### Nuprl Lemma : mu-ge-bound-property

`∀n,m:ℤ. ∀f:{n..m-} ⟶ 𝔹.  ((∃m:{n..m-}. (↑(f m))) `` {(↑(f mu-ge(f;n))) ∧ (∀[i:{n..mu-ge(f;n)-}]. (¬↑(f i)))})`

Proof

Definitions occuring in Statement :  mu-ge: `mu-ge(f;n)` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` guard: `{T}` 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]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` prop: `ℙ` nat: `ℕ` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` and: `P ∧ Q` guard: `{T}` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` cand: `A c∧ B` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` mu-ge: `mu-ge(f;n)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` label: `...\$L... t` rev_uimplies: `rev_uimplies(P;Q)` le: `A ≤ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` has-value: `(a)↓`
Lemmas referenced :  assert_wf int_seg_wf bool_wf istype-int mu-ge-bound nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf 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 subtract-1-ge-0 nat_wf int_seg_properties itermSubtract_wf int_term_value_subtract_lemma lelt_wf le_wf subtract_wf decidable__lt intformnot_wf int_formula_prop_not_lemma decidable__le eqtt_to_assert eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot assert_functionality_wrt_uiff decidable__equal_int intformeq_wf int_formula_prop_eq_lemma int_subtype_base itermAdd_wf int_term_value_add_lemma subtype_rel_function int_seg_subtype istype-false not-le-2 condition-implies-le minus-add minus-one-mul add-swap minus-one-mul-top add-mul-special zero-mul add-zero add-associates add-commutes le-add-cancel le_reflexive subtype_rel_self value-type-has-value int-value-type iff_weakening_uiff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut sqequalRule Error :productIsType,  Error :universeIsType,  because_Cache introduction extract_by_obid sqequalHypSubstitution isectElimination thin applyEquality hypothesisEquality hypothesis Error :functionIsType,  Error :inhabitedIsType,  setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination independent_pairFormation productElimination independent_pairEquality Error :functionIsTypeImplies,  equalityTransitivity equalitySymmetry Error :isect_memberFormation_alt,  applyLambdaEquality functionExtensionality Error :dependent_set_memberEquality_alt,  unionElimination equalityElimination Error :equalityIsType1,  promote_hyp instantiate cumulativity Error :equalityIsType4,  baseApply closedConclusion baseClosed addEquality minusEquality multiplyEquality intEquality callbyvalueReduce

Latex:
\mforall{}n,m:\mBbbZ{}.  \mforall{}f:\{n..m\msupminus{}\}  {}\mrightarrow{}  \mBbbB{}.
((\mexists{}m:\{n..m\msupminus{}\}.  (\muparrow{}(f  m)))  {}\mRightarrow{}  \{(\muparrow{}(f  mu-ge(f;n)))  \mwedge{}  (\mforall{}[i:\{n..mu-ge(f;n)\msupminus{}\}].  (\mneg{}\muparrow{}(f  i)))\})

Date html generated: 2019_06_20-PM-01_16_54
Last ObjectModification: 2018_10_06-AM-11_21_37

Theory : int_2

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