### Nuprl Lemma : mu-ge-bound

`∀[n,m:ℤ]. ∀[f:{n..m-} ⟶ 𝔹].  mu-ge(f;n) ∈ {n..m-} supposing ∃k:{n..m-}. (↑(f k))`

Proof

Definitions occuring in Statement :  mu-ge: `mu-ge(f;n)` int_seg: `{i..j-}` assert: `↑b` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` 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)` guard: `{T}` bnot: `¬bb` assert: `↑b` has-value: `(a)↓` subtype_rel: `A ⊆r B` true: `True`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 exists_wf int_seg_wf assert_wf bool_wf le_wf subtract_wf decidable__le intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma nat_wf int_seg_properties decidable__lt lelt_wf eqtt_to_assert eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot value-type-has-value int-value-type itermAdd_wf int_term_value_add_lemma subtype_rel_dep_function int_seg_subtype subtype_rel_self int_subtype_base assert_elim equal-wf-T-base intformeq_wf int_formula_prop_eq_lemma
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry applyEquality functionExtensionality functionEquality isect_memberFormation productElimination because_Cache unionElimination dependent_set_memberEquality equalityElimination promote_hyp instantiate cumulativity callbyvalueReduce addEquality applyLambdaEquality

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

Date html generated: 2017_04_14-AM-09_18_35
Last ObjectModification: 2017_02_27-PM-03_55_05

Theory : int_2

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