Nuprl Lemma : path-type-ext-eq
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}].
{t:{X ⊢ _:Path(A)}| (t @ 0(𝕀) = a ∈ {X ⊢ _:A}) ∧ (t @ 1(𝕀) = b ∈ {X ⊢ _:A})} ≡ {X ⊢ _:(Path_A a b)}
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
cubicalpath-app: pth @ r,
pathtype: Path(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
cubical-term: {X ⊢ _:A},
path-type: (Path_A a b),
cubical-subset: cubical-subset,
all: ∀x:A. B[x],
cand: A c∧ B,
pathtype: Path(A),
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
uimplies: b supposing a,
squash: ↓T,
true: True,
respects-equality: respects-equality(S;T),
implies: P ⇒ Q,
prop: ℙ,
interval-0: 0(𝕀),
cubicalpath-app: pth @ r,
cubical-term-at: u(a),
cubical-app: app(w; u),
interval-1: 1(𝕀),
guard: {T},
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-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,
cubical-path-app: pth @ r
Lemmas referenced :
cubical-term_wf,
pathtype_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubicalpath-app_wf,
interval-0_wf,
interval-1_wf,
path-type_wf,
cubical-type_wf,
cubical_set_wf,
cubical_type_at_pair_lemma,
I_cube_wf,
fset_wf,
nat_wf,
cubical_type_ap_morph_pair_lemma,
names-hom_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
nh-id_wf,
dM0_wf,
cubical-term-at_wf,
subtype-respects-equality,
cubical-type-at_wf,
subtype_rel-equal,
cube-set-restriction-id,
dM1_wf,
squash_wf,
true_wf,
equal_wf,
istype-universe,
subtype_rel_self,
interval-type_wf,
path-type-subtype,
cubical-path-app-0,
cubical-path-app-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
setIsType,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
productIsType,
equalityIstype,
because_Cache,
productElimination,
independent_pairEquality,
axiomEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
setElimination,
rename,
dependent_set_memberEquality_alt,
dependent_functionElimination,
Error :memTop,
functionExtensionality,
lambdaFormation_alt,
functionIsType,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeEquality,
applyLambdaEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}].
\{t:\{X \mvdash{} \_:Path(A)\}| (t @ 0(\mBbbI{}) = a) \mwedge{} (t @ 1(\mBbbI{}) = b)\} \mequiv{} \{X \mvdash{} \_:(Path\_A a b)\}
Date html generated:
2020_05_20-PM-03_17_17
Last ObjectModification:
2020_04_06-PM-06_36_07
Theory : cubical!type!theory
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