Nuprl Lemma : composition-function-cumulativity
∀Gamma:j⊢. ∀Z:{Gamma ⊢ _}. (composition-function{[i | j]:l, i:l}(Gamma; Z) ⊆r composition-function{j:l,i:l}(Gamma;Z))
Proof
Definitions occuring in Statement :
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
composition-function: composition-function{j:l,i:l}(Gamma;A),
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,
uall: ∀[x:A]. B[x]
Lemmas referenced :
cubical_set_cumulativity-i-j,
subtype_rel_self,
cube_set_map_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_wf,
composition-function_wf,
cubical-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
functionExtensionality,
applyEquality,
hypothesisEquality,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
isectElimination,
because_Cache,
universeIsType
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}Z:\{Gamma \mvdash{} \_\}.
(composition-function\{[i | j]:l, i:l\}(Gamma; Z) \msubseteq{}r composition-function\{j:l,i:l\}(Gamma;Z))
Date html generated:
2020_05_20-PM-04_21_05
Last ObjectModification:
2020_04_14-AM-01_23_55
Theory : cubical!type!theory
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