Nuprl Lemma : composition-op-universe-sq
∀[G:Top]
(G ⊢ CompOp(c𝕌) ~ {comp:I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:G(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:c𝕌}
⟶ {a0:FibrantType(formal-cube(I))|
∀J:fset(ℕ). ∀f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;a0)
= u((i0) ⋅ f)
∈ FibrantType(formal-cube(J)))}
⟶ {a1:FibrantType(formal-cube(I))|
∀J:fset(ℕ). ∀f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;a1)
= u((i1) ⋅ f)
∈ FibrantType(formal-cube(J)))} |
∀I,J:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀j:{j:ℕ| ¬j ∈ J} . ∀g:J ⟶ I. ∀rho:G(I+i). ∀phi:𝔽(I).
∀u:{I+i,s(phi) ⊢ _:c𝕌}. ∀a0:{a0:FibrantType(formal-cube(I))|
∀J:fset(ℕ). ∀f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>;a0)
= u((i0) ⋅ f)
∈ FibrantType(formal-cube(J)))} .
(csm-fibrant-type(formal-cube(I);formal-cube(J);<g>;comp I i rho phi u a0)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi))
csm-fibrant-type(formal-cube(I);formal-cube(J);<g>;a0))
∈ FibrantType(formal-cube(J)))} )
Proof
Definitions occuring in Statement :
cubical-universe: c𝕌,
csm-fibrant-type: csm-fibrant-type(G;H;s;FT),
fibrant-type: FibrantType(X),
composition-op: Gamma ⊢ CompOp(A),
csm-ap-term: (t)s,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
subset-trans: subset-trans(I;J;f;x),
cubical-subset: I,psi,
face-presheaf: 𝔽,
context-map: <rho>,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
nc-e': g,i=j,
nc-1: (i1),
nc-0: (i0),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
composition-op: Gamma ⊢ CompOp(A),
composition-uniformity: composition-uniformity(Gamma;A;comp),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
member: t ∈ T,
fibrant-type: FibrantType(X)
Lemmas referenced :
csm-cubical-universe,
cubical-universe-at,
istype-top,
cubical-universe-ap-morph
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
because_Cache
Latex:
\mforall{}[G:Top]
(G \mvdash{} CompOp(c\mBbbU{}) \msim{} \{comp:I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:G(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} u:\{I+i,s(phi) \mvdash{} \_:c\mBbbU{}\}
{}\mrightarrow{} \{a0:FibrantType(formal-cube(I))|
\mforall{}J:fset(\mBbbN{}). \mforall{}f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>a0) = u((i0) \mcdot{} f))\}
{}\mrightarrow{} \{a1:FibrantType(formal-cube(I))|
\mforall{}J:fset(\mBbbN{}). \mforall{}f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>a1) = u((i1) \mcdot{} f))\} |\000C
\mforall{}I,J:fset(\mBbbN{}). \mforall{}i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} . \mforall{}j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} . \mforall{}g:J {}\mrightarrow{} I. \mforall{}rho:G(I+i).
\mforall{}phi:\mBbbF{}(I). \mforall{}u:\{I+i,s(phi) \mvdash{} \_:c\mBbbU{}\}.
\mforall{}a0:\{a0:FibrantType(formal-cube(I))|
\mforall{}J:fset(\mBbbN{}). \mforall{}f:I,phi(J).
(csm-fibrant-type(formal-cube(I);formal-cube(J);<f>a0) = u((i0) \mcdot{} f))\} \000C.
(csm-fibrant-type(formal-cube(I);formal-cube(J);<g>comp I i rho phi u a0)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi))
csm-fibrant-type(formal-cube(I);formal-cube(J);<g>a0)))\} )
Date html generated:
2020_05_20-PM-07_24_34
Last ObjectModification:
2020_04_25-PM-10_04_19
Theory : cubical!type!theory
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