Nuprl Lemma : csm-ap-type_wf1
∀[Gamma,Delta:j⊢]. ∀[A:{Gamma ⊢j _}]. ∀[s:Delta j⟶ Gamma]. Delta ⊢j (A)s
Proof
Definitions occuring in Statement :
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B
Lemmas referenced :
csm-ap_wf,
I_cube_wf,
fset_wf,
nat_wf,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-ap-restriction,
subtype_rel_self,
iff_weakening_equal,
names-hom_wf,
nh-comp_wf,
cube-set-restriction-comp,
cube_set_map_wf,
nh-id_wf,
cube-set-restriction-id,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
extract_by_obid,
isectElimination,
hypothesis,
instantiate,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
because_Cache,
functionIsType,
lambdaFormation_alt,
independent_pairFormation,
inhabitedIsType,
productIsType,
equalityIstype,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma,Delta:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{}j \_\}]. \mforall{}[s:Delta j{}\mrightarrow{} Gamma]. Delta \mvdash{}j (A)s
Date html generated:
2020_05_20-PM-01_48_59
Last ObjectModification:
2020_04_03-PM-08_26_41
Theory : cubical!type!theory
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