wf,
discrete-family_wf,
csm-discrete-cubical-type,
discrete-function_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cc-snd_wf,
cubical-term-eqcd,
equiv-witness_wf,
cubical-lam_wf,
cubical-lambda_wf,
contractible-type_wf,
cubical-fiber_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cubical-fun_wf,
csm-cubical-fun,
contr-witness_wf,
fiber-point_wf,
discrete-function-inv_wf,
csm-discrete-pi,
path-type_wf,
squash_wf,
true_wf,
istype-cubical-term,
cubical-type_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
cube_set_restriction_pair_lemma,
I_cube_pair_redex_lemma,
cubical-refl_wf,
csm-cubical-fiber,
equal-fiber-discrete,
csm-fiber-point,
equal_wf,
subtype_rel_self,
iff_weakening_equal,
csm-discrete-family,
csm-path-type,
cubical-app_wf_fun,
subset-cubical-term2,
sub_cubical_set_self,
cubical-term_wf,
cube-context-adjoin-I_cube,
istype-cubical-type-at,
cube-set-restriction-id,
fiber-discrete-equal,
fiber-member_wf,
discrete-function-injection,
cubical-term-at_wf,
discrete-function-inv-property,
cubical-fst-pair
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,
comment,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
hyp_replacement,
functionEquality,
cumulativity,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
productElimination,
equalityIstype,
independent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
setElimination,
rename,
dependent_pairEquality_alt
Latex:
\mforall{}A:Type. \mforall{}B:A {}\mrightarrow{} Type. \mforall{}X:j\mvdash{}. \{X \mvdash{} \_:Equiv(\mPi{}discr(A) discrete-family(A;a.B[a]);discr(a:A {}\mrightarrow{} B[a]))\}
Date html generated:
2020_05_20-PM-03_40_20
Last ObjectModification:
2020_04_20-PM-07_21_07
Theory : cubical!type!theory
Home
Index