Nuprl Lemma : discrete-map-is-constant2
∀[T:Type]. ∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[s:I,phi ⟶ discrete-cube(T)]. ∀[f:{f:I ⟶ I| (phi f) = 1} ].
s = (λJ,g. (s I f)) ∈ I,phi ⟶ discrete-cube(T)
supposing ∀[J:fset(ℕ)]. ∀[g:J ⟶ I]. ((phi g) = 1 ⇒ (g = f ⋅ g ∈ J ⟶ I))
Proof
Definitions occuring in Statement :
cubical-subset: I,psi,
name-morph-satisfies: (psi f) = 1,
face-presheaf: 𝔽,
cube_set_map: A ⟶ B,
discrete-cube: discrete-cube(A),
I_cube: A(I),
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
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],
member: t ∈ T,
functor-arrow: arrow(F),
functor-ob: ob(F),
cubical-subset: I,psi,
discrete-cube: discrete-cube(A),
pi1: fst(t),
pi2: snd(t),
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
cat-comp: cat-comp(C),
compose: f o g,
cat-arrow: cat-arrow(C),
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
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,
prop: ℙ,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cat_ob_pair_lemma,
names-hom_wf,
name-morph-satisfies_wf,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
nh-comp_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,
cube_set_map_wf,
cubical-subset_wf,
discrete-cube_wf,
I_cube_wf,
face-presheaf_wf2,
fset_wf,
nat_wf,
istype-universe,
squash_wf,
true_wf,
implies-nh-comp-satisfies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality_alt,
isectIsType,
because_Cache,
universeIsType,
isectElimination,
hypothesisEquality,
functionIsType,
applyEquality,
equalityIstype,
setIsType,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
independent_isectElimination,
universeEquality,
functionExtensionality,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setEquality,
independent_functionElimination,
lambdaFormation_alt,
inhabitedIsType
Latex:
\mforall{}[T:Type]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:\mBbbF{}(I)]. \mforall{}[s:I,phi {}\mrightarrow{} discrete-cube(T)]. \mforall{}[f:\{f:I {}\mrightarrow{} I| (phi f) = 1\} ].
s = (\mlambda{}J,g. (s I f)) supposing \mforall{}[J:fset(\mBbbN{})]. \mforall{}[g:J {}\mrightarrow{} I]. ((phi g) = 1 {}\mRightarrow{} (g = f \mcdot{} g))
Date html generated:
2020_05_20-PM-02_32_05
Last ObjectModification:
2020_04_04-AM-09_47_48
Theory : cubical!type!theory
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