Nuprl Lemma : composition-structure-cumulativity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. (Gamma +⊢ Compositon(A) ⊆r Gamma ⊢ Compositon(A))
Proof
Definitions occuring in Statement :
composition-structure: Gamma ⊢ Compositon(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
composition-structure: Gamma ⊢ Compositon(A),
all: ∀x:A. B[x],
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
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,
prop: ℙ
Lemmas referenced :
composition-function-cumulativity,
cubical_set_cumulativity-i-j,
subtype_rel_self,
cube_set_map_wf,
cube-context-adjoin_wf,
interval-type_wf,
uniform-comp-function_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
applyEquality,
extract_by_obid,
dependent_functionElimination,
hypothesis,
sqequalRule,
lambdaFormation_alt,
instantiate,
isectElimination,
because_Cache,
universeIsType,
inhabitedIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. (Gamma +\mvdash{} Compositon(A) \msubseteq{}r Gamma \mvdash{} Compositon(A))
Date html generated:
2020_05_20-PM-04_22_29
Last ObjectModification:
2020_04_14-AM-01_26_01
Theory : cubical!type!theory
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