### Nuprl Lemma : div-search-lemma

`∀a:ℤ. ∀b:{a + 1...}. ∀f:ℤ ⟶ 𝔹.`
`  ∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))] `
`  supposing ∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))]`

Proof

Definitions occuring in Statement :  int_upper: `{i...}` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  lelt: `i ≤ j < k` sq_stable: `SqStable(P)` squash: `↓T` guard: `{T}` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` cand: `A c∧ B` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` true: `True` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` int_upper: `{i...}` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` int_seg: `{i..j-}` so_apply: `x[s]` sq_exists: `∃x:A [B[x]]`
Lemmas referenced :  sq_stable__and sq_stable__all sq_stable__not sq_stable_from_decidable decidable__assert assert_witness istype-assert int_seg_properties int_upper_properties full-omega-unsat intformnot_wf intformle_wf itermVar_wf int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformand_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_less_lemma itermAdd_wf itermConstant_wf int_term_value_add_lemma int_term_value_constant_lemma divide-and-conquer isect_wf istype-false istype-le member-less_than istype-less_than upper_subtype_upper decidable__le not-le-2 condition-implies-le minus-add istype-void minus-one-mul add-swap minus-one-mul-top add-mul-special zero-mul add-zero add-associates add-commutes le-add-cancel sq_exists_wf int_seg_wf all_wf not_wf assert_wf bool_wf istype-int_upper istype-int
Rules used in proof :  Error :inrFormation_alt,  promote_hyp Error :inlFormation_alt,  Error :isectIsType,  Error :dependent_set_memberFormation_alt,  Error :functionIsTypeImplies,  imageMemberEquality baseClosed imageElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality dependent_functionElimination Error :dependent_set_memberEquality_alt,  independent_pairFormation independent_functionElimination productElimination independent_pairEquality independent_isectElimination Error :productIsType,  unionElimination voidElimination Error :isect_memberEquality_alt,  minusEquality multiplyEquality sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  Error :universeIsType,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality setElimination rename hypothesis sqequalRule Error :lambdaEquality_alt,  productEquality addEquality because_Cache closedConclusion natural_numberEquality applyEquality Error :functionIsType,  Error :inhabitedIsType

Latex:
\mforall{}a:\mBbbZ{}.  \mforall{}b:\{a  +  1...\}.  \mforall{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbB{}.
\mexists{}x:\{a..b\msupminus{}\}  [((\mforall{}y:\{a..x  +  1\msupminus{}\}.  (\mneg{}\muparrow{}(f  y)))  \mwedge{}  (\mforall{}z:\{x  +  1..b  +  1\msupminus{}\}.  (\muparrow{}(f  z))))]
supposing  \mexists{}x:\{a..b\msupminus{}\}  [((\mforall{}y:\{a..x  +  1\msupminus{}\}.  (\mneg{}\muparrow{}(f  y)))  \mwedge{}  (\mforall{}z:\{x  +  1..b  +  1\msupminus{}\}.  (\muparrow{}(f  z))))]

Date html generated: 2019_06_20-PM-02_12_35
Last ObjectModification: 2019_06_20-PM-02_08_48

Theory : int_2

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