Nuprl Lemma : quotient-valueall-type

[A:Type]. ∀[E:A ⟶ A ⟶ ℙ].  (valueall-type(a,b:A//E[a;b])) supposing (valueall-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] valueall-type: valueall-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 valueall-type: valueall-type(T) sq_stable: SqStable(P) implies:  Q 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: so_apply: x[s1;s2] squash: T so_lambda: λ2y.t[x; y] guard: {T} uiff: uiff(P;Q) and: P ∧ Q rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  sq_stable__has-value bool_wf has-value_wf_base is-exception_wf sqle_wf_base equal_wf equal-wf-base quotient_wf base_wf valueall-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 sqequalRule baseApply closedConclusion baseClosed hypothesisEquality hypothesis independent_functionElimination equalityTransitivity equalitySymmetry because_Cache lambdaFormation pointwiseFunctionality callbyvalueReduce isect_memberEquality unionElimination equalityElimination applyEquality divergentSqle sqleReflexivity dependent_functionElimination imageMemberEquality imageElimination cumulativity lambdaEquality functionExtensionality independent_isectElimination axiomSqleEquality functionEquality universeEquality productElimination

\mforall{}[A:Type].  \mforall{}[E:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].
    (valueall-type(a,b:A//E[a;b]))  supposing  (valueall-type(A)  and  EquivRel(A;a,b.E[a;b]))

Date html generated: 2017_04_14-AM-07_39_38
Last ObjectModification: 2017_02_27-PM-03_11_22

Theory : quot_1

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