Nuprl Lemma : paths-are-refl-iff
∀[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 ⊢ _:Path((A)s)}.
∀[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,
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],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
true: True,
rev_implies: P ⇐ Q,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
cubical-path-app: pth @ r,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
cc-fst: p,
interval-type: 𝕀,
csm-comp: G o F,
csm-ap: (s)x,
compose: f o g,
cc-snd: q,
constant-cubical-type: (X),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C)
Lemmas referenced :
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,
interval-type_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
refl-path-app,
term-to-pathtype-eta,
term-to-pathtype_wf,
csm-pathtype,
cube-context-adjoin_wf,
cc-fst_wf,
csm-comp_wf,
csm-ap-term_wf,
equal_functionality_wrt_subtype_rel2,
cc-snd_wf,
csm-cubical-refl,
cubical-path-app_wf,
csm-cubicalpath-app,
csm-interval-0,
subset-cubical-term2,
sub_cubical_set_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectElimination,
axiomEquality,
hypothesis,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
functionIsType,
universeIsType,
extract_by_obid,
instantiate,
applyEquality,
equalityIstype,
because_Cache,
isectIsType,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
applyLambdaEquality,
setElimination,
rename,
cumulativity,
hyp_replacement,
Error :memTop
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 \mvdash{} \_:Path((A)s)\}.
\mforall{}[x,y:\{Z \mvdash{} \_:\mBbbI{}\}].
(p @ x = p @ y))
Date html generated:
2020_05_20-PM-03_43_53
Last ObjectModification:
2020_04_07-PM-06_01_45
Theory : cubical!type!theory
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