Nuprl Lemma : csm-canonical-section-face-type
∀[I,K:fset(ℕ)]. ∀[f:K ⟶ I]. ∀[phi:𝔽(I)].
(canonical-section(();𝔽;K;⋅;f(phi)) = (canonical-section(();𝔽;I;⋅;phi))<f> ∈ {formal-cube(K) ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-type: 𝔽,
csm-ap-term: (t)s,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term: {X ⊢ _:A},
face-presheaf: 𝔽,
context-map: <rho>,
trivial-cube-set: (),
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
unit: Unit,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
trivial-cube-set: (),
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,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
squash: ↓T,
true: True,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
formal-cube: formal-cube(I),
uimplies: b supposing a,
canonical-section: canonical-section(Gamma;A;I;rho;a),
csm-ap-term: (t)s,
context-map: <rho>,
csm-ap: (s)x,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-term-equal2,
formal-cube_wf,
face-type_wf,
csm-face-type,
cubical-term_wf,
canonical-section_wf,
trivial-cube-set_wf,
it_wf,
subtype_rel_self,
I_cube_wf,
cube-set-restriction_wf,
face-presheaf_wf,
cubical-type-at_wf_face-type,
csm-ap-term_wf,
csm-ap-type_wf,
context-map_wf,
I_cube_pair_redex_lemma,
names-hom_wf,
fset_wf,
nat_wf,
cube_set_restriction_pair_lemma,
face-type-ap-morph,
arrow_pair_lemma,
face-type-at,
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,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
fl-morph_wf,
nh-comp_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
squash_wf,
true_wf,
fl-morph-comp2,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
independent_isectElimination,
lambdaFormation,
rename,
axiomEquality,
productEquality,
cumulativity,
setElimination,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[I,K:fset(\mBbbN{})]. \mforall{}[f:K {}\mrightarrow{} I]. \mforall{}[phi:\mBbbF{}(I)].
(canonical-section(();\mBbbF{};K;\mcdot{};f(phi)) = (canonical-section(();\mBbbF{};I;\mcdot{};phi))<f>)
Date html generated:
2018_05_23-AM-09_24_26
Last ObjectModification:
2018_05_20-PM-06_23_18
Theory : cubical!type!theory
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