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]))


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: supposing a uall: [x:A]. B[x] prop: 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: supposing a value-type: value-type(T) sq_stable: SqStable(P) implies:  Q so_lambda: λ2y.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 then else fi  bfalse: ff subtype_rel: A ⊆B prop: 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

\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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