### Nuprl Lemma : split_tail_correct

`∀[A:Type]. ∀[f:A ⟶ 𝔹]. ∀[L:A List].  (∀b∈snd(split_tail(L | ∀x.f[x])).↑f[b])`

Proof

Definitions occuring in Statement :  split_tail: `split_tail(L | ∀x.f[x])` l_all: `(∀x∈L.P[x])` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` pi2: `snd(t)` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` l_all: `(∀x∈L.P[x])` guard: `{T}` or: `P ∨ Q` split_tail: `split_tail(L | ∀x.f[x])` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` pi2: `snd(t)` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` so_apply: `x[s]` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` lelt: `i ≤ j < k` decidable: `Dec(P)` less_than: `a < b` squash: `↓T` colength: `colength(L)` nil: `[]` it: `⋅` sq_type: `SQType(T)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` cand: `A c∧ B` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf assert_witness intformeq_wf int_formula_prop_eq_lemma list-cases list_ind_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf select_wf int_seg_properties length_wf split_tail_wf istype-universe decidable__le intformnot_wf int_formula_prop_not_lemma decidable__lt subtract-1-ge-0 subtype_base_sq set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma list_ind_cons_lemma nat_wf list_wf bool_wf l_member_wf btrue_neq_bfalse member-implies-null-eq-bfalse btrue_wf null_nil_lemma assert_wf nil_wf l_all_iff l_all_cons l_all_wf list_ind_wf cons_wf equal-wf-T-base bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut thin lambdaFormation_alt extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType equalityTransitivity equalitySymmetry applyLambdaEquality functionIsTypeImplies inhabitedIsType unionElimination promote_hyp hypothesis_subsumption productElimination equalityIsType1 because_Cache dependent_set_memberEquality_alt applyEquality imageElimination instantiate equalityIsType4 baseApply closedConclusion baseClosed intEquality functionIsType universeEquality lambdaFormation setEquality cumulativity functionExtensionality lambdaEquality voidEquality setIsType productEquality independent_pairEquality productIsType equalityElimination

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:A  List].    (\mforall{}b\mmember{}snd(split\_tail(L  |  \mforall{}x.f[x])).\muparrow{}f[b])

Date html generated: 2019_10_15-AM-10_54_49
Last ObjectModification: 2018_10_09-AM-10_28_13

Theory : list!

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