Nuprl Lemma : cubical-equiv-by-cases_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(A;B)}].
(cubical-equiv-by-cases(G;B;f) ∈ {G.𝕀, ((q=0) ∨ (q=1)) ⊢ _:Equiv((if (q=0) then (A)p else (B)p);(B)p)})
Proof
Definitions occuring in Statement :
cubical-equiv-by-cases: cubical-equiv-by-cases(G;B;f),
cubical-equiv: Equiv(T;A),
case-type: (if phi then A else B),
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
face-or: (a ∨ b),
interval-type: 𝕀,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
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],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-equiv-by-cases: cubical-equiv-by-cases(G;B;f),
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
guard: {T},
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
same-cubical-type: Gamma ⊢ A = B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
cubical-equiv: Equiv(T;A),
cubical-sigma: Σ A B
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
subset-cubical-type,
context-subset_wf,
context-subset-is-subset,
istype-cubical-term,
face-type_wf,
case-term_wf2,
face-zero_wf,
cc-snd_wf,
face-one_wf,
cubical-equiv_wf,
face-or_wf,
case-type_wf,
same-cubical-type-zero-and-one,
face-0_wf,
cubical-type_wf,
cubical_set_wf,
cubical-term-eqcd,
squash_wf,
true_wf,
thin-context-subset,
case-type-same2,
cubical-equiv-subset,
cubical-id-equiv_wf,
csm-ap-term_wf,
subset-cubical-term2,
csm-cubical-equiv,
case-type-same1,
face-1_wf,
empty-context-subset-lemma3',
empty-context-subset-lemma5,
subset-cubical-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
universeIsType,
inhabitedIsType,
equalityIstype,
independent_functionElimination,
rename,
lambdaEquality_alt,
cumulativity,
universeEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
Error :memTop,
hyp_replacement,
dependent_set_memberEquality_alt,
independent_pairEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(A;B)\}].
(cubical-equiv-by-cases(G;B;f)
\mmember{} \{G.\mBbbI{}, ((q=0) \mvee{} (q=1)) \mvdash{} \_:Equiv((if (q=0) then (A)p else (B)p);(B)p)\})
Date html generated:
2020_05_20-PM-07_25_40
Last ObjectModification:
2020_04_28-PM-02_19_02
Theory : cubical!type!theory
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