Nuprl Lemma : partial-term-0_wf
∀[H:j⊢]. ∀[A:{H.𝕀 ⊢ _}]. ∀[phi:{H ⊢ _:𝔽}]. ∀[u:{H.𝕀, (phi)p ⊢ _:A}]. (u[0] ∈ {H, phi ⊢ _:(A)[0(𝕀)]})
Proof
Definitions occuring in Statement :
partial-term-0: u[0],
context-subset: Gamma, phi,
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
partial-term-0: u[0],
squash: ↓T,
prop: ℙ,
cc-fst: p,
csm-ap-term: (t)s,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
pi1: fst(t),
cubical-term: {X ⊢ _:A},
uimplies: b supposing a,
true: True,
all: ∀x:A. B[x],
cubical-term-at: u(a)
Lemmas referenced :
cubical-term_wf,
context-subset_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
thin-context-subset,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-0,
context-subset-map,
csm-id-adjoin_wf,
interval-0_wf,
squash_wf,
true_wf,
cubical-term-equal,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-at-morph1,
names-hom_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
setElimination,
rename,
independent_isectElimination,
functionExtensionality_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
inhabitedIsType,
functionIsType,
equalityIstype
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[A:\{H.\mBbbI{} \mvdash{} \_\}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}[u:\{H.\mBbbI{}, (phi)p \mvdash{} \_:A\}].
(u[0] \mmember{} \{H, phi \mvdash{} \_:(A)[0(\mBbbI{})]\})
Date html generated:
2020_05_20-PM-03_04_17
Last ObjectModification:
2020_04_06-PM-00_35_24
Theory : cubical!type!theory
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