Nuprl Lemma : composition-type-lemma2
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)].
(A((new-name(I)1)((s(rho);<new-name(I)>))) = (A)[1(𝕀)](rho) ∈ Type)
Proof
Definitions occuring in Statement :
interval-1: 1(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-1: (i1),
nc-s: s,
new-name: new-name(I),
add-name: I+i,
dM_inc: <x>,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical_set_wf,
csm-ap-type-at,
cubical-type-at_wf,
squash_wf,
true_wf,
csm-ap-interval-1-adjoin-lemma,
new-name_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)].
(A((new-name(I)1)((s(rho);<new-name(I)>))) = (A)[1(\mBbbI{})](rho))
Date html generated:
2020_05_20-PM-04_07_12
Last ObjectModification:
2020_04_10-AM-03_44_12
Theory : cubical!type!theory
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