Nuprl Lemma : I-path-morph-id
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I:Cname List. ∀alpha:X(I). ∀w:cubical-path(X;A;a;b;I;alpha).
(I-path-morph(X;A;I;I;1;alpha;w) = w ∈ cubical-path(X;A;a;b;I;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 ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
id-morph: 1,
coordinate_name: Cname,
list: T List,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
cubical-path: cubical-path(X;A;a;b;I;alpha),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
I-path: I-path(X;A;a;b;I;alpha),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
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),
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
named-path: named-path(X;A;a;b;I;alpha;z)
Lemmas referenced :
I-path_wf,
quotient-member-eq,
path-eq_wf,
path-eq-equiv,
I-path-morph_wf,
id-morph_wf,
named-path_wf,
l_member_wf,
coordinate_name_wf,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
squash_wf,
true_wf,
cube-set-restriction-id,
iff_weakening_equal,
equal-wf-base,
cubical-path_wf,
I-cube_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
fresh-cname_wf,
set_wf,
not_wf,
cubical-type-ap-morph-comp,
cons_wf,
extend-name-morph_wf,
rename-one-name_wf,
iota_wf,
cubical-type-at_wf,
cube-set-restriction-comp,
name-comp_wf,
name-morph_wf,
extend-name-morph-iota,
name-comp-id-left,
rename-one-extend-name-morph,
cubical-type-ap-morph_wf,
rename-one-iota,
rename-one-extend-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
dependent_pairEquality,
setElimination,
independent_functionElimination,
voidElimination,
applyEquality,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productEquality,
addLevel,
hyp_replacement,
equalityUniverse,
levelHypothesis,
independent_pairFormation
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}. \mforall{}I:Cname List. \mforall{}alpha:X(I).
\mforall{}w:cubical-path(X;A;a;b;I;alpha).
(I-path-morph(X;A;I;I;1;alpha;w) = w)
Date html generated:
2017_10_05-PM-03_55_31
Last ObjectModification:
2017_07_28-AM-11_28_25
Theory : cubical!sets
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