Nuprl Lemma : empty-context-subset-lemma5
∀[Gamma:j⊢]. ∀[i:{Gamma ⊢ _:𝕀}]. ∀[x,y:Top × Top]. (x = y ∈ {Gamma, (i=0), (i=1) ⊢ _})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
interval-type: 𝕀,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
top: Top,
product: x:A × B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
context-subset: Gamma, phi,
interval-presheaf: 𝕀,
cubical-type: {X ⊢ _},
guard: {T},
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
istype-top,
cubical-term_wf,
interval-type_wf,
cubical_set_wf,
I_cube_wf,
context-subset_wf,
face-zero_wf,
face-one_wf,
subset-cubical-term,
context-subset-is-subset,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
face-zero-eq-1,
face-one-eq-1,
interval-type-at,
dM-0-not-1,
names-hom_wf,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
nh-comp_wf,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
hypothesis,
because_Cache,
productIsType,
introduction,
extract_by_obid,
universeIsType,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaFormation_alt,
independent_functionElimination,
voidElimination,
applyEquality,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
productElimination,
dependent_pairEquality_alt,
functionIsType,
functionExtensionality,
independent_pairFormation,
equalityIstype,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[i:\{Gamma \mvdash{} \_:\mBbbI{}\}]. \mforall{}[x,y:Top \mtimes{} Top]. (x = y)
Date html generated:
2020_05_20-PM-04_13_40
Last ObjectModification:
2020_04_10-AM-04_41_35
Theory : cubical!type!theory
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