### Nuprl Lemma : deq-member-firstn

`∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[L:A List]. ∀[n:ℕ+].`
`  ∀[x:A]. (x ∈b firstn(n;L) ~ x ∈b firstn(n - 1;L) ∨b(eqof(eq) x L[n - 1])) supposing n - 1 < ||L||`

Proof

Definitions occuring in Statement :  firstn: `firstn(n;as)` select: `L[n]` length: `||as||` deq-member: `x ∈b L` list: `T List` eqof: `eqof(d)` deq: `EqDecider(T)` bor: `p ∨bq` nat_plus: `ℕ+` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` subtract: `n - m` natural_number: `\$n` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat_plus: `ℕ+` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` iff: `P `⇐⇒` Q` uiff: `uiff(P;Q)` rev_implies: `P `` Q` sq_type: `SQType(T)` guard: `{T}` subtype_rel: `A ⊆r B`
Lemmas referenced :  subtype_base_sq bool_wf bool_subtype_base iff_imp_equal_bool deq-member_wf firstn_wf bor_wf select_wf nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf iff_transitivity assert_wf or_wf l_member_wf equal_wf iff_weakening_uiff assert_of_bor subtract_wf assert-deq-member safe-assert-deq less_than_wf length_wf nat_plus_wf list_wf deq_wf firstn_decomp nat_plus_subtype_nat decidable__lt member_append cons_wf nil_wf member_singleton
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesis independent_isectElimination hypothesisEquality setElimination rename because_Cache applyEquality dependent_functionElimination unionElimination natural_numberEquality dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll lambdaFormation independent_functionElimination orFunctionality productElimination equalityTransitivity equalitySymmetry sqequalAxiom universeEquality addLevel promote_hyp

Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[L:A  List].  \mforall{}[n:\mBbbN{}\msupplus{}].
\mforall{}[x:A].  (x  \mmember{}\msubb{}  firstn(n;L)  \msim{}  x  \mmember{}\msubb{}  firstn(n  -  1;L)  \mvee{}\msubb{}(eqof(eq)  x  L[n  -  1]))  supposing  n  -  1  <  ||L||

Date html generated: 2017_09_29-PM-06_04_21
Last ObjectModification: 2017_07_26-PM-02_53_03

Theory : decidable!equality

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