Nuprl Lemma : csm-path-type-id-adjoin
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a,b:{X.B ⊢ _:(A)p}]. ∀[u:{X ⊢ _:B}].
(((Path_(A)p a b))[u] = (X ⊢ Path_A (a)[u] (b)[u]) ∈ {X ⊢ _})
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
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 ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
csm-path-type,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
csm-id-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
path-type_wf,
csm-ap-term_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-term_wf,
cubical_set_wf,
csm-ap-type-fst-id-adjoin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
sqequalRule,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType,
Error :memTop,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X.B \mvdash{} \_:(A)p\}]. \mforall{}[u:\{X \mvdash{} \_:B\}].
(((Path\_(A)p a b))[u] = (X \mvdash{} Path\_A (a)[u] (b)[u]))
Date html generated:
2020_05_20-PM-03_15_32
Last ObjectModification:
2020_04_06-PM-05_58_06
Theory : cubical!type!theory
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