Nuprl Lemma : discrete-path-endpoints
∀[X:j⊢]. ∀[T:Type].
∀a:{X ⊢ _:discr(T)}. ∀[b:{X ⊢ _:discr(T)}]. ∀[p:{X ⊢ _:(Path_discr(T) a b)}]. (a = b ∈ {X ⊢ _:discr(T)})
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
uimplies: b supposing a,
discrete-cubical-type: discr(T),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
interval-type: 𝕀,
cube-context-adjoin: X.A,
cubical-type-at: A(a),
interval-presheaf: 𝕀,
constant-cubical-type: (X),
pi1: fst(t),
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap: (s)x,
interval-0: 0(𝕀),
csm-id: 1(X),
csm-adjoin: (s;u),
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
names-hom: I ⟶ J,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
names: names(I),
prop: ℙ,
squash: ↓T,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pi2: snd(t),
cubical-path-app: pth @ r
Lemmas referenced :
path-eta_wf,
discrete-cubical-type_wf,
path-type-subtype,
csm-discrete-cubical-type,
cubical-term_wf,
path-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
istype-universe,
cubical_set_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
csm-ap-term_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
cubical_type_at_pair_lemma,
csm-ap-term-at,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
cubical_type_ap_morph_pair_lemma,
istype-void,
istype-le,
fset-singleton_wf,
dM_inc_wf,
member-fset-singleton,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
fset-member_wf,
names_wf,
equal_wf,
squash_wf,
true_wf,
dM0_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM1_wf,
subtype_rel_self,
iff_weakening_equal,
cube-set-restriction_wf,
dM-lift-0,
dM-lift-1,
names-hom_wf,
dM-lift-inc,
cubical-path-app-0,
cubical-path-app-1,
path-eta-0,
path-eta-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
Error :memTop,
universeIsType,
instantiate,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
cumulativity,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
universeEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
independent_functionElimination,
functionExtensionality,
independent_isectElimination,
setElimination,
rename,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
intEquality,
productElimination,
imageElimination,
dependent_pairEquality_alt,
productEquality,
isectEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:Type].
\mforall{}a:\{X \mvdash{} \_:discr(T)\}. \mforall{}[b:\{X \mvdash{} \_:discr(T)\}]. \mforall{}[p:\{X \mvdash{} \_:(Path\_discr(T) a b)\}]. (a = b)
Date html generated:
2020_05_20-PM-03_36_05
Last ObjectModification:
2020_04_08-PM-09_54_51
Theory : cubical!type!theory
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