Nuprl Lemma : cubical-subset_functionality_wrt_le
∀J:fset(ℕ). ∀a,b:Point(face_lattice(J)). (a ≤ b ⇒ sub_cubical_set{j:l}(J,a; J,b))
Proof
Definitions occuring in Statement :
cubical-subset: I,psi,
face_lattice: face_lattice(I),
sub_cubical_set: Y ⊆ X,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
lattice-le: a ≤ b,
lattice-point: Point(l)
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
true: True,
unit: Unit,
trivial-cube-set: (),
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term-at: u(a),
formal-cube: formal-cube(I),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
uiff: uiff(P;Q)
Lemmas referenced :
lattice-le_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-point_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
fset_wf,
nat_wf,
subtype_rel_self,
I_cube_wf,
face-presheaf_wf2,
face-term-implies-subset,
formal-cube_wf1,
canonical-section_wf,
trivial-cube-set_wf,
face-type_wf,
it_wf,
cubical-type-at_wf_face-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
context-map_wf,
csm-face-type,
sub_cubical_set_wf,
squash_wf,
true_wf,
cubical_set_wf,
cubical-subset-is-context-subset-canonical,
iff_weakening_equal,
I_cube_pair_redex_lemma,
fl-morph_wf,
lattice-1_wf,
names-hom_wf,
face-type-ap-morph,
lattice-hom-le,
bdd-distributive-lattice-subtype-bdd-lattice,
lattice-1-le-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
independent_isectElimination,
inhabitedIsType,
natural_numberEquality,
Error :memTop,
dependent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
equalityIstype,
setElimination,
rename
Latex:
\mforall{}J:fset(\mBbbN{}). \mforall{}a,b:Point(face\_lattice(J)). (a \mleq{} b {}\mRightarrow{} sub\_cubical\_set\{j:l\}(J,a; J,b))
Date html generated:
2020_05_20-PM-02_52_23
Last ObjectModification:
2020_04_19-PM-07_00_47
Theory : cubical!type!theory
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