Nuprl Lemma : cubical-fun-as-cubical-pi
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ((X ⊢ A ⟶ B) = X ⊢ ΠA (B)p ∈ {X ⊢ _})
Proof
Definitions occuring in Statement :
cubical-fun: (A ⟶ B),
cubical-pi: ΠA B,
cc-fst: p,
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],
member: t ∈ T,
cubical_set: CubicalSet,
cubical-fun: (A ⟶ B),
presheaf-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
cube-cat: CubeCat,
all: ∀x:A. B[x],
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),
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),
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-ap: (s)x,
pscm-ap: (s)x,
cc-fst: p,
psc-fst: p,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cube-context-adjoin: X.A,
psc-adjoin: X.A,
I_cube: A(I),
I_set: A(I)
Lemmas referenced :
presheaf-fun-as-presheaf-pi,
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,
isectElimination,
thin,
hypothesis,
sqequalRule,
Error :memTop,
dependent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A,B:\{X \mvdash{} \_\}]. ((X \mvdash{} A {}\mrightarrow{} B) = X \mvdash{} \mPi{}A (B)p)
Date html generated:
2020_05_20-PM-02_24_31
Last ObjectModification:
2020_04_03-PM-08_34_44
Theory : cubical!type!theory
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