Nuprl Lemma : paths-are-refl-iff2
∀[X:j⊢]. ∀[A:{X ⊢ _}].
uiff(∀Z:j⊢. ∀s:Z j⟶ X. ∀p:{Z ⊢ _:Path((A)s)}. (p = refl(p @ 0(𝕀)) ∈ {Z ⊢ _:Path((A)s)});∀Z:j⊢. ∀s:Z j⟶ X.
∀p:{Z.𝕀 ⊢ _:((A)s)p}.
∀[x,y:{Z ⊢ _:𝕀}].
((p)[x]
= (p)[y]
∈ {Z ⊢ _:(A)s}))
Proof
Definitions occuring in Statement :
cubical-refl: refl(a),
cubicalpath-app: pth @ r,
pathtype: Path(A),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
csm-id: 1(X),
csm-ap: (s)x,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
paths-are-refl-iff,
cube_set_map_wf,
cubical-term_wf,
pathtype_wf,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-refl_wf,
cubicalpath-app_wf,
interval-0_wf,
path-type-subtype,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf,
csm-ap-term_wf,
csm-id-adjoin_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm_id_adjoin_fst_type_lemma,
cubical-type_wf,
cubical_set_wf,
term-to-path_wf,
csm-id-adjoin_wf-interval-1,
equal_wf,
squash_wf,
true_wf,
istype-universe,
term-to-path-beta,
path-eta_wf,
path-eta-id-adjoin
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairFormation,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
universeIsType,
because_Cache,
sqequalRule,
lambdaEquality_alt,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
functionIsType,
instantiate,
applyEquality,
equalityIstype,
isectIsType,
Error :memTop,
setElimination,
rename,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}].
uiff(\mforall{}Z:j\mvdash{}. \mforall{}s:Z j{}\mrightarrow{} X. \mforall{}p:\{Z \mvdash{} \_:Path((A)s)\}. (p = refl(p @ 0(\mBbbI{})));\mforall{}Z:j\mvdash{}. \mforall{}s:Z j{}\mrightarrow{} X.
\mforall{}p:\{Z.\mBbbI{} \mvdash{} \_:((A)s)p\}.
\mforall{}[x,y:\{Z \mvdash{} \_:\mBbbI{}\}].
((p)[x] = (p)[y]))
Date html generated:
2020_05_20-PM-03_44_05
Last ObjectModification:
2020_04_07-PM-06_03_56
Theory : cubical!type!theory
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