Nuprl Lemma : path-contraction-0
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}].
((path-contraction(X;pth))[0(𝕀)] = refl(a) ∈ {X ⊢ _:(Path_A a a)})
Proof
Definitions occuring in Statement :
path-contraction: path-contraction(X;pth),
cubical-refl: refl(a),
path-type: (Path_A a b),
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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-contraction: path-contraction(X;pth),
term-to-pathtype: <>a,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
cubical-refl: refl(a),
csm-ap-term: (t)s,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm+: tau+,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-comp: G o F,
pi1: fst(t),
compose: f o g,
pi2: snd(t),
interval-meet: r ∧ s,
cubical-term-at: u(a)
Lemmas referenced :
cubical-term_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
cubical_set_cumulativity-i-j,
csm-path-type,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
csm-ap-type_wf,
cc-fst_wf,
cubical-type-cumulativity2,
csm-ap-term_wf,
path-point_wf,
path-type_wf,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm_id_adjoin_fst_term_lemma,
csm-id_wf,
path-point-0,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-type,
csm-ap-id-term,
path-contraction_wf,
cubical_set_wf,
cubical-refl_wf,
path-type-subtype,
paths-equal,
csm-term-to-pathtype,
pathtype_wf,
csm-id-adjoin_wf,
interval-0_wf,
cubical-path-app_wf,
interval-meet_wf,
csm-interval-type,
cc-snd_wf,
path-type-p,
cube_set_map_wf,
term-to-pathtype_wf,
csm-cubical-path-app,
cubical-path-app-0,
csm-interval-meet,
interval-meet-comm,
interval-meet-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :memTop,
hyp_replacement,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
applyLambdaEquality,
lambdaFormation_alt,
equalityIstype
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[pth:\{X \mvdash{} \_:(Path\_A a b)\}].
((path-contraction(X;pth))[0(\mBbbI{})] = refl(a))
Date html generated:
2020_05_20-PM-03_28_35
Last ObjectModification:
2020_04_07-PM-05_38_59
Theory : cubical!type!theory
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