Nuprl Lemma : cubical-pi-family-comp
∀X,Delta:CubicalSet. ∀s:Delta ⟶ X. ∀I,J:Cname List. ∀f:name-morph(I;J). ∀a:Delta(I). ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}.
∀w:cubical-pi-family(X;A;B;I;(s)a).
(λK,g. (w K (f o g)) ∈ cubical-pi-family(X;A;B;J;(s)f(a)))
Proof
Definitions occuring in Statement :
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
csm-ap: (s)x,
I-cube: X(I),
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
top: Top,
cubical-type-ap-morph: (u a f),
cand: A c∧ B,
cubical-type: {X ⊢ _},
pi2: snd(t),
prop: ℙ,
cubical-type-at: A(a),
pi1: fst(t)
Lemmas referenced :
list_wf,
coordinate_name_wf,
name-comp_wf,
subtype_rel_dep_function,
cubical-type-at_wf,
cube-set-restriction_wf,
csm-ap_wf,
cube-context-adjoin_wf,
cc-adjoin-cube_wf,
subtype_rel-equal,
equal_wf,
I-cube_wf,
csm-ap-restriction,
iff_weakening_equal,
cube-set-restriction-comp,
name-morph_wf,
cubical-type-ap-morph_wf,
cc-adjoin-cube-restriction,
all_wf,
cubical-pi-family_wf,
cubical-type_wf,
cube-set-map_wf,
squash_wf,
true_wf,
subtype_rel_self,
cubical-set_wf,
name-comp-assoc,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
because_Cache,
sqequalRule,
dependent_functionElimination,
independent_isectElimination,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
universeEquality,
instantiate,
hyp_replacement,
functionEquality,
applyLambdaEquality
Latex:
\mforall{}X,Delta:CubicalSet. \mforall{}s:Delta {}\mrightarrow{} X. \mforall{}I,J:Cname List. \mforall{}f:name-morph(I;J). \mforall{}a:Delta(I). \mforall{}A:\{X \mvdash{} \_\}.
\mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}w:cubical-pi-family(X;A;B;I;(s)a).
(\mlambda{}K,g. (w K (f o g)) \mmember{} cubical-pi-family(X;A;B;J;(s)f(a)))
Date html generated:
2018_05_23-PM-06_29_38
Last ObjectModification:
2018_05_20-PM-04_11_33
Theory : cubical!sets
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