Nuprl Lemma : canonical-section-cubical-path-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)].
∀[u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}]. ∀[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)].
(canonical-section(Gamma;A;I;(i0)(rho);a0)
∈ {formal-cube(I) ⊢ _:((A)<rho> o cube+(I;i))[0(𝕀)][canonical-section(();𝔽;I;⋅;phi) |⟶ ((u)cube+(I;i))[0(𝕀)]]})
Proof
Definitions occuring in Statement :
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cube+: cube+(I;i),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
trivial-cube-set: (),
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-0: (i0),
nc-s: s,
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term-at: u(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
formal-cube: formal-cube(I),
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
subset-iota: iota,
csm-comp: G o F,
csm-ap: (s)x,
context-map: <rho>,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
interval-type: 𝕀,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
unit: Unit,
trivial-cube-set: (),
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
context-subset: Gamma, phi,
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
name-morph-satisfies: (psi f) = 1,
bdd-distributive-lattice: BoundedDistributiveLattice,
csm-ap-term: (t)s,
cubical-type-ap-morph: (u a f),
sq_type: SQType(T),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cube+: cube+(I;i),
nc-0: (i0),
names: names(I),
bool: 𝔹,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical_set_wf,
context-subset-is-cubical-subset,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
canonical-section_wf,
face-type_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
cubical-subset_wf,
subtype_rel_self,
iff_weakening_equal,
csm-ap-term-cube+,
cubical-path-0_wf,
cubical-type-cumulativity2,
cubical-term_wf,
face-presheaf_wf2,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
cubical-type_wf,
fl-morph-id,
face-type-ap-morph,
nc-0_wf,
csm-ap-comp-type,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
cube+_wf,
cube_set_map_wf,
csm-comp-assoc,
context-map-comp2,
cube+_interval-0,
context-subset-map,
trivial-cube-set_wf,
it_wf,
context-subset_wf,
csm-canonical-section-face-type-0,
cubical-subset-is-context-subset-canonical,
I_cube_pair_redex_lemma,
context-map_wf_cubical-subset,
csm-equal2,
arrow_pair_lemma,
cubical-subset-I_cube,
cube-set-restriction-comp,
csm-ap-term_wf,
cubical-type-subtype-cubical-subset,
equal_functionality_wrt_subtype_rel2,
cubical-term-equal2,
cubical-type-at_wf_face-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-face-type,
thin-context-subset,
cubical-term-subtype-cubical-subset,
name-morph-satisfies_wf,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-type-at_wf,
csm-ap-type-at,
csm-ap_wf,
cubical-type-ap-morph_wf,
csm-ap-term-at,
subtype_base_sq,
base_wf,
csm-ap-csm-comp,
cubical-term-at_wf,
nh-comp_wf,
name-morph-satisfies-comp,
names-hom_wf,
nh-comp-assoc,
s-comp-nc-0,
fl-morph_wf,
nh-id-right,
context-subset-term-subtype,
constrained-cubical-term_wf,
nh-comp-sq,
names_wf,
dM0-sq-empty,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
dM-lift-0,
not-added-name,
dM-lift-inc,
cubical-term-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
productElimination,
setIsType,
functionIsType,
intEquality,
inhabitedIsType,
hyp_replacement,
lambdaFormation_alt,
cumulativity,
productEquality,
isectEquality,
productIsType,
equalityIstype,
applyLambdaEquality,
baseApply,
closedConclusion,
functionExtensionality,
equalityElimination,
promote_hyp
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[rho:Gamma(I+i)]. \mforall{}[phi:\mBbbF{}(I)].
\mforall{}[u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}]. \mforall{}[a0:cubical-path-0(Gamma;A;I;i;rho;phi;u)].
(canonical-section(Gamma;A;I;(i0)(rho);a0)
\mmember{} \{formal-cube(I) \mvdash{} \_:((A)<rho> o cube+(I;i))[0(\mBbbI{})][canonical-section(();\mBbbF{};I;\mcdot{};phi)
|{}\mrightarrow{} ((u)cube+(I;i))[0(\mBbbI{})]]\})
Date html generated:
2020_05_20-PM-04_28_16
Last ObjectModification:
2020_04_11-AM-10_19_56
Theory : cubical!type!theory
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