Nuprl Lemma : term-to-pathtype-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀pth:{X ⊢ _:Path(A)}. (<>(pth)p @ q = pth ∈ {X ⊢ _:Path(A)})
Proof
Definitions occuring in Statement :
term-to-pathtype: <>a,
cubical-path-app: pth @ r,
pathtype: Path(A),
cc-snd: q,
cc-fst: p,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
term-to-pathtype: <>a,
pathtype: Path(A),
cubical-path-app: pth @ r,
term-to-path: <>(a),
cubicalpath-app: pth @ r,
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
cubical-term_wf,
pathtype_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
cubical-fun-as-cubical-pi,
interval-type_wf,
cubical-eta,
csm-ap-type_wf,
cube-context-adjoin_wf,
cc-fst_wf,
subset-cubical-term2,
sub_cubical_set_self,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}pth:\{X \mvdash{} \_:Path(A)\}. (<>(pth)p @ q = pth)
Date html generated:
2020_05_20-PM-03_20_12
Last ObjectModification:
2020_04_06-PM-06_37_01
Theory : cubical!type!theory
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