Nuprl Lemma : cubical-term-at-comp-is-1
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[K:fset(ℕ)]. ∀[g:K ⟶ J].
((phi(f(rho)) = 1 ∈ Point(face_lattice(J))) ⇒ (phi(f ⋅ g(rho)) = 1 ∈ Point(face_lattice(K))))
Proof
Definitions occuring in Statement :
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
face_lattice: face_lattice(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
lattice-1: 1,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
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,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
compose: f o g,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term-at-morph,
face-type_wf,
subtype_rel_self,
iff_weakening_equal,
lattice-point_wf,
face_lattice_wf,
nh-comp_wf,
lattice-1_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-term-at_wf,
cube-set-restriction_wf,
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical_set_wf,
face-type-ap-morph,
fl-morph-comp,
fl-morph_wf,
fl-morph-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
equalityIstype,
productEquality,
cumulativity,
isectEquality,
setElimination,
rename,
inhabitedIsType,
Error :memTop
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
\mforall{}[K:fset(\mBbbN{})]. \mforall{}[g:K {}\mrightarrow{} J].
((phi(f(rho)) = 1) {}\mRightarrow{} (phi(f \mcdot{} g(rho)) = 1))
Date html generated:
2020_05_20-PM-02_53_32
Last ObjectModification:
2020_04_04-PM-05_07_57
Theory : cubical!type!theory
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