Nuprl Lemma : div_induction
∀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 :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_apply: x[s],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
ge: i ≥ j ,
int_seg: {i..j-},
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
int_upper: {i...},
less_than: a < b,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
int_nzero_wf,
int_nzero_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int_subtype_base,
subtype_rel_self,
less_than_wf,
uniform-comp-nat-induction,
all_wf,
absval_wf,
nat_wf,
decidable__equal_int,
subtype_base_sq,
nat_properties,
set_subtype_base,
set-value-type,
equal_wf,
int-value-type,
int_seg_wf,
absval_ifthenelse,
le_wf,
lt_int_wf,
assert_wf,
bnot_wf,
not_wf,
intformnot_wf,
itermMinus_wf,
int_formula_prop_not_lemma,
int_term_value_minus_lemma,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
nequal_wf,
subtract_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
decidable__lt,
iff_weakening_equal,
absval_div_decreases,
subtype_rel_sets,
add_nat_wf,
istype-false,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
cut,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
introduction,
extract_by_obid,
hypothesis,
applyEquality,
hypothesisEquality,
divideEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
because_Cache,
isectElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :equalityIsType4,
Error :inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
instantiate,
universeEquality,
Error :setIsType,
setEquality,
intEquality,
unionElimination,
cumulativity,
cutEval,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType1,
Error :isectIsType,
productElimination,
imageMemberEquality,
imageElimination,
Error :productIsType,
addEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp
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:
2019_06_20-PM-02_33_00
Last ObjectModification:
2019_03_19-AM-10_48_21
Theory : num_thy_1
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