Nuprl Lemma : csm-id-fiber-contraction
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}].
(id-fiber-contraction(K;(A)tau)
= (id-fiber-contraction(G;A))tau++
∈ {K.(A)tau.Σ ((A)tau)p (Path_(((A)tau)p)p (q)p q) ⊢ _
:(Path_(Σ ((A)tau)p (Path_(((A)tau)p)p (q)p q))p (id-fiber-center(K;(A)tau))p q)})
Proof
Definitions occuring in Statement :
id-fiber-contraction: id-fiber-contraction(X;T),
id-fiber-center: id-fiber-center(X;T),
path-type: (Path_A a b),
cubical-sigma: Σ A B,
csm+: tau+,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
pi1: fst(t),
compose: f o g,
csm-adjoin: (s;u),
csm-ap: (s)x,
csm-comp: G o F,
csm+: tau+,
csm-ap-type: (AF)s,
cc-fst: p,
cc-snd: q,
cubical-type: {X ⊢ _},
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
all: ∀x:A. B[x],
prop: ℙ,
cat-comp: cat-comp(C),
names-hom: I ⟶ J,
type-cat: TypeCat,
pi2: snd(t),
cat-arrow: cat-arrow(C),
quotient: x,y:A//B[x; y],
fset: fset(T),
cube-cat: CubeCat,
spreadn: spread4,
op-cat: op-cat(C),
cat-ob: cat-ob(C),
nat-trans: nat-trans(C;D;F;G),
psc_map: A ⟶ B,
cube_set_map: A ⟶ B,
cube-context-adjoin: X.A,
cc-adjoin-cube: (v;u),
cube-set-restriction: f(s),
I_cube: A(I),
ps_context: __⊢,
cubical_set: CubicalSet,
functor-ob: ob(F),
so_apply: x[s],
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
btrue: tt,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
free-dist-lattice: free-dist-lattice(T; eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dM: dM(I),
record-select: r.x,
lattice-point: Point(l),
interval-presheaf: 𝕀,
constant-cubical-type: (X),
interval-type: 𝕀,
cubical-type-at: A(a),
spreadn: spread3,
sigma-elim-csm: SigmaElim,
cubical-fst: p.1,
cubical-app: app(w; u),
csm-ap-term: (t)s,
cubicalpath-app: pth @ r,
cubical-lambda: (λb),
cubical-pair: cubical-pair(u;v),
cubical-path-app: pth @ r,
term-to-path: <>(a),
singleton-contraction: singleton-contraction(X;pth),
path-eta: path-eta(pth),
id-fiber-contraction: id-fiber-contraction(X;T),
cubical-snd: p.2,
path-contraction: path-contraction(X;pth),
cubical-term-at: u(a),
interval-meet: r ∧ s
Lemmas referenced :
paths-equal-eta,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical-sigma_wf,
cc-fst_wf,
path-type_wf,
csm-ap-term_wf,
cc-snd_wf,
id-fiber-center_wf,
id-fiber-contraction_wf,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
p-csm+-type,
csm-comp-type,
q-csm+,
csm-comp-term,
csm-comp_wf,
path-type-q-csm-adjoin,
iff_weakening_equal,
csm-cubical-sigma,
equal_wf,
csm+_wf,
subtype_rel-equal,
cube_set_map_cumulativity-i-j,
csm+_wf+,
csm-path-type,
istype-universe,
true_wf,
squash_wf,
subtype_rel_self,
cubical-term-eqcd,
path-type-sub-pathtype,
cubical-term_wf,
pathtype_wf,
cubical-type-cumulativity,
path-eta_wf,
cubical-sigma-p-p,
interval-type_wf,
cubical-term-equal,
cube_set_restriction_pair_lemma,
I_cube_pair_redex_lemma,
cubical-sigma-at,
path-type-at,
path-type-ap-morph,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
csm-cubical-type-ap-morph,
I_cube_wf,
cat-ob_wf,
small-category-cumulativity-2,
type-cat_wf,
cube-cat_wf,
op-cat_wf,
functor-ob_wf,
nh-id-left,
nh-id_wf,
nh-comp_wf,
cube-set-restriction-when-id,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
dM_wf,
lattice-point_wf,
cc_fst_adjoin_cube_lemma,
nat_wf,
fset_wf,
cc-fst_wf_interval,
cubical-fst_wf,
csm-adjoin_wf,
csm-adjoin-id-adjoin,
cubical-snd_wf,
subtype_rel_universe1,
equal_functionality_wrt_subtype_rel2,
sub_cubical_set_self,
subset-cubical-term2,
interval-type-at,
interval-type-ap-morph,
csm-ap-type-at,
csm-ap_wf,
cubical-type-at_wf,
dM-lift_wf2,
cubical-sigma-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
Error :memTop,
rename,
setElimination,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaEquality_alt,
dependent_functionElimination,
universeEquality,
hyp_replacement,
functionExtensionality,
functionEquality,
isectEquality,
cumulativity,
productEquality,
lambdaFormation_alt,
equalityIstype
Latex:
\mforall{}[G,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} G]. \mforall{}[A:\{G \mvdash{} \_\}].
(id-fiber-contraction(K;(A)tau) = (id-fiber-contraction(G;A))tau++)
Date html generated:
2020_05_20-PM-03_32_33
Last ObjectModification:
2020_05_01-PM-06_35_02
Theory : cubical!type!theory
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