Nuprl Lemma : quotient-value-type
∀[A:Type]. ∀[E:A ⟶ A ⟶ ℙ]. (value-type(a,b:A//E[a;b])) supposing (value-type(A) and EquivRel(A;a,b.E[a;b]))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
value-type: value-type(T),
sq_stable: SqStable(P),
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
has-value: (a)↓,
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
sq_stable__has-value,
quotient_wf,
bool_wf,
value-type-has-value,
has-value_wf_base,
is-exception_wf,
sqle_wf_base,
equal_wf,
equal-wf-base,
base_wf,
value-type_wf,
equiv_rel_wf,
isect2_wf,
isect2_subtype_rel,
subtype_rel_functionality_wrt_iff,
quotient-isect-base,
ext-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
lambdaFormation,
pointwiseFunctionality,
callbyvalueReduce,
isect_memberEquality,
unionElimination,
equalityElimination,
because_Cache,
divergentSqle,
sqleReflexivity,
baseClosed,
dependent_functionElimination,
imageMemberEquality,
imageElimination,
axiomSqleEquality,
functionEquality,
universeEquality,
productElimination
Latex:
\mforall{}[A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(value-type(a,b:A//E[a;b])) supposing (value-type(A) and EquivRel(A;a,b.E[a;b]))
Date html generated:
2017_04_14-AM-07_39_36
Last ObjectModification:
2017_02_27-PM-03_11_17
Theory : quot_1
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