Nuprl Lemma : cubical-type-restriction-eq
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[g:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ A(alpha)].
cubical-type-restriction(X;T;I,alpha,t.(g I alpha t) = a(alpha) ∈ A(alpha))
supposing ∀I,J:fset(ℕ). ∀f:J ⟶ I. ∀alpha:X(I). ∀t:T(alpha).
((g J f(alpha) (t alpha f)) = (g I alpha t alpha f) ∈ A(f(alpha)))
Proof
Definitions occuring in Statement :
cubical-type-restriction: cubical-type-restriction,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
cubical-type-restriction: cubical-type-restriction,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-at_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-term-at-morph,
subtype_rel_self,
iff_weakening_equal,
istype-cubical-type-at,
cubical-term-at_wf,
I_cube_wf,
names-hom_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
equalityIstype,
functionIsType,
inhabitedIsType,
dependent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[g:I:fset(\mBbbN{}) {}\mrightarrow{} alpha:X(I) {}\mrightarrow{} T(alpha) {}\mrightarrow{} A(alpha)].
cubical-type-restriction(X;T;I,alpha,t.(g I alpha t) = a(alpha))
supposing \mforall{}I,J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}alpha:X(I). \mforall{}t:T(alpha).
((g J f(alpha) (t alpha f)) = (g I alpha t alpha f))
Date html generated:
2020_05_20-PM-03_13_23
Last ObjectModification:
2020_04_06-PM-05_16_51
Theory : cubical!type!theory
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