Nuprl Lemma : discrete-pathtype
∀[T:Type]. ∀[X:j⊢]. ∀[pth:{X ⊢ _:Path(discr(T))}]. (pth = refl(pth @ 0(𝕀)) ∈ {X ⊢ _:Path(discr(T))})
Proof
Definitions occuring in Statement :
cubical-refl: refl(a),
cubical-path-app: pth @ r,
pathtype: Path(A),
interval-0: 0(𝕀),
discrete-cubical-type: discr(T),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-term-at: u(a),
uimplies: b supposing a,
pathtype: Path(A),
cubical-fun: (A ⟶ B),
all: ∀x:A. B[x],
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
squash: ↓T,
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,
discrete-cubical-type: discr(T),
cubical-refl: refl(a),
term-to-path: <>(a),
cubical-lambda: (λb),
cubical-path-app: pth @ r,
csm-ap-term: (t)s,
cubicalpath-app: pth @ r,
cubical-app: app(w; u),
interval-0: 0(𝕀),
cubical-term: {X ⊢ _:A},
prop: ℙ,
true: True,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
names-hom: I ⟶ J,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
names: names(I),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
DeMorgan-algebra: DeMorganAlgebra,
dma-neg: ¬(x),
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
dM0: 0,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
opposite-lattice: opposite-lattice(L),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
dM1: 1,
or: P ∨ Q,
lattice-meet: a ∧ b,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
fset-union: x ⋃ y,
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
lattice-join: a ∨ b,
so_lambda: λ2x y.t[x; y],
lattice-fset-meet: /\(s)
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
pathtype_wf,
discrete-cubical-type_wf,
cubical-term_wf,
cubical_set_wf,
istype-universe,
cubical-term-at_wf,
cubical_type_at_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,
interval-type-at,
I_cube_pair_redex_lemma,
cubical_type_ap_morph_pair_lemma,
cc_fst_adjoin_cube_lemma,
nh-id_wf,
dM0_wf,
subtype_rel_self,
equal_wf,
squash_wf,
true_wf,
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,
interval-type-ap-morph,
iff_weakening_equal,
dM-lift_wf2,
dM-lift-0,
dM1_wf,
nh-id-left,
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,
dM-lift-1,
dM_opp_wf,
dM-lift-inc,
dM-lift-opp,
neg-dM1,
one-dimensional-dM,
dM-lift-meet,
dM-lift-join,
lattice-join-1,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
dma-neg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
instantiate,
cumulativity,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeEquality,
dependent_functionElimination,
Error :memTop,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
functionIsType,
because_Cache,
equalityIstype,
applyEquality,
hyp_replacement,
lambdaEquality_alt,
natural_numberEquality,
independent_pairFormation,
lambdaFormation_alt,
voidElimination,
intEquality,
productElimination,
independent_functionElimination,
productEquality,
isectEquality,
promote_hyp,
unionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[X:j\mvdash{}]. \mforall{}[pth:\{X \mvdash{} \_:Path(discr(T))\}]. (pth = refl(pth @ 0(\mBbbI{})))
Date html generated:
2020_05_20-PM-03_36_27
Last ObjectModification:
2020_04_07-PM-04_29_09
Theory : cubical!type!theory
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