### Nuprl Lemma : div_induction-ext

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

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` 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 :  member: `t ∈ T` div_induction uniform-comp-nat-induction decidable__equal_int decidable__int_equal uall: `∀[x:A]. B[x]` so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` uimplies: `b supposing a` strict4: `strict4(F)` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` has-value: `(a)↓` prop: `ℙ` guard: `{T}` or: `P ∨ Q` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` iff_weakening_equal genrec-ap: genrec-ap
Lemmas referenced :  div_induction lifting-strict-int_eq top_wf equal_wf has-value_wf_base base_wf is-exception_wf lifting-strict-spread uniform-comp-nat-induction decidable__equal_int decidable__int_equal iff_weakening_equal
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution isectElimination baseClosed isect_memberEquality voidElimination voidEquality independent_isectElimination independent_pairFormation lambdaFormation callbyvalueDecide hypothesisEquality equalityTransitivity equalitySymmetry unionEquality unionElimination sqleReflexivity dependent_functionElimination independent_functionElimination baseApply closedConclusion decideExceptionCases inrFormation because_Cache imageMemberEquality imageElimination exceptionSqequal inlFormation callbyvalueApply applyExceptionCases

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

Date html generated: 2018_05_21-PM-07_49_13
Last ObjectModification: 2017_07_26-PM-05_27_01

Theory : general

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