Nuprl Lemma : csm-path-type-sub-pathtype
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[tau:H j⟶ X].  ({H ⊢ _:((Path_A a b))tau} ⊆r {H ⊢ _:(Path(A))tau})
Proof
Definitions occuring in Statement : 
path-type: (Path_A a b)
, 
pathtype: Path(A)
, 
cubical-term: {X ⊢ _:A}
, 
csm-ap-type: (AF)s
, 
cubical-type: {X ⊢ _}
, 
cube_set_map: A ⟶ B
, 
cubical_set: CubicalSet
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
cube_set_map: A ⟶ B
, 
psc_map: A ⟶ B
, 
nat-trans: nat-trans(C;D;F;G)
, 
cat-ob: cat-ob(C)
, 
pi1: fst(t)
, 
op-cat: op-cat(C)
, 
spreadn: spread4, 
cube-cat: CubeCat
, 
fset: fset(T)
, 
quotient: x,y:A//B[x; y]
, 
cat-arrow: cat-arrow(C)
, 
pi2: snd(t)
, 
type-cat: TypeCat
, 
all: ∀x:A. B[x]
, 
names-hom: I ⟶ J
, 
cat-comp: cat-comp(C)
, 
compose: f o g
, 
true: True
, 
squash: ↓T
, 
prop: ℙ
, 
uimplies: b supposing a
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
Lemmas referenced : 
cube_set_map_wf, 
cubical-term_wf, 
cubical-type-cumulativity2, 
cubical_set_cumulativity-i-j, 
subtype_rel_self, 
path-type-subtype, 
csm-ap-type_wf, 
csm-ap-term_wf, 
subtype_rel_wf, 
squash_wf, 
true_wf, 
istype-universe, 
csm-path-type, 
csm-pathtype, 
iff_weakening_equal
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
sqequalRule, 
axiomEquality, 
hypothesis, 
universeIsType, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
inhabitedIsType, 
because_Cache, 
instantiate, 
applyEquality, 
natural_numberEquality, 
lambdaEquality_alt, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
dependent_functionElimination, 
imageMemberEquality, 
baseClosed, 
independent_isectElimination, 
productElimination, 
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[a,b:\{X  \mvdash{}  \_:A\}].  \mforall{}[H:j\mvdash{}].  \mforall{}[tau:H  j{}\mrightarrow{}  X].
    (\{H  \mvdash{}  \_:((Path\_A  a  b))tau\}  \msubseteq{}r  \{H  \mvdash{}  \_:(Path(A))tau\})
Date html generated:
2020_05_20-PM-03_17_40
Last ObjectModification:
2020_04_06-PM-06_32_56
Theory : cubical!type!theory
Home
Index