Nuprl Lemma : I-path-morph-comp
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I,J,K:Cname List. ∀f:name-morph(I;J). ∀g:name-morph(J;K). ∀alpha:X(I).
∀u:cubical-path(X;A;a;b;I;alpha).
(I-path-morph(X;A;I;K;(f o g);alpha;u)
= I-path-morph(X;A;J;K;g;f(alpha);I-path-morph(X;A;I;J;f;alpha;u))
∈ cubical-path(X;A;a;b;K;(f o g)(alpha)))
Proof
Definitions occuring in Statement :
cubical-path: cubical-path(X;A;a;b;I;alpha),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
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],
equal: s = t ∈ T
Definitions unfolded in proof :
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
path-eq: path-eq(X;A;I;alpha;p;q),
I-path: I-path(X;A;a;b;I;alpha),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-path: cubical-path(X;A;a;b;I;alpha),
all: ∀x:A. B[x],
cand: A c∧ B,
named-path: named-path(X;A;a;b;I;alpha;z),
not: ¬A,
false: False
Lemmas referenced :
name-morph_wf,
I-cube_wf,
fresh-cname_wf,
iff_weakening_equal,
subtype_rel_self,
cube-set-restriction-comp,
istype-universe,
true_wf,
squash_wf,
equal_wf,
subtype_rel-equal,
I-path-morph_wf,
path-eq-equiv,
path-eq_wf,
I-path_wf,
quotient-member-eq,
name-comp_wf,
cube-set-restriction_wf,
cubical-path_wf,
rename-one-name_wf,
cubical-type-ap-morph-id,
iota_wf,
coordinate_name_wf,
cons_wf,
cubical-type-at_wf,
cubical-type-ap-morph_wf,
extend-name-morph_wf,
list_wf,
extend-name-morph-iota,
l_member_wf,
rename-one-same,
name-comp-assoc,
cubical-type-ap-morph-comp,
cubical-type_wf,
cubical-set_wf,
extend-name-morph-rename-one,
rename-one-iota,
extend-name-morph-comp,
cube-set-restriction-id
Rules used in proof :
sqequalBase,
productIsType,
equalityIstype,
setElimination,
independent_functionElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
imageElimination,
instantiate,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
universeIsType,
because_Cache,
lambdaEquality_alt,
rename,
inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
productElimination,
promote_hyp,
pertypeElimination,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyLambdaEquality,
hyp_replacement,
independent_pairFormation,
voidElimination,
dependent_set_memberEquality_alt
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}. \mforall{}I,J,K:Cname List. \mforall{}f:name-morph(I;J).
\mforall{}g:name-morph(J;K). \mforall{}alpha:X(I). \mforall{}u:cubical-path(X;A;a;b;I;alpha).
(I-path-morph(X;A;I;K;(f o g);alpha;u)
= I-path-morph(X;A;J;K;g;f(alpha);I-path-morph(X;A;I;J;f;alpha;u)))
Date html generated:
2020_05_21-AM-11_06_26
Last ObjectModification:
2020_01_15-PM-01_26_08
Theory : cubical!sets
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