Nuprl Lemma : div_nat_induction-ext
∀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 :
squash: ↓T,
or: P ∨ Q,
guard: {T},
prop: ℙ,
has-value: (a)↓,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
strict4: strict4(F),
uimplies: b supposing a,
so_apply: x[s],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2;s3;s4],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
uall: ∀[x:A]. B[x],
decidable__int_equal,
decidable__equal_int,
so_apply: x[s1;s2],
div_nat_induction,
member: t ∈ T
Lemmas referenced :
is-exception_wf,
base_wf,
has-value_wf_base,
equal_wf,
top_wf,
lifting-strict-int_eq,
div_nat_induction,
decidable__int_equal,
decidable__equal_int
Rules used in proof :
inlFormation,
exceptionSqequal,
imageElimination,
imageMemberEquality,
because_Cache,
inrFormation,
decideExceptionCases,
closedConclusion,
baseApply,
independent_functionElimination,
dependent_functionElimination,
sqleReflexivity,
unionElimination,
unionEquality,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
callbyvalueDecide,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
voidEquality,
voidElimination,
isect_memberEquality,
baseClosed,
isectElimination,
sqequalHypSubstitution,
thin,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
cut,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
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_34
Last ObjectModification:
2018_05_19-AM-07_44_27
Theory : general
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