Nuprl Lemma : compU-wf-lemma1
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}].
(λE,A. encode(Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))] decode(A);
cfun-to-cop(G;Glue [phi ⊢→ (decode((E)[1(𝕀)]);equiv-fun(equivU(G, phi;(decode(E))-;(compOp(E))-)))]
decode(A)
;comp(Glue [phi ⊢→ (decode((E)[1(𝕀)]), equivU(G, phi;(decode(E))-;(compOp(E))-))] decode(A)) ))
∈ u:{G, phi.𝕀 ⊢ _:c𝕌} ⟶ a0:{G ⊢ _:c𝕌[phi |⟶ (u)[0(𝕀)]]} ⟶ {G ⊢ _:c𝕌[phi |⟶ (u)[1(𝕀)]]})
Proof
Definitions occuring in Statement :
equivU: equivU(G;E;cE),
universe-comp-op: compOp(t),
universe-decode: decode(t),
universe-encode: encode(T;cT),
cubical-universe: c𝕌,
glue-comp: comp(Glue [phi ⊢→ (T, f)] A) ,
glue-type: Glue [phi ⊢→ (T;w)] A,
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
comp-op-to-comp-fun: cop-to-cfun(cA),
rev-type-line-comp: (cA)-,
rev-type-line: (A)-,
csm-composition: (comp)sigma,
equiv-fun: equiv-fun(f),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
same-cubical-type: Gamma ⊢ A = B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
comp-fun-to-comp-op: cfun-to-cop(Gamma;A;comp),
comp-fun-to-comp-op1: comp-fun-to-comp-op1(Gamma;A;comp),
csm-comp: G o F,
context-map: <rho>,
context-subset: Gamma, phi,
functor-arrow: arrow(F),
pi2: snd(t),
cube-set-restriction: f(s),
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-type: {X ⊢ _},
universe-encode: encode(T;cT),
guard: {T},
and: P ∧ Q
Lemmas referenced :
cubical-term_wf,
cube-context-adjoin_wf,
context-subset_wf,
interval-type_wf,
cubical-universe_wf,
subtype_rel_universe1,
csm-ap-term_wf,
csm-id-adjoin_wf,
interval-0_wf,
csm-cubical-universe,
constrained-cubical-term_wf,
istype-cubical-term,
face-type_wf,
cubical_set_wf,
rev-type-line_wf,
universe-decode_wf,
equivU_wf,
rev-type-line-comp_wf,
universe-comp-op_wf,
interval-1_wf,
rev-type-line-0,
rev-type-line-1,
csm-universe-decode,
cubical-equiv_wf,
squash_wf,
true_wf,
cubical-type_wf,
cubical-term-eqcd,
equiv-fun_wf,
thin-context-subset,
glue-type_wf,
glue-type-constraint,
comp-op-to-comp-fun_wf2,
cubical_set_cumulativity-i-j,
csm-composition_wf,
glue-comp_wf2,
subset-cubical-term,
context-subset-is-subset,
universe-encode_wf,
comp-fun-to-comp-op_wf,
universe-encode-decode,
comp-op-to-comp-fun-inverse,
composition-op_wf,
cubical-type-cumulativity2,
equal_wf,
istype-universe,
cubical_type_ap_morph_pair_lemma,
composition-structure_wf,
subset-cubical-type,
sub_cubical_set_self,
glue-comp-agrees,
csm-universe-comp-op,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cubical-universe-at,
csm-ap-type_wf,
cube_set_map_wf,
formal-cube_wf1,
context-map-subset,
subtype_rel-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
because_Cache,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
universeIsType,
setElimination,
rename,
applyLambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
cumulativity,
universeEquality,
hyp_replacement,
dependent_functionElimination,
dependent_set_memberEquality_alt,
equalityIstype,
lambdaFormation_alt,
independent_functionElimination,
productElimination,
functionExtensionality,
dependent_pairEquality_alt,
independent_pairFormation,
productIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}].
(\mlambda{}E,A. encode(Glue [phi \mvdash{}\mrightarrow{} (decode((E)[1(\mBbbI{})]);equiv-fun(equivU(G, phi;(decode(E))-;
(compOp(E))-)))] decode(A);
cfun-to-cop(G;Glue [phi \mvdash{}\mrightarrow{} (decode((E)[1(\mBbbI{})]);equiv-fun(equivU(G, phi;(decode(E))-;
(compOp(E))-)))]
decode(A)
;comp(Glue [phi \mvdash{}\mrightarrow{} (decode((E)[1(\mBbbI{})]), equivU(G, phi;(decode(E))-;
(compOp(E))-))] decode(A)) ))
\mmember{} u:\{G, phi.\mBbbI{} \mvdash{} \_:c\mBbbU{}\} {}\mrightarrow{} a0:\{G \mvdash{} \_:c\mBbbU{}[phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\} {}\mrightarrow{} \{G \mvdash{} \_:c\mBbbU{}[phi |{}\mrightarrow{} (u)[1(\mBbbI{})]]\})
Date html generated:
2020_05_20-PM-07_22_22
Last ObjectModification:
2020_04_27-PM-05_53_24
Theory : cubical!type!theory
Home
Index