### Nuprl Lemma : decidable__exists_iseg

`∀[T:Type]. ∀[P:(T List) ⟶ ℙ].  ((∀L:T List. Dec(P[L])) `` (∀L:T List. Dec(∃L':T List. (L' ≤ L ∧ P[L']))))`

Proof

Definitions occuring in Statement :  iseg: `l1 ≤ l2` list: `T List` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` sq_type: `SQType(T)` nat: `ℕ` ge: `i ≥ j ` less_than: `a < b` squash: `↓T` cons: `[a / b]` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` int_iseg: `{i...j}` cand: `A c∧ B` btrue: `tt` iseg: `l1 ≤ l2` less_than': `less_than'(a;b)` true: `True` last: `last(L)` subtract: `n - m` length: `||as||` list_ind: list_ind nil: `[]` it: `⋅`
Lemmas referenced :  int_seg_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf istype-int int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf int_seg_wf decidable__equal_int subtract_wf subtype_base_sq set_subtype_base lelt_wf int_subtype_base intformnot_wf intformeq_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma decidable__le decidable__lt istype-le istype-less_than subtype_rel_self non_neg_length nat_properties length_wf decidable__assert null_wf list-cases product_subtype_list null_cons_lemma last-lemma-sq pos_length iff_transitivity not_wf equal-wf-T-base list_wf assert_wf bnot_wf iff_weakening_uiff assert_of_null istype-assert nil_wf length_of_nil_lemma istype-void assert_of_bnot firstn_wf length_firstn le_wf decidable_wf iseg_wf primrec-wf2 itermAdd_wf int_term_value_add_lemma istype-nat length_wf_nat istype-universe iseg_weakening null_nil_lemma iseg_nil last_wf cons_wf append_wf iseg_append append_back_nil iseg_append_iff iseg_single length_of_cons_lemma equal_wf list_extensionality length-append select_wf squash_wf true_wf select_append_front select_firstn iff_weakening_equal length_firstn_eq less_than_wf add_functionality_wrt_eq select_append_back nat_wf select-nthtl subtype_rel_list top_wf nth_tl_decomp nth_tl_is_nil minus-add minus-minus add-associates minus-one-mul add-swap add-mul-special add-commutes zero-add zero-mul select0 hd_wf istype-false
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination setElimination rename productElimination hypothesis hypothesisEquality natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination Error :memTop,  sqequalRule independent_pairFormation universeIsType voidElimination unionElimination applyEquality instantiate cumulativity intEquality inhabitedIsType equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality_alt because_Cache productIsType promote_hyp hypothesis_subsumption imageElimination baseClosed functionIsType equalityIstype functionEquality productEquality setIsType addEquality universeEquality inrFormation_alt inlFormation_alt hyp_replacement imageMemberEquality multiplyEquality

Latex:
\mforall{}[T:Type].  \mforall{}[P:(T  List)  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}L:T  List.  Dec(P[L]))  {}\mRightarrow{}  (\mforall{}L:T  List.  Dec(\mexists{}L':T  List.  (L'  \mleq{}  L  \mwedge{}  P[L']))))

Date html generated: 2020_05_19-PM-09_48_58
Last ObjectModification: 2019_12_31-PM-00_13_48

Theory : list_1

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