Nuprl Lemma : subset-trans_wf
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[psi:𝔽(I)]. (subset-trans(I;J;f;psi) ∈ J,(psi)<f> j⟶ I,psi)
Proof
Definitions occuring in Statement :
subset-trans: subset-trans(I;J;f;x),
cubical-subset: I,psi,
face-presheaf: 𝔽,
fl-morph: <f>,
cube_set_map: A ⟶ B,
I_cube: A(I),
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subset-trans: subset-trans(I;J;f;x),
cube_set_map: A ⟶ B,
cube-cat: CubeCat,
psc_map: A ⟶ B,
type-cat: TypeCat,
op-cat: op-cat(C),
nat-trans: nat-trans(C;D;F;G),
spreadn: spread4,
all: ∀x:A. B[x],
functor-arrow: arrow(F),
functor-ob: ob(F),
cubical-subset: I,psi,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
pi1: fst(t),
pi2: snd(t),
cat-comp: cat-comp(C),
compose: f o g,
cat-arrow: cat-arrow(C),
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
I_cube: A(I),
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,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
name-morph-satisfies: (psi f) = 1,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
face-presheaf_wf,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_ob_pair_lemma,
I_cube_wf,
small_cubical_set_subtype,
names-hom_wf,
fset_wf,
nat_wf,
name-morph-satisfies_wf,
fl-morph_wf,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
nh-comp_wf,
lattice-1_wf,
squash_wf,
true_wf,
istype-universe,
fl-morph-comp,
iff_weakening_equal,
nh-comp-assoc,
fl-morph-1
Rules used in proof :
cut,
instantiate,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
dependent_set_memberEquality_alt,
universeIsType,
isectElimination,
applyEquality,
hypothesisEquality,
because_Cache,
lambdaEquality_alt,
lambdaFormation_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
isectEquality,
independent_isectElimination,
setIsType,
equalityIstype,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
functionExtensionality,
setEquality,
functionIsType
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[psi:\mBbbF{}(I)]. (subset-trans(I;J;f;psi) \mmember{} J,(psi)<f> j{}\mrightarrow{} I,psi)
Date html generated:
2020_05_20-PM-01_45_46
Last ObjectModification:
2020_04_03-PM-07_16_14
Theory : cubical!type!theory
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