`∀[n:ℕ+]. ∀[x:ℝ].  reg-seq-adjust(n;x) ∈ {f:ℕ+ ⟶ ℤ| if (n =z 1) then 1 else 4 fi -regular-seq(f)}  supposing ∀i:ℕ+. (i <\000C n `` (|x i| ≤ 4))`

Proof

Definitions occuring in Statement :  reg-seq-adjust: `reg-seq-adjust(n;x)` real: `ℝ` regular-int-seq: `k-regular-seq(f)` absval: `|i|` nat_plus: `ℕ+` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` real: `ℝ` reg-seq-adjust: `reg-seq-adjust(n;x)` nat_plus: `ℕ+` less_than: `a < b` and: `P ∧ Q` less_than': `less_than'(a;b)` true: `True` squash: `↓T` top: `Top` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat: `ℕ` so_apply: `x[s]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` le: `A ≤ B` decidable: `Dec(P)` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` absval: `|i|` subtract: `n - m`
Lemmas referenced :  top_wf less_than_wf nat_plus_wf eq_int_wf bool_wf uiff_transitivity equal-wf-T-base assert_wf eqtt_to_assert assert_of_eq_int iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot equal_wf regular-int-seq_wf ifthenelse_wf all_wf le_wf absval_wf nat_wf real_wf lt_int_wf assert_of_lt_int nat_plus_properties full-omega-unsat intformand_wf intformless_wf itermVar_wf itermConstant_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_eq_lemma int_formula_prop_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot bdd-diff-regular-int-seq false_wf subtract_wf decidable__le intformnot_wf intformle_wf int_formula_prop_not_lemma int_formula_prop_le_lemma decidable__equal_int itermAdd_wf int_term_value_add_lemma add-is-int-iff itermSubtract_wf int_term_value_subtract_lemma and_wf le_functionality le_weakening int-triangle-inequality add_functionality_wrt_le squash_wf true_wf minus-one-mul add-mul-special zero-mul
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution setElimination thin rename dependent_set_memberEquality sqequalRule lambdaEquality hypothesisEquality hypothesis lessCases independent_pairFormation isectElimination baseClosed natural_numberEquality equalityTransitivity equalitySymmetry imageMemberEquality axiomSqEquality extract_by_obid isect_memberEquality because_Cache voidElimination voidEquality lambdaFormation imageElimination productElimination independent_functionElimination applyEquality unionElimination equalityElimination intEquality independent_isectElimination impliesFunctionality dependent_functionElimination axiomEquality functionEquality functionExtensionality approximateComputation dependent_pairFormation int_eqEquality promote_hyp instantiate cumulativity hyp_replacement applyLambdaEquality addEquality minusEquality pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[x:\mBbbR{}].
reg-seq-adjust(n;x)  \mmember{}  \{f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}|  if  (n  =\msubz{}  1)  then  1  else  4  fi  -regular-seq(f)\}    supposing  \mforall{}i:\mBbbN{}\msupplus{}.  \000C(i  <  n  {}\mRightarrow{}  (|x  i|  \mleq{}  4))

Date html generated: 2019_10_16-PM-03_07_18
Last ObjectModification: 2018_08_20-PM-09_45_00

Theory : reals

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