Nuprl Lemma : cubical-type-restriction-and
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi,phi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ].
(cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
⇒ cubical-type-restriction(X;T;I,a,t.phi[I;a;t])
⇒ cubical-type-restriction(X;T;I,a,t.psi[I;a;t] ∧ phi[I;a;t]))
Proof
Definitions occuring in Statement :
cubical-type-restriction: cubical-type-restriction,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
cubical-type-restriction: cubical-type-restriction,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
so_apply: x[s1;s2;s3],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: so_lambda3,
guard: {T}
Lemmas referenced :
istype-cubical-type-at,
I_cube_wf,
names-hom_wf,
cubical-type-restriction_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
productIsType,
universeIsType,
applyEquality,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
instantiate,
lambdaEquality_alt,
cumulativity,
inhabitedIsType,
functionIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[psi,phi:I:fset(\mBbbN{}) {}\mrightarrow{} alpha:X(I) {}\mrightarrow{} T(alpha) {}\mrightarrow{} \mBbbP{}].
(cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
{}\mRightarrow{} cubical-type-restriction(X;T;I,a,t.phi[I;a;t])
{}\mRightarrow{} cubical-type-restriction(X;T;I,a,t.psi[I;a;t] \mwedge{} phi[I;a;t]))
Date html generated:
2020_05_20-PM-03_13_11
Last ObjectModification:
2020_04_06-PM-05_17_17
Theory : cubical!type!theory
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