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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