Nuprl Lemma : I-path-morph_functionality
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I,K:Cname List. ∀f:name-morph(I;K). ∀alpha:X(I).
∀p,q:I-path(X;A;a;b;I;alpha).
(path-eq(X;A;I;alpha;p;q) ⇒ path-eq(X;A;K;f(alpha);I-path-morph(X;A;I;K;f;alpha;p);I-path-morph(X;A;I;K;f;alpha;q)))
Proof
Definitions occuring in Statement :
path-eq: path-eq(X;A;I;alpha;p;q),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
I-path: I-path(X;A;a;b;I;alpha),
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
I-path: I-path(X;A;a;b;I;alpha),
I-path-morph: I-path-morph(X;A;I;K;f;alpha;p),
path-eq: path-eq(X;A;I;alpha;p;q),
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
named-path: named-path(X;A;a;b;I;alpha;z),
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
not: ¬A,
false: False
Lemmas referenced :
fresh-cname_wf,
set_wf,
coordinate_name_wf,
not_wf,
l_member_wf,
rename-one-same,
equal_wf,
path-eq_wf,
I-path_wf,
I-cube_wf,
name-morph_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
cubical-type-at_wf,
cons_wf,
cube-set-restriction_wf,
iota_wf,
cubical-type-ap-morph_wf,
extend-name-morph_wf,
subtype_rel-equal,
cube-set-restriction-comp,
iff_weakening_equal,
squash_wf,
true_wf,
extend-name-morph-iota,
name-comp_wf,
rename-one-name_wf,
cube-set-restriction-when-id,
id-morph_wf,
cubical-type-ap-morph-id,
cube-set-restriction-id,
cubical-type-ap-morph-comp,
extend-name-morph-rename-one,
rename-one-iota
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
setElimination,
rename,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
independent_isectElimination,
applyEquality,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairFormation,
voidElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}. \mforall{}I,K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}alpha:X(I).
\mforall{}p,q:I-path(X;A;a;b;I;alpha).
(path-eq(X;A;I;alpha;p;q)
{}\mRightarrow{} path-eq(X;A;K;f(alpha);I-path-morph(X;A;I;K;f;alpha;p);I-path-morph(X;A;I;K;f;alpha;q)))
Date html generated:
2017_10_05-PM-03_55_09
Last ObjectModification:
2017_07_28-AM-11_28_10
Theory : cubical!sets
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