Nuprl Lemma : discrete-pathtype-uniform
∀[T:Type]. ∀[X,Z:j⊢]. ∀[s:Z j⟶ X]. ∀[pth:{Z ⊢ _:(Path(discr(T)))s}].
(pth = refl(pth @ 0(𝕀)) ∈ {Z ⊢ _:(Path(discr(T)))s})
Proof
Definitions occuring in Statement :
cubical-refl: refl(a),
cubical-path-app: pth @ r,
pathtype: Path(A),
interval-0: 0(𝕀),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
discrete-pathtype,
cubical_set_cumulativity-i-j,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
pathtype_wf,
discrete-cubical-type_wf,
equal_wf,
cubical-type_wf,
csm-pathtype,
iff_weakening_equal,
csm-discrete-cubical-type,
squash_wf,
true_wf,
subtype_rel_self,
cubical-term_wf,
cube_set_map_wf,
cubical_set_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
Error :memTop,
universeIsType,
universeEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[T:Type]. \mforall{}[X,Z:j\mvdash{}]. \mforall{}[s:Z j{}\mrightarrow{} X]. \mforall{}[pth:\{Z \mvdash{} \_:(Path(discr(T)))s\}]. (pth = refl(pth @ 0(\mBbbI{})))
Date html generated:
2020_05_20-PM-03_36_39
Last ObjectModification:
2020_04_06-PM-07_02_19
Theory : cubical!type!theory
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