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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