Nuprl Lemma : unit-pathtype
∀[G:j⊢]. ∀a,b:{G ⊢ _:Path(1)}. (a = b ∈ {G ⊢ _:Path(1)})
Proof
Definitions occuring in Statement :
pathtype: Path(A),
cubical-unit: 1,
cubical-term: {X ⊢ _:A},
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],
uimplies: b supposing a,
cubical-term: {X ⊢ _:A},
pathtype: Path(A),
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
interval-presheaf: 𝕀,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
cubical-unit: 1,
discrete-cubical-type: discr(T),
unit: Unit,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
pathtype_wf,
cubical-unit_wf,
cubical-term_wf,
cubical_set_wf,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
names-hom_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
interval-type_wf,
cubical-type-ap-morph_wf,
nh-comp_wf,
subtype_rel-equal,
cubical-type-at_wf,
subtype_rel_self,
unit_wf2,
equal-unit,
sq_stable__all,
equal_wf,
sq_stable__equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inhabitedIsType,
universeIsType,
instantiate,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
setElimination,
rename,
Error :memTop,
dependent_set_memberEquality_alt,
functionIsType,
because_Cache,
equalityIstype,
applyEquality,
independent_functionElimination,
functionEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}a,b:\{G \mvdash{} \_:Path(1)\}. (a = b)
Date html generated:
2020_05_20-PM-03_34_36
Last ObjectModification:
2020_04_06-PM-06_59_17
Theory : cubical!type!theory
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