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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