Nuprl Lemma : cubical-refl-p-p
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[C:{X.B ⊢ _}].
(refl(((a)p)p) = ((refl(a))p)p ∈ {X.B.C ⊢ _:(((Path_A a a))p)p})
Proof
Definitions occuring in Statement :
cubical-refl: refl(a),
path-type: (Path_A a b),
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,
prop: ℙ,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-refl-p,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subset-cubical-term2,
sub_cubical_set_self,
path-type_wf,
cubical-type_wf,
path-type-p,
subtype_rel_self,
iff_weakening_equal,
cubical-term_wf,
cubical_set_wf,
cube_set_map_wf,
csm-cubical-refl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
inhabitedIsType,
applyLambdaEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[C:\{X.B \mvdash{} \_\}]. (refl(((a)p)p) = ((refl(a))p)p)
Date html generated:
2020_05_20-PM-03_21_47
Last ObjectModification:
2020_04_07-PM-03_21_31
Theory : cubical!type!theory
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