Nuprl Lemma : subset-trans-iota-lemma
∀Gamma:j⊢. ∀I:fset(ℕ). ∀rho:Gamma(I). ∀phi:𝔽(I). ∀J:fset(ℕ). ∀f:J ⟶ I.
(<rho> o iota o subset-trans(I;J;f;phi) = <f(rho)> o iota ∈ J,(phi)<f> j⟶ Gamma)
Proof
Definitions occuring in Statement :
subset-trans: subset-trans(I;J;f;x),
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
fl-morph: <f>,
csm-comp: G o F,
context-map: <rho>,
cube_set_map: A ⟶ B,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
context-map: <rho>,
subset-iota: iota,
csm-comp: G o F,
subset-trans: subset-trans(I;J;f;x),
compose: f o g,
functor-arrow: arrow(F),
cube-set-restriction: f(s)
Lemmas referenced :
names-hom_wf,
I_cube_wf,
face-presheaf_wf2,
fset_wf,
nat_wf,
csm-equal2,
cubical-subset_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,
csm-comp_wf,
subset-trans_wf,
formal-cube_wf,
subset-iota_wf,
context-map_wf,
cube-set-restriction_wf,
cubical-subset-I_cube,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
inhabitedIsType,
because_Cache,
instantiate,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productEquality,
cumulativity,
isectEquality,
independent_isectElimination,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}I:fset(\mBbbN{}). \mforall{}rho:Gamma(I). \mforall{}phi:\mBbbF{}(I). \mforall{}J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I.
(<rho> o iota o subset-trans(I;J;f;phi) = <f(rho)> o iota)
Date html generated:
2020_05_20-PM-01_45_54
Last ObjectModification:
2020_04_03-PM-07_56_37
Theory : cubical!type!theory
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