Nuprl Lemma : member-cubical-path-0-0
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[u:Top]. ∀[x:A((i0)(rho))].
(x ∈ cubical-path-0(Gamma;A;I;i;rho;0;u))
Proof
Definitions occuring in Statement :
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
face_lattice: face_lattice(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-0: (i0),
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]} ,
lattice-0: 0
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
subtype_rel: A ⊆r B,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-term: {X ⊢ _:A},
name-morph-satisfies: (psi f) = 1,
cube-set-restriction: f(s),
pi2: snd(t),
face-presheaf: 𝔽,
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
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,
lattice-0: 0,
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,
empty-fset: {},
nil: [],
it: ⋅,
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
lattice-point: Point(l),
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t)
Lemmas referenced :
cubical-path-condition-0,
cubical-path-condition_wf,
lattice-0_wf,
face_lattice_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
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,
nc-0_wf,
istype-top,
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,
cubical_set_wf,
cubical-subset-I_cube-member,
nc-s_wf,
f-subset-add-name,
equal_wf,
lattice-point_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,
lattice-1_wf,
squash_wf,
true_wf,
istype-universe,
fl-morph-0,
subtype_rel_self,
iff_weakening_equal,
face-lattice-0-not-1,
cubical-subset_wf,
face-presheaf_wf2,
csm-ap-type-at,
names-hom_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
cubical-type-ap-morph_wf,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
universeIsType,
instantiate,
because_Cache,
applyEquality,
sqequalRule,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality,
functionExtensionality,
productElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
productEquality,
cumulativity,
isectEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
lambdaFormation_alt,
equalityIstype
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{}[u:Top].
\mforall{}[x:A((i0)(rho))].
(x \mmember{} cubical-path-0(Gamma;A;I;i;rho;0;u))
Date html generated:
2020_05_20-PM-03_46_22
Last ObjectModification:
2020_04_09-AM-11_04_41
Theory : cubical!type!theory
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