Nuprl Lemma : composition-function_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (composition-function{j:l,i:l}(Gamma;A) ∈ 𝕌{[i' | j'']})
Proof
Definitions occuring in Statement :
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
composition-function: composition-function{j:l,i:l}(Gamma;A),
member: t ∈ T,
subtype_rel: A ⊆r B,
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,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
guard: {T},
squash: ↓T,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
true: True,
interval-1: 1(𝕀)
Lemmas referenced :
cubical_set_wf,
cube_set_map_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical-term_wf,
face-type_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
context-subset_wf,
csm-context-subset-subtype3,
subtype_rel_self,
cubical-type-cumulativity2,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
csm-id-adjoin_wf,
interval-0_wf,
csm-context-subset-subtype2,
csm-ap-term_wf,
csm-id-adjoin_wf-interval-1,
interval-1_wf,
cubical-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
functionEquality,
cumulativity,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
lambdaEquality_alt,
universeIsType,
universeEquality,
sqequalRule,
because_Cache,
imageElimination,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. (composition-function\{j:l,i:l\}(Gamma;A) \mmember{} \mBbbU{}\{[i' | j'']\})
Date html generated:
2020_05_20-PM-04_20_51
Last ObjectModification:
2020_04_11-PM-04_57_27
Theory : cubical!type!theory
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