Nuprl Lemma : universe-comp-op_wf
∀[X:j⊢]. ∀[t:{X ⊢ _:c𝕌}]. (compOp(t) ∈ X ⊢ CompOp(decode(t)))
Proof
Definitions occuring in Statement :
universe-comp-op: compOp(t),
universe-decode: decode(t),
cubical-universe: c𝕌,
composition-op: Gamma ⊢ CompOp(A),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-op: Gamma ⊢ CompOp(A),
prop: ℙ,
all: ∀x:A. B[x],
universe-comp-op: compOp(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,
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
csm-ap-type: (AF)s,
cubical-term-at: u(a),
subset-iota: iota,
csm-comp: G o F,
universe-decode: decode(t),
csm-ap: (s)x,
compose: f o g,
universe-type: universe-type(t;I;a),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
formal-cube: formal-cube(I),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
I_cube: A(I),
functor-ob: ob(F),
names-hom: I ⟶ J,
cubical-universe: c𝕌,
closed-cubical-universe: cc𝕌,
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
closed-type-to-type: closed-type-to-type(T),
context-map: <rho>,
functor-arrow: arrow(F),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
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,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
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;a1),
composition-uniformity: composition-uniformity(Gamma;A;comp),
cubical-type-at: A(a),
fibrant-type: FibrantType(X),
label: ...$L... t,
cubical-type: {X ⊢ _},
csm-id: 1(X),
subset-trans: subset-trans(I;J;f;x),
cube-set-restriction: f(s),
name-morph-satisfies: (psi f) = 1,
csm-composition: (comp)sigma
Lemmas referenced :
composition-uniformity_wf,
universe-decode_wf,
istype-cubical-universe-term,
cubical_set_wf,
cubical-term-at_wf,
cubical-universe_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,
cubical-universe-at,
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,
face-presheaf_wf2,
cubical-term-eqcd,
cubical-subset_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
csm-universe-decode,
context-map-1,
cubical-type_wf,
formal-cube_wf1,
universe-type_wf,
cubical-type-subtype-cubical-subset,
equal_wf,
csm-ap-id-type,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
csm-ap-type_wf,
cube_set_map_wf,
csm-subtype-cubical-subset,
subtype_rel_self,
universe-decode-type,
cubical-type-at_wf,
I_cube_pair_redex_lemma,
nh-id_wf,
subtype_rel_universe1,
cubical-universe-cumulativity,
nc-0_wf,
universe-decode-restriction,
cubical-subset-I_cube,
universe-type-at,
names-hom_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cube-set-restriction-comp,
nh-comp_wf,
formal-cube-restriction,
nh-id-right,
cubical_type_ap_morph_pair_lemma,
cubical-term-at-morph,
pi2_wf,
composition-op_wf,
pi1_wf_top,
context-map_wf,
csm-composition_wf,
istype-cubical-type-at,
cube_set_restriction_pair_lemma,
subtype_rel-equal,
csm-ap-type-at,
s-comp-if-lemma1,
csm-ap_wf,
cubical-type-ap-morph_wf,
arrow_pair_lemma,
nh-id-left,
csm-cubical-type-ap-morph,
nh-comp-assoc,
csm-comp_wf,
subset-iota_wf,
name-morph-satisfies-comp,
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,
s-comp-nc-0,
nc-1_wf,
s-comp-nc-1,
istype-cubical-term,
equal_functionality_wrt_subtype_rel2,
cubical-type-cumulativity,
istype-top,
subtype_rel_product,
top_wf,
cubical_type_at_pair_lemma,
nc-e'_wf,
cubical-path-0_wf,
cubical-term_wf,
fl-morph-restriction,
nc-e'-lemma3,
fl-morph_wf,
subset-trans_wf,
csm-ap-term_wf,
cubical-path-0-ap-morph,
csm-ap-context-map,
context-map_wf_cubical-subset,
cubical-path-condition_wf,
nc-e'-lemma2,
trivial-equal,
nc-e'-lemma1,
cube-set-restriction-id,
member_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
universeIsType,
dependent_functionElimination,
instantiate,
lambdaEquality_alt,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
voidElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
setIsType,
functionIsType,
applyEquality,
intEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
hyp_replacement,
functionEquality,
cumulativity,
universeEquality,
setEquality,
independent_pairEquality,
dependent_pairEquality_alt,
productIsType,
closedConclusion,
productEquality,
isectEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[t:\{X \mvdash{} \_:c\mBbbU{}\}]. (compOp(t) \mmember{} X \mvdash{} CompOp(decode(t)))
Date html generated:
2020_05_20-PM-07_15_46
Last ObjectModification:
2020_04_27-PM-01_32_47
Theory : cubical!type!theory
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