Nuprl Lemma : csm-comp-structure_wf2
∀[Gamma,Delta:j⊢]. ∀[tau:Delta j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma +⊢ Compositon(A)].
  ((cA)tau ∈ Delta +⊢ Compositon((A)tau))
Proof
Definitions occuring in Statement : 
csm-comp-structure: (cA)tau
, 
composition-structure: Gamma ⊢ Compositon(A)
, 
csm-ap-type: (AF)s
, 
cubical-type: {X ⊢ _}
, 
cube_set_map: A ⟶ B
, 
cubical_set: CubicalSet
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
csm-comp-structure: (cA)tau
, 
interval-type: 𝕀
, 
csm-comp: G o F
, 
compose: f o g
Lemmas referenced : 
csm-comp-structure_wf, 
cubical_set_cumulativity-i-j, 
cube_set_map_cumulativity-i-j, 
composition-structure_wf, 
cubical-type_wf, 
cube_set_map_wf, 
cubical_set_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
thin, 
instantiate, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
applyEquality, 
hypothesis, 
sqequalRule, 
because_Cache, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
universeIsType, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
inhabitedIsType
Latex:
\mforall{}[Gamma,Delta:j\mvdash{}].  \mforall{}[tau:Delta  j{}\mrightarrow{}  Gamma].  \mforall{}[A:\{Gamma  \mvdash{}  \_\}].  \mforall{}[cA:Gamma  +\mvdash{}  Compositon(A)].
    ((cA)tau  \mmember{}  Delta  +\mvdash{}  Compositon((A)tau))
Date html generated:
2020_05_20-PM-04_35_36
Last ObjectModification:
2020_04_23-PM-01_10_44
Theory : cubical!type!theory
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