Nuprl Lemma : discrete-function-injection
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[X:j⊢].
∀f,g:{X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])}.
f = g ∈ {X ⊢ _:Πdiscr(A) discrete-family(A;a.B[a])}
supposing discrete-function(f) = discrete-function(g) ∈ {X ⊢ _:discr(a:A ⟶ B[a])}
Proof
Definitions occuring in Statement :
discrete-function: discrete-function(f),
discrete-family: discrete-family(A;a.B[a]),
discrete-cubical-type: discr(T),
cubical-pi: ΠA B,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
cubical-term-at: u(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-pi: ΠA B,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
squash: ↓T,
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
discrete-cubical-type: discr(T),
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
respects-equality: respects-equality(S;T),
cc-adjoin-cube: (v;u),
discrete-family: discrete-family(A;a.B[a]),
pi2: snd(t),
discrete-function: discrete-function(f),
cube-context-adjoin: X.A
Lemmas referenced :
cubical-term-at_wf,
cubical-pi_wf,
discrete-cubical-type_wf,
discrete-family_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical-term_wf,
discrete-function_wf,
cubical_set_wf,
istype-universe,
cubical_type_at_pair_lemma,
cubical-type-at_wf,
cube-set-restriction_wf,
names-hom_wf,
istype-cubical-type-at,
cube-context-adjoin_wf,
cc-adjoin-cube_wf,
cubical-type-ap-morph_wf,
nh-comp_wf,
subtype_rel-equal,
subtype-respects-equality,
cc-adjoin-cube-restriction,
equal_wf,
squash_wf,
true_wf,
cube-set-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
nh-id_wf,
cube_set_restriction_pair_lemma,
cubical_type_ap_morph_pair_lemma,
nh-id-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
applyEquality,
universeIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
equalityIstype,
instantiate,
cumulativity,
functionEquality,
inhabitedIsType,
dependent_functionElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsTypeImplies,
functionIsType,
universeEquality,
Error :memTop,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
because_Cache,
productElimination,
independent_functionElimination,
natural_numberEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[X:j\mvdash{}].
\mforall{}f,g:\{X \mvdash{} \_:\mPi{}discr(A) discrete-family(A;a.B[a])\}.
f = g supposing discrete-function(f) = discrete-function(g)
Date html generated:
2020_05_20-PM-03_39_32
Last ObjectModification:
2020_04_07-PM-04_29_43
Theory : cubical!type!theory
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