### 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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