Nuprl Lemma : cubical-pi-p
∀X:j⊢. ∀T,A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ((ΠA B)p = X.T ⊢ Π(A)p (B)(p o p;q) ∈ {X.T ⊢ _})
Proof
Definitions occuring in Statement :
cubical-pi: ΠA B,
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
cube-context-adjoin: X.A,
psc-adjoin: X.A,
I_cube: A(I),
I_set: A(I),
cubical-type-at: A(a),
presheaf-type-at: A(a),
cube-set-restriction: f(s),
psc-restriction: f(s),
cubical-type-ap-morph: (u a f),
presheaf-type-ap-morph: (u a f),
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
cubical-pi: ΠA B,
presheaf-pi: ΠA B,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
cube-cat: CubeCat,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
csm-ap: (s)x,
pscm-ap: (s)x,
cc-fst: p,
psc-fst: p,
csm-adjoin: (s;u),
pscm-adjoin: (s;u),
csm-comp: G o F,
pscm-comp: G o F,
cc-snd: q,
psc-snd: q
Lemmas referenced :
presheaf-pi-p,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
isectElimination,
Error :memTop
Latex:
\mforall{}X:j\mvdash{}. \mforall{}T,A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. ((\mPi{}A B)p = X.T \mvdash{} \mPi{}(A)p (B)(p o p;q))
Date html generated:
2020_05_20-PM-02_00_24
Last ObjectModification:
2020_04_03-PM-08_33_28
Theory : cubical!type!theory
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