Nuprl Lemma : cubical-type-ap-morph-comp-eq
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[I,J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[a:X(I)]. ∀[b:X(J)]. ∀[u:A(a)].
((u a f) b g) = (u a f ⋅ g) ∈ A(f ⋅ g(a)) supposing b = f(a) ∈ X(J)
Proof
Definitions occuring in Statement :
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,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-at_wf,
cube-set-restriction_wf,
nh-comp_wf,
cubical-type-ap-morph-comp-eq-general,
cubical-type-ap-morph_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
subtype_rel_self,
iff_weakening_equal,
istype-cubical-type-at,
I_cube_wf,
names-hom_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
independent_isectElimination,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
because_Cache,
productElimination,
independent_functionElimination,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I,J,K:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[g:K {}\mrightarrow{} J]. \mforall{}[a:X(I)]. \mforall{}[b:X(J)]. \mforall{}[u:A(a)].
((u a f) b g) = (u a f \mcdot{} g) supposing b = f(a)
Date html generated:
2020_05_20-PM-01_48_32
Last ObjectModification:
2020_04_03-PM-08_26_09
Theory : cubical!type!theory
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