Nuprl Lemma : path-type-q-id-adjoin
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,u:{X ⊢ _:A}]. (((Path_(A)p (a)p q))[u] = (X ⊢ Path_A a u) ∈ {X ⊢ _})
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
csm-id-adjoin: [u],
cc-snd: q,
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 ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
true: True,
all: ∀x:A. B[x],
squash: ↓T,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
csm-ap-term_wf,
cube-context-adjoin_wf,
cc-fst_wf,
cc-snd_wf,
path-type_wf,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
equal_wf,
csm-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
csm-ap-id-type,
csm-ap-id-term,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
csm-path-type-id-adjoin,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
inhabitedIsType,
hypothesisEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality_alt,
isectElimination,
thin,
axiomEquality,
isectIsTypeImplies,
universeIsType,
instantiate,
extract_by_obid,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
imageElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
universeEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,u:\{X \mvdash{} \_:A\}]. (((Path\_(A)p (a)p q))[u] = (X \mvdash{} Path\_A a u))
Date html generated:
2020_05_20-PM-03_15_43
Last ObjectModification:
2020_04_06-PM-06_31_07
Theory : cubical!type!theory
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