Nuprl Lemma : csm-term-to-pathtype
∀[G:j⊢]. ∀[A:{G ⊢ _}].
∀a:{G.𝕀 ⊢ _:(A)p}. ∀[H:j⊢]. ∀[sigma:H j⟶ G]. ((<>a)sigma = <>(a)sigma+ ∈ {H ⊢ _:Path((A)sigma)})
Proof
Definitions occuring in Statement :
term-to-pathtype: <>a,
pathtype: Path(A),
interval-type: 𝕀,
csm+: tau+,
cc-fst: p,
cube-context-adjoin: X.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,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
term-to-pathtype: <>a,
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,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g
Lemmas referenced :
csm-term-to-path,
cube_set_map_wf,
cubical-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
path-type-subtype,
csm-ap-term_wf,
csm+_wf,
subtype_rel_self,
csm-id-adjoin_wf,
csm-interval-type,
interval-0_wf,
interval-1_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
universeIsType,
inhabitedIsType,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
Error :memTop
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}a:\{G.\mBbbI{} \mvdash{} \_:(A)p\}. \mforall{}[H:j\mvdash{}]. \mforall{}[sigma:H j{}\mrightarrow{} G]. ((<>a)sigma = <>(a)sigma+)
Date html generated:
2020_05_20-PM-03_20_24
Last ObjectModification:
2020_04_07-PM-00_59_22
Theory : cubical!type!theory
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