Nuprl Lemma : uniform-comp-function_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:composition-function{j:l,i:l}(Gamma;A)].
(uniform-comp-function{j:l, i:l}(Gamma; A; comp) ∈ ℙ{[i' | j'']})
Proof
Definitions occuring in Statement :
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
csm-id-adjoin: [u],
csm-id: 1(X),
composition-function: composition-function{j:l,i:l}(Gamma;A),
implies: P ⇒ Q,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-1: 1(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-comp: G o F,
csm-ap: (s)x,
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-term: (t)s,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
interval-0: 0(𝕀)
Lemmas referenced :
composition-function_wf,
cubical_set_wf,
cube_set_map_wf,
cube-context-adjoin_wf,
interval-type_wf,
face-type_wf,
cubical-term-eqcd,
thin-context-subset-adjoin,
csm-ap-type_wf,
context-subset_wf,
constrained-cubical-term_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
csm-ap-term_wf,
equal_wf,
csm-id-adjoin_wf-interval-1,
csm-face-type,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
context-subset-map,
cubical-type_wf,
csm-constrained-cubical-term,
subtype_rel_self,
csm-context-subset-subtype3,
cubical-term_wf,
squash_wf,
true_wf,
subtype_rel_wf,
istype-universe,
subset-cubical-type,
context-subset-is-subset,
equal_functionality_wrt_subtype_rel2,
iff_weakening_equal,
subset-cubical-term,
sub_cubical_set_self,
csm-comp_wf,
csm+_wf_interval,
csm-ap-comp-type-sq,
interval-0_wf,
sub_cubical_set_functionality,
csm-id-adjoin-subset
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
functionEquality,
cumulativity,
instantiate,
because_Cache,
applyEquality,
lambdaEquality_alt,
universeIsType,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:composition-function\{j:l,i:l\}(Gamma;A)].
(uniform-comp-function\{j:l, i:l\}(Gamma; A; comp) \mmember{} \mBbbP{}\{[i' | j'']\})
Date html generated:
2020_05_20-PM-04_21_37
Last ObjectModification:
2020_04_19-PM-01_54_50
Theory : cubical!type!theory
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