Nuprl Lemma : case-endpoints_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a,b:{G ⊢ _:A}]. ∀[r:{G ⊢ _:𝕀}]. ([r=0 ⊢→ a; r=1 ⊢→ b] ∈ {G, ((r=0) ∨ (r=1)) ⊢ _:A})
Proof
Definitions occuring in Statement :
case-endpoints: [r=0 ⊢→ a; r=1 ⊢→ b],
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
face-or: (a ∨ b),
interval-type: 𝕀,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
case-endpoints: [r=0 ⊢→ a; r=1 ⊢→ b],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
same-cubical-term: X ⊢ u=v:A,
squash: ↓T,
prop: ℙ,
guard: {T},
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
case-term_wf,
face-zero_wf,
face-one_wf,
thin-context-subset,
face-or_wf,
context-subset-term-subtype,
empty-context-subset-lemma3,
subtype_rel-equal,
cubical-term_wf,
context-subset_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
face-0_wf,
face-type_wf,
face-zero-and-one,
face-and_wf,
subtype_rel_self,
iff_weakening_equal,
interval-type_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
sqequalRule,
independent_isectElimination,
Error :memTop,
instantiate,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[a,b:\{G \mvdash{} \_:A\}]. \mforall{}[r:\{G \mvdash{} \_:\mBbbI{}\}].
([r=0 \mvdash{}\mrightarrow{} a; r=1 \mvdash{}\mrightarrow{} b] \mmember{} \{G, ((r=0) \mvee{} (r=1)) \mvdash{} \_:A\})
Date html generated:
2020_05_20-PM-04_15_17
Last ObjectModification:
2020_04_10-AM-04_45_16
Theory : cubical!type!theory
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