Nuprl Lemma : cubical-fun-equal2
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[f:{X ⊢ _:(A ⟶ B)}].
∀[g:I:fset(ℕ) ⟶ a:X(I) ⟶ J:fset(ℕ) ⟶ h:J ⟶ I ⟶ u:A(h(a)) ⟶ B(h(a))].
f = g ∈ {X ⊢ _:(A ⟶ B)}
supposing ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[h:J ⟶ I]. ∀[u:A(h(a))]. ((f(a) J h u) = (g(a) J h u) ∈ B(h(a)))
Proof
Definitions occuring in Statement :
cubical-fun: (A ⟶ B),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: 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),
I_cube: A(I),
I_set: A(I),
cubical-term-at: u(a),
presheaf-term-at: u(a)
Lemmas referenced :
presheaf-fun-equal2,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cubical-term-sq-presheaf-term
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{} \_\}]. \mforall{}[f:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}]. \mforall{}[g:I:fset(\mBbbN{})
{}\mrightarrow{} a:X(I)
{}\mrightarrow{} J:fset(\mBbbN{})
{}\mrightarrow{} h:J {}\mrightarrow{} I
{}\mrightarrow{} u:A(h(a))
{}\mrightarrow{} B(h(a))].
f = g
supposing \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[h:J {}\mrightarrow{} I]. \mforall{}[u:A(h(a))].
((f(a) J h u) = (g(a) J h u))
Date html generated:
2020_05_20-PM-02_23_52
Last ObjectModification:
2020_04_03-PM-08_34_14
Theory : cubical!type!theory
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