Nuprl Lemma : face_comp_wf
∀[G:j⊢]. (face_comp() ∈ G ⊢ CompOp(𝔽))
Proof
Definitions occuring in Statement :
face_comp: face_comp(),
composition-op: Gamma ⊢ CompOp(A),
face-type: 𝔽,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
face_comp: face_comp(),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
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;a1),
bdd-distributive-lattice: BoundedDistributiveLattice,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
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-subset: I,psi,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
cat-arrow: cat-arrow(C),
cube-cat: CubeCat,
pi2: snd(t),
names-hom: I ⟶ J,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cube-set-restriction: f(s),
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,
respects-equality: respects-equality(S;T),
composition-op: Gamma ⊢ CompOp(A),
composition-uniformity: composition-uniformity(Gamma;A;comp),
face-type: 𝔽,
constant-cubical-type: (X),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
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,
fset: fset(T),
quotient: x,y:A//B[x; y],
type-cat: TypeCat,
name-morph-satisfies: (psi f) = 1,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
cat-comp: cat-comp(C),
compose: f o g,
functor-arrow: arrow(F),
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
subset-trans: subset-trans(I;J;f;x),
csm-ap-term: (t)s,
csm-ap: (s)x,
cubical-term: {X ⊢ _:A}
Lemmas referenced :
cubical_set_wf,
csm-face-type,
cubical-path-0_wf,
face-type_wf,
cubical-term_wf,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_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,
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,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
face-type-at,
face-type-ap-morph,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
nc-1_wf,
extend-face-term_wf,
cube_set_restriction_pair_lemma,
I_cube_pair_redex_lemma,
cat_arrow_triple_lemma,
nh-comp_wf,
subtype_rel_self,
names-hom_wf,
name-morph-satisfies_wf,
name-morph-satisfies-comp,
squash_wf,
true_wf,
istype-universe,
nh-comp-assoc,
iff_weakening_equal,
nh-id-right,
s-comp-nc-1,
fl-morph-comp2,
fl-morph_wf,
cubical-term-at_wf,
extend-face-term-property,
cubical-subset-I_cube-member,
respects-equality_weakening,
composition-uniformity_wf,
istype-nat,
istype-void,
cubical_type_ap_morph_pair_lemma,
cubical-type_wf,
nc-e'_wf,
nc-e'-lemma3,
csm-ap-term_wf,
subset-trans_wf,
cube_set_map_wf,
subtype_rel-equal,
extend-face-term-morph,
nh-id_wf,
nh-id-left,
cubical-term-equal,
nc-e'-lemma1,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
instantiate,
introduction,
extract_by_obid,
hypothesis,
functionExtensionality,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
Error :memTop,
hypothesisEquality,
applyEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
setEquality,
intEquality,
productEquality,
cumulativity,
isectEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
lambdaFormation_alt,
inhabitedIsType,
productElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
equalityIstype,
functionIsType,
setIsType
Latex:
\mforall{}[G:j\mvdash{}]. (face\_comp() \mmember{} G \mvdash{} CompOp(\mBbbF{}))
Date html generated:
2020_05_20-PM-05_24_05
Last ObjectModification:
2020_04_10-PM-01_23_12
Theory : cubical!type!theory
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