Nuprl Lemma : singleton-contraction_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[a,b:{X ⊢ _:T}]. ∀[pth:{X ⊢ _:(Path_T a b)}].
(singleton-contraction(X;pth) ∈ {X ⊢ _:(Path_Σ T (Path_(T)p (a)p q) cubical-pair(a;refl(a)) cubical-pair(b;pth))})
Proof
Definitions occuring in Statement :
singleton-contraction: singleton-contraction(X;pth),
cubical-refl: refl(a),
path-type: (Path_A a b),
cubical-pair: cubical-pair(u;v),
cubical-sigma: Σ A B,
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 ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
singleton-contraction: singleton-contraction(X;pth),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
path-point: path-point(pth),
cubical-type: {X ⊢ _},
cc-fst: p,
csm-ap-type: (AF)s,
interval-type: 𝕀,
csm-id: 1(X),
csm-ap: (s)x,
cc-snd: q,
csm-comp: G o F,
constant-cubical-type: (X),
compose: f o g,
csm-adjoin: (s;u),
pi1: fst(t),
csm-ap-term: (t)s,
same-cubical-term: X ⊢ u=v:A
Lemmas referenced :
cubical-refl_wf,
cubical-term_wf,
squash_wf,
true_wf,
equal_wf,
istype-universe,
cubical-type_wf,
cubical_set_cumulativity-i-j,
path-type_wf,
cubical-type-cumulativity2,
csm-path-type,
cube-context-adjoin_wf,
csm-id-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cc-snd_wf,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
csm-id_wf,
csm-ap-id-term,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-type,
term-to-path-wf,
cubical-sigma_wf,
cubical-pair_wf,
cubical_set_wf,
path-contraction_wf,
path-point_wf,
interval-type_wf,
csm-cubical-sigma,
csm-adjoin_wf,
csm-comp_wf,
csm-ap-term-snd-adjoin,
csm-cubical-pair,
cubical-path-ap-id-adjoin,
cubical-path-app-1,
path-contraction-1,
cubical-path-app_wf,
interval-1_wf,
path-type-q-id-adjoin,
cubical-path-app-0,
path-contraction-0,
interval-0_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
because_Cache,
hypothesis,
universeIsType,
equalityTransitivity,
equalitySymmetry,
universeEquality,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :memTop,
hyp_replacement,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
setElimination,
rename
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:T\}]. \mforall{}[pth:\{X \mvdash{} \_:(Path\_T a b)\}].
(singleton-contraction(X;pth) \mmember{} \{X \mvdash{} \_:(Path\_\mSigma{} T (Path\_(T)p (a)p q) cubical-pair(a;refl(a))
cubical-pair(b;pth))\})
Date html generated:
2020_05_20-PM-03_29_03
Last ObjectModification:
2020_04_07-PM-05_36_34
Theory : cubical!type!theory
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