Nuprl Lemma : path-restriction
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}].
cubical-type-restriction(X;Path(A);I,a1,p.((p I 1 0) = a(a1) ∈ A(a1)) ∧ ((p I 1 1) = b(a1) ∈ A(a1)))
Proof
Definitions occuring in Statement :
pathtype: Path(A),
cubical-type-restriction: cubical-type-restriction,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
nh-id: 1,
dM1: 1,
dM0: 0,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: so_lambda3,
pathtype: Path(A),
cubical-fun: (A ⟶ B),
all: ∀x:A. B[x],
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
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,
so_apply: x[s1;s2;s3],
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
cubical-type-restriction-and,
pathtype_wf,
equal_wf,
cubical-type-at_wf,
cubical_type_at_pair_lemma,
nh-id_wf,
dM0_wf,
subtype_rel_self,
interval-type_wf,
cube-set-restriction_wf,
cubical-term-at_wf,
istype-cubical-type-at,
I_cube_wf,
dM1_wf,
cubical-type-restriction-eq,
names-hom_wf,
interval-type-at,
I_cube_pair_redex_lemma,
interval-type-ap-morph,
cubical_type_ap_morph_pair_lemma,
squash_wf,
true_wf,
istype-universe,
fset_wf,
nat_wf,
cube-set-restriction-id,
iff_weakening_equal,
cubical-type-ap-morph_wf,
subtype_rel-equal,
nh-id-left,
nh-comp_wf,
dM-lift_wf2,
nh-id-right,
dM-lift-0-sq,
dM-lift-1-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
because_Cache,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
independent_functionElimination,
independent_isectElimination,
lambdaFormation_alt,
equalitySymmetry,
hyp_replacement,
imageElimination,
equalityTransitivity,
universeEquality,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination,
natural_numberEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}].
cubical-type-restriction(X;Path(A);I,a1,p.((p I 1 0) = a(a1)) \mwedge{} ((p I 1 1) = b(a1)))
Date html generated:
2020_05_20-PM-03_14_25
Last ObjectModification:
2020_04_06-PM-05_37_03
Theory : cubical!type!theory
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