Nuprl Lemma : discrete-fun-bijection
∀[A,B:Type]. Bij({() ⊢ _:(discr(A) ⟶ discr(B))};{() ⊢ _:discr(A ⟶ B)};λf.discrete-fun(f))
Proof
Definitions occuring in Statement :
discrete-fun: discrete-fun(f),
discrete-cubical-type: discr(T),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
trivial-cube-set: (),
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
surject: Surj(A;B;f),
cubical-term-at: u(a),
uimplies: b supposing a,
discrete-cubical-type: discr(T),
cubical-fun: (A ⟶ B),
top: Top,
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
discrete-fun: discrete-fun(f),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cubical-term: {X ⊢ _:A},
cc-snd: q,
cube-context-adjoin: X.A,
pi1: fst(t),
pi2: snd(t),
cubical-lam: cubical-lam(X;b),
cubical-lambda: (λb),
cc-adjoin-cube: (v;u)
Lemmas referenced :
equal_wf,
cubical-term_wf,
trivial-cube-set_wf,
discrete-cubical-type_wf,
discrete-fun_wf,
cubical-fun_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical-term-at_wf,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
all_wf,
names-hom_wf,
nh-comp_wf,
squash_wf,
true_wf,
cubical-type-at_wf,
nh-id_wf,
nh-id-right,
iff_weakening_equal,
cubical-lam_wf,
csm-discrete-cubical-type,
I_cube_pair_redex_lemma,
pi1_wf_top,
pi2_wf,
cube_set_restriction_pair_lemma,
cube-set-restriction_wf,
cube-context-adjoin_wf,
cubical-type-ap-morph_wf,
cube-set-restriction-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalRule,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
rename,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
applyLambdaEquality,
setElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality,
lambdaEquality,
because_Cache,
applyEquality,
hyp_replacement,
natural_numberEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation,
independent_pairEquality,
productEquality
Latex:
\mforall{}[A,B:Type]. Bij(\{() \mvdash{} \_:(discr(A) {}\mrightarrow{} discr(B))\};\{() \mvdash{} \_:discr(A {}\mrightarrow{} B)\};\mlambda{}f.discrete-fun(f))
Date html generated:
2017_10_05-AM-02_12_38
Last ObjectModification:
2017_03_02-PM-11_22_08
Theory : cubical!type!theory
Home
Index