### Nuprl Lemma : union-deq_wf

`∀[A,B:Type]. ∀[a:EqDecider(A)]. ∀[b:EqDecider(B)].  (union-deq(A;B;a;b) ∈ EqDecider(A + B))`

Proof

Definitions occuring in Statement :  union-deq: `union-deq(A;B;a;b)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` member: `t ∈ T` union: `left + right` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` union-deq: `union-deq(A;B;a;b)` deq: `EqDecider(T)` all: `∀x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` and: `P ∧ Q` prop: `ℙ` sq_stable: `SqStable(P)` squash: `↓T` sumdeq: `sumdeq(a;b)` eqof: `eqof(d)` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` uimplies: `b supposing a` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` false: `False` sq_type: `SQType(T)` guard: `{T}` true: `True`
Lemmas referenced :  sumdeq_wf all_wf iff_wf equal_wf assert_wf deq_wf squash_wf sq_stable__all sq_stable__iff sq_stable__equal sq_stable_from_decidable decidable__assert assert_witness safe-assert-deq subtype_base_sq int_subtype_base false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut dependent_set_memberEquality extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality hypothesis lambdaFormation because_Cache unionEquality sqequalRule lambdaEquality applyEquality functionExtensionality axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality universeEquality setElimination rename independent_functionElimination dependent_functionElimination productElimination independent_pairEquality imageMemberEquality baseClosed imageElimination unionElimination independent_pairFormation independent_isectElimination applyLambdaEquality inlEquality natural_numberEquality instantiate intEquality voidElimination promote_hyp inrEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[a:EqDecider(A)].  \mforall{}[b:EqDecider(B)].    (union-deq(A;B;a;b)  \mmember{}  EqDecider(A  +  B))

Date html generated: 2017_04_14-AM-07_39_20
Last ObjectModification: 2017_02_27-PM-03_10_59

Theory : equality!deciders

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