### Nuprl Lemma : can-find-first2

`∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹.  ((∃x:T [first-member(T;x;L;P)]) ∨ (∀x∈L.¬↑(P x)))`

Proof

Definitions occuring in Statement :  first-member: `first-member(T;x;L;P)` l_all: `(∀x∈L.P[x])` l_member: `(x ∈ l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` not: `¬A` or: `P ∨ Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_exists: `∃x:A [B[x]]` first-member: `first-member(T;x;L;P)` exists: `∃x:A. B[x]` and: `P ∧ Q` cand: `A c∧ B` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` l_member: `(x ∈ l)` le: `A ≤ B` less_than': `less_than'(a;b)` nat: `ℕ` ge: `i ≥ j `
Lemmas referenced :  can-find-first1-ext l_member_wf list-subtype bool_wf list_wf l_all_wf2 not_wf assert_wf equal_functionality_wrt_subtype_rel2 equal_wf select_wf int_seg_properties length_wf 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 all_wf int_seg_wf exists_wf int_seg_subtype_nat false_wf less_than_wf nat_properties sq_exists_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin setEquality cumulativity hypothesisEquality hypothesis dependent_functionElimination because_Cache equalityTransitivity equalitySymmetry rename functionEquality universeEquality unionElimination inlEquality sqequalRule lambdaEquality applyEquality functionExtensionality setElimination dependent_set_memberEquality inrEquality productElimination dependent_pairFormation independent_isectElimination independent_functionElimination independent_pairFormation productEquality natural_numberEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageElimination

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.    ((\mexists{}x:T  [first-member(T;x;L;P)])  \mvee{}  (\mforall{}x\mmember{}L.\mneg{}\muparrow{}(P  x)))

Date html generated: 2018_05_21-PM-06_34_16
Last ObjectModification: 2017_07_26-PM-04_52_29

Theory : general

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