Nuprl Lemma : named-path-morph_wf
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I,K:Cname List. ∀f:name-morph(I;K). ∀alpha:X(I). ∀z:{z:Cname| ¬(z ∈ I)} .
∀w:named-path(X;A;a;b;I;alpha;z). ∀x:{x:Cname| ¬(x ∈ K)} .
(named-path-morph(X;A;I;K;z;x;f;alpha;w) ∈ named-path(X;A;a;b;K;f(alpha);x))
Proof
Definitions occuring in Statement :
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
named-path: named-path(X;A;a;b;I;alpha;z),
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,
l_member: (x ∈ l),
list: T List,
all: ∀x:A. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
named-path: named-path(X;A;a;b;I;alpha;z),
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
named-path-morph: named-path-morph(X;A;I;K;z;x;f;alpha;w),
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,
name-path-endpoints: name-path-endpoints(X;A;a;b;I;alpha;z;omega),
cubical-term-at: u(a),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b
Lemmas referenced :
set_wf,
coordinate_name_wf,
not_wf,
l_member_wf,
named-path_wf,
I-cube_wf,
name-morph_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
cubical-type-at_wf,
squash_wf,
true_wf,
cons_wf,
equal_wf,
cube-set-restriction-comp,
iota_wf,
extend-name-morph_wf,
cube-set-restriction_wf,
iff_weakening_equal,
extend-name-morph-iota,
name-comp_wf,
cubical-type-ap-morph_wf,
cubical-term-at-morph,
cubical-term-at_wf,
cubical-type-ap-morph-comp,
face-map_wf,
false_wf,
lelt_wf,
name-comp-assoc,
iota-identity,
name-comp-id-right,
subtype_rel-equal,
name-comp-id-left,
cube-set-restriction-id,
extend-name-morph-face-map,
name-path-endpoints_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesisEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
voidElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
imageElimination,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_pairFormation,
applyLambdaEquality,
instantiate
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{}z:\{z:Cname| \mneg{}(z \mmember{} I)\} . \mforall{}w:named-path(X;A;a;b;I;alpha;z). \mforall{}x:\{x:Cname| \mneg{}(x \mmember{} K)\} .
(named-path-morph(X;A;I;K;z;x;f;alpha;w) \mmember{} named-path(X;A;a;b;K;f(alpha);x))
Date html generated:
2017_10_05-PM-03_54_35
Last ObjectModification:
2017_07_28-AM-11_27_39
Theory : cubical!sets
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