Nuprl Lemma : discrete-comp_wf
∀[G:j⊢]. ∀[T:Type]. (discrete-comp(G;T) ∈ G ⊢ CompOp(discr(T)))
Proof
Definitions occuring in Statement :
discrete-comp: discrete-comp(G;T),
composition-op: Gamma ⊢ CompOp(A),
discrete-cubical-type: discr(T),
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
discrete-comp: discrete-comp(G;T),
composition-op: Gamma ⊢ CompOp(A),
composition-uniformity: composition-uniformity(Gamma;A;comp),
all: ∀x:A. B[x],
cubical-type-ap-morph: (u a f),
pi2: snd(t),
discrete-cubical-type: discr(T),
nat: ℕ,
ge: i ≥ j ,
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,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
squash: ↓T,
true: True,
cubical-subset: I,psi,
cube-cat: CubeCat,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
I_cube: A(I),
functor-ob: ob(F),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
guard: {T},
iff: P ⇐⇒ Q,
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
compose: f o g,
uiff: uiff(P;Q),
cand: A c∧ B,
irr_face: irr_face(I;as;bs),
lattice-fset-meet: /\(s),
lattice-meet: a ∧ b,
fset-constrained-ac-glb: glb(P;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
union-deq: union-deq(A;B;a;b),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
names: names(I),
rev_implies: P ⇐ Q,
nc-s: s,
rev_uimplies: rev_uimplies(P;Q),
nc-1: (i1),
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nc-0: (i0),
cubical-term-at: u(a),
discrete-cubical-term: discr(t)
Lemmas referenced :
cubical-type-ap-morph_wf,
discrete-cubical-type_wf,
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,
subtype_rel-equal,
cubical-type-at_wf,
nc-1_wf,
cubical-path-0_wf,
cubical-term_wf,
cubical-subset_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
names-hom_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,
composition-uniformity_wf,
istype-universe,
cubical_set_wf,
csm-discrete-cubical-type,
cubical-path-condition'_wf,
subset-cubical-term2,
sub_cubical_set_self,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
equal_wf,
squash_wf,
true_wf,
I_cube_pair_redex_lemma,
cat_arrow_triple_lemma,
face_lattice-basis,
subtype_rel_self,
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,
name-morph-satisfies_wf,
iff_weakening_equal,
fl-morph-fset-join,
fset-image_wf,
names_wf,
assert_wf,
fset-disjoint_wf,
names-deq_wf,
pi1_wf_top,
pi2_wf,
product-deq_wf,
deq-fset_wf,
strong-subtype-set2,
face_lattice-deq_wf,
irr_face_wf,
subtype_rel_product,
top_wf,
istype-assert,
face_lattice-fset-join-eq-1,
fl-morph_wf,
deq_wf,
fset-image-compose,
member-fset-image-iff,
subset-cubical-term,
lattice-le_wf,
lattice-fset-join-is-lub,
bdd-distributive-lattice-subtype-bdd-lattice,
cube_set_restriction_pair_lemma,
lattice-hom-le,
cubical-subset_functionality_wrt_le,
discrete-cubical-term-is-constant-on-irr-face,
fset-subtype,
names-subtype,
cubical-type_wf,
face_lattice-hom-fixes-sublattice,
fl-morph-fl0,
fl0_wf,
dM-to-FL_wf,
neg-dM_inc,
dM-to-FL-opp,
fl-morph-fl1,
fl1_wf,
dM-to-FL-inc,
cubical-term-at_wf,
nh-comp_wf,
name-morph-satisfies-comp,
face_lattice-hom-fixes-sublattice2,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
subtype_base_sq,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
because_Cache,
applyEquality,
instantiate,
cumulativity,
setIsType,
functionIsType,
intEquality,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
isectEquality,
productElimination,
setEquality,
independent_pairEquality,
productIsType,
equalityIstype,
applyLambdaEquality,
equalityElimination,
promote_hyp
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[T:Type]. (discrete-comp(G;T) \mmember{} G \mvdash{} CompOp(discr(T)))
Date html generated:
2020_05_20-PM-05_21_22
Last ObjectModification:
2020_04_10-PM-00_01_47
Theory : cubical!type!theory
Home
Index