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,
prop: ℙ,
so_lambda: λ2x.t[x],
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,
subtype_rel: A ⊆r B,
guard: {T},
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 :
all_wf,
int_nzero_wf,
int_nzero_properties,
satisfiable-full-omega-tt,
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,
int_subtype_base,
equal-wf-T-base,
set_wf,
less_than_wf,
uniform-comp-nat-induction,
absval_wf,
nat_wf,
decidable__equal_int,
subtype_base_sq,
nat_properties,
set_subtype_base,
equal_wf,
set-value-type,
int-value-type,
uall_wf,
int_seg_wf,
absval_ifthenelse,
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,
lelt_wf,
iff_weakening_equal,
absval_div_decreases,
subtype_rel_sets,
le_wf,
add_nat_wf,
false_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
intEquality,
divideEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
independent_functionElimination,
universeEquality,
cumulativity,
setEquality,
unionElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
cutEval,
dependent_set_memberEquality,
productElimination,
impliesFunctionality,
imageMemberEquality,
imageElimination,
applyLambdaEquality,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
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_04
Last ObjectModification:
2017_07_26-PM-05_26_51
Theory : general
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