### Nuprl Lemma : div_nat_induction

`∀b:{b:ℤ| 1 < b} . ∀[P:ℕ ⟶ ℙ]. (P[0] `` (∀i:ℕ+. (P[i ÷ b] `` P[i])) `` (∀i:ℕ. P[i]))`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` divide: `n ÷ m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` nat_plus: `ℕ+` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` true: `True` subtract: `n - m` int_seg: `{i..j-}` lelt: `i ≤ j < k` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  decidable__equal_int subtype_base_sq int_subtype_base nat_properties full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_wf equal-wf-base equal-wf-T-base equal_wf set-value-type int-value-type all_wf int_seg_wf nat_wf int_seg_subtype_nat false_wf natrec_wf nat_plus_wf divide_wf nat_plus_subtype_nat decidable__lt not-lt-2 less-iff-le add_functionality_wrt_le add-swap add-commutes add-associates zero-add le-add-cancel less_than_wf le_wf set_wf not-equal-2 add-zero condition-implies-le minus-add minus-zero div_bounds_1 div_mono1 subtype_rel_sets sq_stable__less_than decidable__le intformnot_wf intformle_wf int_formula_prop_not_lemma int_formula_prop_le_lemma lelt_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut thin introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination setElimination rename because_Cache hypothesis unionElimination instantiate isectElimination cumulativity intEquality independent_isectElimination independent_functionElimination divideEquality hypothesisEquality natural_numberEquality approximateComputation dependent_pairFormation lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation equalityTransitivity equalitySymmetry applyEquality baseClosed cutEval dependent_set_memberEquality functionExtensionality functionEquality productElimination addEquality universeEquality minusEquality setEquality imageMemberEquality imageElimination

Latex:
\mforall{}b:\{b:\mBbbZ{}|  1  <  b\}  .  \mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  (P[0]  {}\mRightarrow{}  (\mforall{}i:\mBbbN{}\msupplus{}.  (P[i  \mdiv{}  b]  {}\mRightarrow{}  P[i]))  {}\mRightarrow{}  (\mforall{}i:\mBbbN{}.  P[i]))

Date html generated: 2018_05_21-PM-07_49_24
Last ObjectModification: 2017_11_20-PM-01_54_54

Theory : general

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