Nuprl Lemma : case-term-0'
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u:Top]. ∀[v,x:{Gamma ⊢ _:A}].
(Gamma ⊢ (u ∨ v)=x:A) supposing ((x = v ∈ {Gamma ⊢ _:A}) and (phi = 0(𝔽) ∈ {Gamma ⊢ _:𝔽}))
Proof
Definitions occuring in Statement :
case-term: (u ∨ v),
same-cubical-term: X ⊢ u=v:A,
face-0: 0(𝔽),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
same-cubical-term: X ⊢ u=v:A,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
case-term-0,
same-cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
equal_wf,
iff_weakening_equal,
face-0_wf,
cubical-term_wf,
istype-top,
cubical-type_wf,
face-type_wf,
cubical_set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
because_Cache,
sqequalRule,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
axiomEquality,
equalityIstype,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[u:Top]. \mforall{}[v,x:\{Gamma \mvdash{} \_:A\}].
(Gamma \mvdash{} (u \mvee{} v)=x:A) supposing ((x = v) and (phi = 0(\mBbbF{})))
Date html generated:
2020_05_20-PM-04_15_03
Last ObjectModification:
2020_04_10-PM-03_46_40
Theory : cubical!type!theory
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