Nuprl Lemma : cc-fst-csm-adjoin
∀[Gamma,Delta:CubicalSet]. ∀[A:{Gamma ⊢ _}]. ∀[sigma:Delta ⟶ Gamma]. ∀[u:{Delta ⊢ _:(A)sigma}].
(p o (sigma;u) = sigma ∈ Delta ⟶ Gamma)
Proof
Definitions occuring in Statement :
csm-adjoin: (s;u),
cc-fst: p,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:AF},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cube-set-map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-comp: cat-comp(C),
compose: f o g,
trans-comp: t1 o t2,
csm-ap: (s)x,
pi1: fst(t),
pi2: snd(t),
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
functor-ob: ob(F),
csm-comp: G o F,
csm-adjoin: (s;u),
cc-fst: p,
cubical-set: CubicalSet,
and: P ∧ Q,
list: T List,
name-cat: NameCat,
cat-ob: cat-ob(C),
subtype_rel: A ⊆r B,
functor-arrow: arrow(F),
all: ∀x:A. B[x]
Lemmas referenced :
cubical-term_wf,
csm-ap-type_wf,
cube-set-map_wf,
cubical-type_wf,
cubical-set_wf,
list_wf,
coordinate_name_wf,
name-morph_wf,
compose_wf,
subtype_rel_self,
cat-ob_wf,
name-cat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
lemma_by_obid,
functionExtensionality,
productElimination,
applyEquality,
functionIsType,
because_Cache,
equalityIsType1
Latex:
\mforall{}[Gamma,Delta:CubicalSet]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[sigma:Delta {}\mrightarrow{} Gamma]. \mforall{}[u:\{Delta \mvdash{} \_:(A)sigma\}].
(p o (sigma;u) = sigma)
Date html generated:
2019_11_05-PM-00_26_12
Last ObjectModification:
2018_11_10-PM-03_06_01
Theory : cubical!sets
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