Nuprl Lemma : csm-term-to-path
∀[G:j⊢]. ∀[A:{G ⊢ _}].
∀a:{G.𝕀 ⊢ _:(A)p}
∀[H:j⊢]. ∀[sigma:H j⟶ G].
((<>(a))sigma = H ⊢ <>((a)sigma+) ∈ {H ⊢ _:(Path_(A)sigma ((a)sigma+)[0(𝕀)] ((a)sigma+)[1(𝕀)])})
Proof
Definitions occuring in Statement :
term-to-path: <>(a),
path-type: (Path_A a b),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
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],
all: ∀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,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
interval-type: 𝕀,
csm+: tau+,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-comp: G o F,
interval-1: 1(𝕀),
term-to-path: <>(a),
pathtype: Path(A),
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
paths-equal,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cubical-type-cumulativity2,
csm-ap-term_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf,
csm+_wf,
subtype_rel_self,
cube_set_map_wf,
csm-id-adjoin_wf,
csm-interval-type,
interval-0_wf,
interval-1_wf,
cubical-term_wf,
cubical-type_wf,
cubical_set_wf,
squash_wf,
true_wf,
csm-path-type,
csm-id-adjoin_wf-interval-0,
subset-cubical-term2,
sub_cubical_set_self,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
csm-id-adjoin_wf-interval-1,
path-type_wf,
term-to-path_wf,
p-csm+-type,
equal_wf,
istype-universe,
cubical-pi_wf,
cubical-fun-as-cubical-pi,
iff_weakening_equal,
cubical-lambda_wf,
csm-cubical-lambda,
csm-ap-type-fst-adjoin,
csm-comp-type,
csm-cubical-pi
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
Error :memTop,
independent_isectElimination,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
universeEquality,
productElimination,
independent_functionElimination
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 = H \mvdash{} <>((a)sigma+))
Date html generated:
2020_05_20-PM-03_18_30
Last ObjectModification:
2020_04_06-PM-06_36_37
Theory : cubical!type!theory
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