Nuprl Lemma : csm-ap-term-cube+
∀[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}].
((u)cube+(I;i) ∈ {formal-cube(I), canonical-section(();𝔽;I;⋅;phi).𝕀 ⊢ _:(A)<rho> o cube+(I;i)})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cube+: cube+(I;i),
interval-type: 𝕀,
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-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,
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,
prop: ℙ,
subtype_rel: A ⊆r B,
unit: Unit,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
trivial-cube-set: (),
face-presheaf: 𝔽,
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,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
squash: ↓T,
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],
cubical-type: {X ⊢ _},
interval-type: 𝕀,
csm-comp: G o F,
csm-ap-type: (AF)s,
subset-iota: iota,
compose: f o g,
csm-ap: (s)x,
csm-ap-term: (t)s,
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cube-context-adjoin: X.A,
interval-presheaf: 𝕀,
cc-fst: p,
cube+: cube+(I;i),
nc-s: s,
DeMorgan-algebra: DeMorganAlgebra,
names: names(I),
bool: 𝔹,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
csm-face-type,
face-type_wf,
formal-cube_wf1,
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,
cubical-term-eqcd,
canonical-section_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
trivial-cube-set_wf,
it_wf,
subtype_rel_self,
I_cube_wf,
cubical-type-at_wf_face-type,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical_set_wf,
context-subset-is-cubical-subset,
iff_weakening_equal,
cubical-subset_wf,
face-presheaf_wf2,
istype-cubical-term,
csm-ap-type_wf,
csm-comp_wf,
subset-iota_wf,
context-map_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,
context-subset-map,
cube-context-adjoin_wf,
interval-type_wf,
cube+_wf,
csm-ap-term_wf,
context-subset_wf,
cubical_set_cumulativity-i-j,
subtype_rel-equal,
cube_set_map_wf,
subset-cubical-term,
context-adjoin-subset2,
sub_cubical_set_wf,
cubical-term-equal2,
cc-fst_wf_interval,
cube_set_restriction_pair_lemma,
fl-morph-comp2,
csm-ap_wf,
names-hom_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,
fl-morph_wf,
I_cube_pair_redex_lemma,
interval-type-at,
nh-comp-sq,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
dM-lift-inc,
eq_int_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
names_wf,
names-subtype,
eqtt_to_assert,
assert_of_eq_int,
int_subtype_base,
subset-cubical-type,
context-subset-is-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
hyp_replacement,
instantiate,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
setIsType,
functionIsType,
intEquality,
inhabitedIsType,
cumulativity,
lambdaFormation_alt,
equalityIstype,
productEquality,
isectEquality,
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\}].
((u)cube+(I;i) \mmember{} \{formal-cube(I), canonical-section(();\mBbbF{};I;\mcdot{};phi).\mBbbI{} \mvdash{} \_:(A)<rho> o cube+(I;i)\})
Date html generated:
2020_05_20-PM-04_27_35
Last ObjectModification:
2020_04_21-AM-00_49_57
Theory : cubical!type!theory
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