Nuprl Lemma : pi_comp_wf_fun
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X +⊢ Compositon(B)].
  (pi_comp(X;A;cA;(cB)p) ∈ X ⊢ Compositon((A ⟶ B)))
Proof
Definitions occuring in Statement : 
pi_comp: pi_comp(X;A;cA;cB)
, 
csm-comp-structure: (cA)tau
, 
composition-structure: Gamma ⊢ Compositon(A)
, 
cubical-fun: (A ⟶ B)
, 
cc-fst: p
, 
cube-context-adjoin: X.A
, 
cubical-type: {X ⊢ _}
, 
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
, 
squash: ↓T
, 
true: True
Lemmas referenced : 
pi_comp_wf2, 
csm-ap-type_wf, 
cube-context-adjoin_wf, 
cubical-type-cumulativity2, 
cubical_set_cumulativity-i-j, 
cc-fst_wf, 
csm-comp-structure_wf2, 
composition-structure_wf, 
cubical-fun-as-cubical-pi, 
cubical-type_wf, 
cubical_set_wf
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
instantiate, 
applyEquality, 
because_Cache, 
sqequalRule, 
lambdaEquality_alt, 
imageElimination, 
equalitySymmetry, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
equalityTransitivity, 
hyp_replacement, 
universeIsType, 
inhabitedIsType
Latex:
\mforall{}[X:j\mvdash{}].  \mforall{}[A,B:\{X  \mvdash{}  \_\}].  \mforall{}[cA:X  +\mvdash{}  Compositon(A)].  \mforall{}[cB:X  +\mvdash{}  Compositon(B)].
    (pi\_comp(X;A;cA;(cB)p)  \mmember{}  X  \mvdash{}  Compositon((A  {}\mrightarrow{}  B)))
Date html generated:
2020_05_20-PM-05_08_05
Last ObjectModification:
2020_04_27-PM-01_52_27
Theory : cubical!type!theory
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