### Nuprl Lemma : mu-ge-property

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

Proof

Definitions occuring in Statement :  mu-ge: `mu-ge(f;n)` int_upper: `{i...}` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` guard: `{T}` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` guard: `{T}` and: `P ∧ Q` implies: `P `` Q` not: `¬A` false: `False` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` squash: `↓T` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` int_upper: `{i...}` decidable: `Dec(P)` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` assert: `↑b` bnot: `¬bb` sq_type: `SQType(T)` bfalse: `ff` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` mu-ge: `mu-ge(f;n)` true: `True` less_than': `less_than'(a;b)` le: `A ≤ B` subtract: `n - m` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` has-value: `(a)↓` cand: `A c∧ B` nat: `ℕ`
Lemmas referenced :  assert_witness mu-ge_wf istype-int_upper istype-assert bool_wf istype-int int_seg_wf int_seg_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma istype-void int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf int_seg_subtype_upper le_reflexive decidable__le intformnot_wf int_formula_prop_not_lemma istype-le istype-less_than primrec-wf2 istype-nat subtract_wf assert_wf uall_wf all_wf exists_wf assert-bnot bool_subtype_base subtype_base_sq bool_cases_sqequal eqff_to_assert eqtt_to_assert le_wf int_upper_wf lelt_wf int_term_value_subtract_lemma itermSubtract_wf decidable__lt int_formula_prop_eq_lemma intformeq_wf btrue_neq_bfalse not_assert_elim int_subtype_base assert_elim decidable__equal_int subtype_rel_self le-add-cancel add-commutes add-associates add-zero zero-mul add-mul-special minus-one-mul-top add-swap minus-one-mul minus-add condition-implies-le not-le-2 istype-false upper_subtype_upper subtype_rel_function int-value-type value-type-has-value set_subtype_base int_upper_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality extract_by_obid isectElimination applyEquality hypothesisEquality independent_isectElimination hypothesis independent_functionElimination Error :isect_memberEquality_alt,  Error :lambdaEquality_alt,  dependent_functionElimination because_Cache Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :isectIsTypeImplies,  Error :productIsType,  Error :functionIsType,  Error :universeIsType,  voidElimination Error :lambdaFormation_alt,  addEquality natural_numberEquality setElimination rename imageElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality independent_pairFormation Error :dependent_set_memberEquality_alt,  unionElimination Error :setIsType,  productEquality functionEquality cumulativity instantiate promote_hyp Error :equalityIsType1,  equalitySymmetry equalityTransitivity equalityElimination functionExtensionality applyLambdaEquality hyp_replacement Error :equalityIsType4,  multiplyEquality minusEquality intEquality callbyvalueReduce dependent_set_memberEquality dependent_pairFormation lambdaEquality isect_memberEquality voidEquality

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

Date html generated: 2019_06_20-PM-01_16_43
Last ObjectModification: 2019_03_05-PM-03_39_24

Theory : int_2

Home Index