Nuprl Lemma : csm-cubical-pi-typed
∀X,Delta:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀s:Delta ij⟶ X. ((ΠA B)s = Delta ⊢ Π(A)s (B)(s)dep ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
csm-dependent: (s)dep,
cubical-pi: ΠA B,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
csm-dependent: (s)dep,
typed-cc-snd: tq,
typed-cc-fst: tp{i:l}
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-dependent_wf,
cube_set_map_wf,
cubical-type_wf,
cubical_set_wf,
csm-cubical-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
dependent_functionElimination,
universeIsType,
inhabitedIsType
Latex:
\mforall{}X,Delta:j\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}s:Delta ij{}\mrightarrow{} X. ((\mPi{}A B)s = Delta \mvdash{} \mPi{}(A)s (B)(s)dep)
Date html generated:
2020_05_20-PM-02_00_43
Last ObjectModification:
2020_04_04-AM-09_52_39
Theory : cubical!type!theory
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