Nuprl Lemma : csm-composition-exists
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀s:Delta j⟶ Gamma. (Gamma ⊢ CompOp(A) ⇒ Delta ⊢ CompOp((A)s))
Proof
Definitions occuring in Statement :
composition-op: Gamma ⊢ CompOp(A),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g
Lemmas referenced :
csm-composition_wf,
subtype_rel_self,
cube_set_map_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
rename,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
universeIsType,
inhabitedIsType
Latex:
\mforall{}[Gamma,Delta:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}].
\mforall{}s:Delta j{}\mrightarrow{} Gamma. (Gamma \mvdash{} CompOp(A) {}\mRightarrow{} Delta \mvdash{} CompOp((A)s))
Date html generated:
2020_05_20-PM-03_51_23
Last ObjectModification:
2020_04_09-PM-01_11_24
Theory : cubical!type!theory
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