Nuprl Lemma : sub-cubical-set-and
∀[X:j⊢]. ∀[P,Q:I:fset(ℕ) ⟶ X(I) ⟶ ℙ].
X | I,rho.P[I;rho] | I,rho.Q[I;rho] ≡ X | I,rho.P[I;rho] ∧ Q[I;rho]
supposing cs-predicate(X;I,rho.P[I;rho]) ∧ cs-predicate(X;I,rho.Q[I;rho])
Proof
Definitions occuring in Statement :
sub-cubical-set: X | I,rho.P[I; rho],
cs-predicate: cs-predicate(X;I,rho.P[I; rho]),
I_cube: A(I),
ext-eq-cs: X ≡ Y,
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
cube-cat: CubeCat,
all: ∀x:A. B[x],
I_cube: A(I),
I_set: A(I),
cs-predicate: cs-predicate(X;I,rho.P[I; rho]),
ext-eq-cs: X ≡ Y,
sub-cubical-set: X | I,rho.P[I; rho],
sub-presheaf-set: X | I,rho.P[I; rho]
Lemmas referenced :
sub-presheaf-set-and,
cube-cat_wf,
cat_ob_pair_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule,
dependent_functionElimination,
Error :memTop
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[P,Q:I:fset(\mBbbN{}) {}\mrightarrow{} X(I) {}\mrightarrow{} \mBbbP{}].
X | I,rho.P[I;rho] | I,rho.Q[I;rho] \mequiv{} X | I,rho.P[I;rho] \mwedge{} Q[I;rho]
supposing cs-predicate(X;I,rho.P[I;rho]) \mwedge{} cs-predicate(X;I,rho.Q[I;rho])
Date html generated:
2020_05_20-PM-01_39_45
Last ObjectModification:
2020_04_03-PM-03_33_01
Theory : cubical!type!theory
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