Nuprl Lemma : term-to-path-app-snd
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ((<>((a)p))p @ q = (a)p ∈ {X.𝕀 ⊢ _:(A)p})
Proof
Definitions occuring in Statement :
term-to-path: <>(a),
cubical-path-app: pth @ r,
interval-type: 𝕀,
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-term-at: u(a),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cube-context-adjoin: X.A,
all: ∀x:A. B[x],
cc-fst: p,
csm-ap: (s)x,
pi1: fst(t),
term-to-path: <>(a),
cubical-path-app: pth @ r,
cubicalpath-app: pth @ r,
cubical-lambda: (λb),
cubical-app: app(w; u),
cc-snd: q,
cc-adjoin-cube: (v;u),
csm-ap-term: (t)s,
pi2: snd(t),
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-term-at,
I_cube_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
fset_wf,
nat_wf,
cubical-term-equal,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
cubical_type_at_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term-at_wf,
cube-set-restriction-id,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
functionExtensionality,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
instantiate,
hypothesisEquality,
applyEquality,
because_Cache,
equalityTransitivity,
independent_isectElimination,
universeIsType,
dependent_functionElimination,
productElimination,
setElimination,
rename,
lambdaEquality_alt,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. ((<>((a)p))p @ q = (a)p)
Date html generated:
2020_05_20-PM-03_19_27
Last ObjectModification:
2020_04_06-PM-06_35_39
Theory : cubical!type!theory
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