### Nuprl Lemma : member-values-for-distinct

`∀[A,V:Type].`
`  ∀eq:EqDecider(A). ∀L:(A × V) List. ∀a:A.`
`    ((a ∈ map(λp.(fst(p));L)) `` (∃v:V. ((v ∈ values-for-distinct(eq;L)) ∧ (<a, v> ∈ L))))`

Proof

Definitions occuring in Statement :  values-for-distinct: `values-for-distinct(eq;L)` l_member: `(x ∈ l)` map: `map(f;as)` list: `T List` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` pi1: `fst(t)` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` lambda: `λx.A[x]` pair: `<a, b>` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` prop: `ℙ` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` uimplies: `b supposing a` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}`
Lemmas referenced :  and_wf iff_weakening_equal true_wf squash_wf select_member int_formula_prop_eq_lemma int_formula_prop_less_lemma intformeq_wf intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties values-for-distinct_wf select_wf lelt_wf values-for-distinct-property deq_wf list_wf l_member_wf pi1_wf map_wf member-remove-repeats
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin because_Cache dependent_functionElimination hypothesisEquality productEquality lambdaEquality sqequalRule hypothesis productElimination independent_functionElimination universeEquality setElimination rename dependent_set_memberEquality independent_pairFormation equalityTransitivity equalitySymmetry dependent_pairFormation cumulativity independent_isectElimination natural_numberEquality unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll applyEquality imageElimination independent_pairEquality imageMemberEquality baseClosed

Latex:
\mforall{}[A,V:Type].
\mforall{}eq:EqDecider(A).  \mforall{}L:(A  \mtimes{}  V)  List.  \mforall{}a:A.
((a  \mmember{}  map(\mlambda{}p.(fst(p));L))  {}\mRightarrow{}  (\mexists{}v:V.  ((v  \mmember{}  values-for-distinct(eq;L))  \mwedge{}  (<a,  v>  \mmember{}  L))))

Date html generated: 2016_05_14-PM-03_28_01
Last ObjectModification: 2016_01_14-PM-11_21_40

Theory : decidable!equality

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