Nuprl Lemma : csm-cubical-refl
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[H:j⊢]. ∀[s:H j⟶ X]. ((refl(a))s = refl((a)s) ∈ {H ⊢ _:(Path_(A)s (a)s (a)s)})
Proof
Definitions occuring in Statement :
cubical-refl: refl(a),
path-type: (Path_A a b),
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-refl: refl(a),
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
true: True,
cubical-type: {X ⊢ _},
csm-ap-term: (t)s,
csm-ap-type: (AF)s,
cc-fst: p,
interval-type: 𝕀,
csm+: tau+,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
interval-0: 0(𝕀),
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
cc-snd: q,
constant-cubical-type: (X),
csm-comp: G o F,
pi1: fst(t),
compose: f o g,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
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)
Lemmas referenced :
csm-term-to-path,
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cc-fst_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube_set_map_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
path-type_wf,
csm-ap-type_wf,
csm-path-type,
subtype_rel_self,
cubical-refl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
instantiate,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
setElimination,
rename,
productElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} X]. ((refl(a))s = refl((a)s))
Date html generated:
2020_05_20-PM-03_21_22
Last ObjectModification:
2020_04_07-PM-03_23_06
Theory : cubical!type!theory
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