Nuprl Lemma : quotient-dep-function-subtype
∀[X:Type]
∀[A:X ⟶ Type]. ∀[E:x:X ⟶ A[x] ⟶ A[x] ⟶ ℙ].
((∀x:X. EquivRel(A[x];a,b.E[x;a;b]))
⇒ ((x:X ⟶ (a,b:A[x]//E[x;a;b])) ⊆r (f,g:x:X ⟶ A[x]//(∀x:X. (↓E[x;f x;g x])))))
supposing X ⊆r Base
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
prop: ℙ,
subtype_rel: A ⊆r B,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
squash: ↓T,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
guard: {T},
bfalse: ff,
ext-eq: A ≡ B,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
isect2: T1 ⋂ T2,
so_lambda: λ2x.t[x],
true: True,
quotient: x,y:A//B[x; y]
Lemmas referenced :
quotient-squash,
equiv_rel_squash,
equiv_rel_wf,
subtype_rel_wf,
base_wf,
istype-universe,
squash_wf,
quotient_wf,
subtype_rel_weakening,
isect2_wf,
subtype_rel_transitivity,
isect2_subtype_rel,
quotient-isect-base,
void_wf,
all_wf,
quotient-member-eq,
member_wf,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
Error :universeIsType,
because_Cache,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
Error :functionIsType,
axiomEquality,
Error :functionIsTypeImplies,
universeEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
closedConclusion,
baseApply,
Error :equalityIsType1,
equalityElimination,
unionElimination,
isect_memberEquality,
Error :functionExtensionality_alt,
voidElimination,
functionExtensionality,
functionEquality,
natural_numberEquality,
applyLambdaEquality,
pertypeElimination,
promote_hyp,
Error :productIsType,
sqequalBase
Latex:
\mforall{}[X:Type]
\mforall{}[A:X {}\mrightarrow{} Type]. \mforall{}[E:x:X {}\mrightarrow{} A[x] {}\mrightarrow{} A[x] {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:X. EquivRel(A[x];a,b.E[x;a;b]))
{}\mRightarrow{} ((x:X {}\mrightarrow{} (a,b:A[x]//E[x;a;b])) \msubseteq{}r (f,g:x:X {}\mrightarrow{} A[x]//(\mforall{}x:X. (\mdownarrow{}E[x;f x;g x])))))
supposing X \msubseteq{}r Base
Date html generated:
2019_06_20-PM-00_32_33
Last ObjectModification:
2018_11_26-AM-00_13_30
Theory : quot_1
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