Nuprl Lemma : term-to-path-beta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[b:{X.𝕀 ⊢ _:(A)p}]. ∀[r:{X ⊢ _:𝕀}]. (<>(b) @ r = (b)[r] ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
term-to-path: <>(a),
cubicalpath-app: pth @ r,
interval-type: 𝕀,
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],
term-to-path: <>(a),
cubicalpath-app: pth @ r,
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
cubical-beta,
interval-type_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
cc-fst_wf,
csm_id_adjoin_fst_type_lemma,
cubical-term_wf,
squash_wf,
true_wf,
csm-ap-id-type,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[b:\{X.\mBbbI{} \mvdash{} \_:(A)p\}]. \mforall{}[r:\{X \mvdash{} \_:\mBbbI{}\}]. (<>(b) @ r = (b)[r])
Date html generated:
2020_05_20-PM-03_18_41
Last ObjectModification:
2020_04_06-PM-06_33_50
Theory : cubical!type!theory
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