Nuprl Lemma : last_with_property

`∀[T:Type]`
`  ∀L:T List`
`    ∀[P:ℕ||L|| ⟶ ℙ]`
`      ((∀x:ℕ||L||. Dec(P x)) `` (∃i:ℕ||L||. (P i)) `` (∃i:ℕ||L||. ((P i) ∧ (∀j:ℕ||L||. ¬(P j) supposing i < j))))`

Proof

Definitions occuring in Statement :  length: `||as||` list: `T List` int_seg: `{i..j-}` less_than: `a < b` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` exists: `∃x:A. B[x]` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` cand: `A c∧ B` subtype_rel: `A ⊆r B` interleaving_occurence: `interleaving_occurence(T;L1;L2;L;f1;f2)` nat: `ℕ` ge: `i ≥ j ` sq_type: `SQType(T)`
Lemmas referenced :  interleaving_split exists_wf int_seg_wf length_wf all_wf decidable_wf list_wf int_seg_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_wf subtract_wf decidable__le itermSubtract_wf int_term_value_subtract_lemma lelt_wf subtype_rel_self less_than_wf isect_wf not_wf increasing_implies length_wf_nat nat_properties intformeq_wf int_formula_prop_eq_lemma subtype_base_sq int_subtype_base decidable__equal_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_functionElimination independent_functionElimination hypothesis productElimination natural_numberEquality sqequalRule lambdaEquality applyEquality functionEquality cumulativity universeEquality setElimination rename unionElimination imageElimination independent_isectElimination approximateComputation dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation functionExtensionality dependent_set_memberEquality because_Cache instantiate productEquality equalityTransitivity equalitySymmetry applyLambdaEquality

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List
\mforall{}[P:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||.  Dec(P  x))
{}\mRightarrow{}  (\mexists{}i:\mBbbN{}||L||.  (P  i))
{}\mRightarrow{}  (\mexists{}i:\mBbbN{}||L||.  ((P  i)  \mwedge{}  (\mforall{}j:\mBbbN{}||L||.  \mneg{}(P  j)  supposing  i  <  j))))

Date html generated: 2019_10_15-AM-10_57_23
Last ObjectModification: 2018_09_17-PM-06_30_33

Theory : list!

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