Nuprl Lemma : term-to-path-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u,v:{X ⊢ _:A}].
∀a:{X.𝕀 ⊢ _:(A)p}
(∀[b:{X.𝕀 ⊢ _:(A)p}]. X ⊢ <>(a) = X ⊢ <>(b) ∈ {X ⊢ _:(Path_A u v)} supposing a = b ∈ {X.𝕀 ⊢ _:(A)p}) supposing
(X ⊢ (a)[0(𝕀)]=u:A and
X ⊢ (a)[1(𝕀)]=v:A)
Proof
Definitions occuring in Statement :
term-to-path: <>(a),
path-type: (Path_A a b),
same-cubical-term: X ⊢ u=v:A,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
same-cubical-term: X ⊢ u=v:A
Lemmas referenced :
term-to-path-wf,
same-cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-id-adjoin_wf-interval-0,
subset-cubical-term2,
sub_cubical_set_self,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
csm-id-adjoin_wf-interval-1,
cubical-term_wf,
cubical-type_wf,
cubical_set_wf,
term-to-path_wf,
squash_wf,
true_wf,
path-type_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
independent_isectElimination,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
applyEquality,
because_Cache,
Error :memTop,
equalityIstype,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u,v:\{X \mvdash{} \_:A\}].
\mforall{}a:\{X.\mBbbI{} \mvdash{} \_:(A)p\}
(\mforall{}[b:\{X.\mBbbI{} \mvdash{} \_:(A)p\}]. X \mvdash{} <>(a) = X \mvdash{} <>(b) supposing a = b) supposing
(X \mvdash{} (a)[0(\mBbbI{})]=u:A and
X \mvdash{} (a)[1(\mBbbI{})]=v:A)
Date html generated:
2020_05_20-PM-03_19_50
Last ObjectModification:
2020_04_07-PM-00_59_31
Theory : cubical!type!theory
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