### Nuprl Lemma : l_member-first

`∀[A:Type]. ∀d:A List. ∀x:A. ∀eq:EqDecider(A).  ((x ∈ d) `` (∃i:ℕ||d||. ((∀j:ℕi. (¬(d[j] = x ∈ A))) ∧ (d[i] = x ∈ A))))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` select: `L[n]` length: `||as||` list: `T List` deq: `EqDecider(T)` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` and: `P ∧ Q` int_seg: `{i..j-}` uimplies: `b supposing a` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` so_apply: `x[s]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` eqof: `eqof(d)` deq: `EqDecider(T)` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` le: `A ≤ B` less_than': `less_than'(a;b)` nat_plus: `ℕ+` true: `True` select: `L[n]` cons: `[a / b]` cand: `A c∧ B` ge: `i ≥ j ` iff: `P `⇐⇒` Q` subtract: `n - m` subtype_rel: `A ⊆r B` rev_implies: `P `` Q`
Lemmas referenced :  list_induction all_wf deq_wf l_member_wf exists_wf int_seg_wf length_wf not_wf equal_wf select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma list_wf null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse eqof_wf bool_wf eqtt_to_assert safe-assert-deq length_of_cons_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot cons_wf false_wf add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma lelt_wf non_neg_length cons_member add-member-int_seg2 subtract_wf itermSubtract_wf int_term_value_subtract_lemma select-cons-tl add-subtract-cancel decidable__equal_int int_subtype_base squash_wf true_wf select_cons_tl iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity because_Cache hypothesis functionEquality natural_numberEquality productEquality setElimination rename independent_isectElimination productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination independent_functionElimination equalityTransitivity equalitySymmetry applyEquality equalityElimination promote_hyp instantiate universeEquality dependent_set_memberEquality imageMemberEquality baseClosed applyLambdaEquality pointwiseFunctionality baseApply closedConclusion addEquality

Latex:
\mforall{}[A:Type]
\mforall{}d:A  List.  \mforall{}x:A.  \mforall{}eq:EqDecider(A).
((x  \mmember{}  d)  {}\mRightarrow{}  (\mexists{}i:\mBbbN{}||d||.  ((\mforall{}j:\mBbbN{}i.  (\mneg{}(d[j]  =  x)))  \mwedge{}  (d[i]  =  x))))

Date html generated: 2017_04_17-AM-09_15_43
Last ObjectModification: 2017_02_27-PM-05_21_32

Theory : decidable!equality

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