Nuprl Lemma : term-to-path-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. (X ⊢ <>((pth)p @ q) = pth ∈ {X ⊢ _:(Path_A a b)})
Proof
Definitions occuring in Statement :
term-to-path: <>(a),
cubical-path-app: pth @ r,
path-type: (Path_A a b),
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
cubical-path-app: pth @ r,
path-eta: path-eta(pth),
all: ∀x:A. B[x],
term-to-path: <>(a),
pathtype: Path(A),
cubicalpath-app: pth @ r,
guard: {T},
cubical-fun: (A ⟶ B),
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
paths-equal,
cubical-term_wf,
path-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
path-type-subtype,
term-to-path_wf,
path-eta_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,
pathtype_wf,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
dependent_functionElimination,
equalityTransitivity,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[pth:\{X \mvdash{} \_:(Path\_A a b)\}]. (X \mvdash{} <>((pth)p @ q) = pth)
Date html generated:
2020_05_20-PM-03_18_53
Last ObjectModification:
2020_04_06-PM-06_34_34
Theory : cubical!type!theory
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