Nuprl Lemma : csm-paths-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[p:{X ⊢ _:(Path_A a b)}]. ∀[H:j⊢]. ∀[tau:H j⟶ X]. ∀[q:{H ⊢ _:(Path(A))tau}].
(p)tau = q ∈ {H ⊢ _:((Path_A a b))tau} supposing (p)tau = q ∈ {H ⊢ _:(Path(A))tau}
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
pathtype: Path(A),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
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,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-term_wf,
pathtype_wf,
path-type-subtype,
cubical-term_wf,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cube_set_map_wf,
path-type_wf,
cubical-type_wf,
cubical_set_wf,
squash_wf,
true_wf,
csm-pathtype,
paths-equal,
subset-cubical-term2,
sub_cubical_set_self,
csm-path-type,
subtype_rel_self,
equal_wf,
istype-universe,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
equalityIstype,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
universeIsType,
instantiate,
inhabitedIsType,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
hyp_replacement,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[p:\{X \mvdash{} \_:(Path\_A a b)\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[tau:H j{}\mrightarrow{} X].
\mforall{}[q:\{H \mvdash{} \_:(Path(A))tau\}].
(p)tau = q supposing (p)tau = q
Date html generated:
2020_05_20-PM-03_18_03
Last ObjectModification:
2020_04_07-PM-03_16_17
Theory : cubical!type!theory
Home
Index