Nuprl Lemma : csm-cubical-equiv-by-cases
∀G:j⊢. ∀A,B:{G ⊢ _}. ∀f:{G ⊢ _:Equiv(A;B)}. ∀H:j⊢. ∀s:H j⟶ G.
((cubical-equiv-by-cases(G;B;f))s+
= cubical-equiv-by-cases(H;(B)s;(f)s)
∈ {H.𝕀, ((q=0) ∨ (q=1)) ⊢ _:Equiv((if (q=0) then ((A)s)p else ((B)s)p);((B)s)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,
csm+: tau+,
cc-snd: q,
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 ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-equiv-by-cases: cubical-equiv-by-cases(G;B;f),
squash: ↓T,
prop: ℙ,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
guard: {T},
true: True,
csm+: tau+,
csm-comp: G o F,
csm-ap-term: (t)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
pi2: snd(t),
cubical-type: {X ⊢ _},
pi1: fst(t),
compose: f o g,
same-cubical-type: Gamma ⊢ A = B,
implies: P ⇒ Q,
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,
csm-case-term,
cube_set_map_wf,
cubical-equiv_wf,
cubical-type_wf,
cubical_set_wf,
case-term_wf2,
squash_wf,
true_wf,
face-or_wf,
face-term-implies-subset,
face-term-implies-or1,
subset-cubical-term,
sub_cubical_set_functionality2,
context-subset-subtype-or2,
constrained-cubical-term-eqcd,
csm-face-zero,
q-csm+,
face-zero_wf,
cc-snd_wf,
face-one_wf,
case-type_wf,
same-cubical-type-zero-and-one,
face-0_wf,
csm-cubical-id-equiv,
csm-ap-term_wf,
csm+_wf_interval,
csm-face-type,
context-subset-map,
csm-face-one,
case-type-same2,
subset-cubical-term2,
sub_cubical_set_self,
thin-context-subset,
face-term-implies-or2,
cubical-equiv-subset,
csm-cubical-equiv,
cubical-term-eqcd,
case-type-same1,
face-1_wf,
empty-context-subset-lemma3',
empty-context-subset-lemma5
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
Error :memTop,
universeIsType,
inhabitedIsType,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
setElimination,
rename,
productElimination,
equalityElimination,
equalityIstype,
independent_functionElimination,
cumulativity,
universeEquality,
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)\}. \mforall{}H:j\mvdash{}. \mforall{}s:H j{}\mrightarrow{} G.
((cubical-equiv-by-cases(G;B;f))s+ = cubical-equiv-by-cases(H;(B)s;(f)s))
Date html generated:
2020_05_20-PM-07_26_02
Last ObjectModification:
2020_04_28-PM-04_24_35
Theory : cubical!type!theory
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