Nuprl Lemma : div_absval_bound
∀[M:ℕ+]. ∀[z:ℤ]. ∀[n:ℕ]. |z ÷ M| ≤ n supposing |z| ≤ (n * M)
Proof
Definitions occuring in Statement :
absval: |i|,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
divide: n ÷ m,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
absval: |i|,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
int_lower: {...i},
gt: i > j
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_properties,
nat_plus_properties,
decidable__le,
absval_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
absval_le_zero,
zero-div-rem,
nat_plus_inc_int_nzero,
istype-false,
intformeq_wf,
int_formula_prop_eq_lemma,
mul_preserves_le,
nat_plus_subtype_nat,
le_witness_for_triv,
istype-le,
istype-nat,
nat_plus_wf,
mul_cancel_in_le,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
absval_mul,
iff_weakening_equal,
absval_pos,
div_rem_sum,
div_rem_sum2,
rem_bounds_1,
absval_unfold,
subtract_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
itermMinus_wf,
int_term_value_minus_lemma,
rem_bounds_2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
natural_numberEquality,
hypothesis,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
setElimination,
rename,
applyEquality,
sqequalRule,
productElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
Error :lambdaFormation_alt,
minusEquality,
Error :inhabitedIsType,
multiplyEquality,
Error :isectIsTypeImplies,
divideEquality,
Error :equalityIstype,
baseClosed,
sqequalBase,
imageElimination,
imageMemberEquality,
Error :dependent_set_memberEquality_alt,
remainderEquality,
equalityElimination,
lessCases,
axiomSqEquality,
closedConclusion,
promote_hyp
Latex:
\mforall{}[M:\mBbbN{}\msupplus{}]. \mforall{}[z:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. |z \mdiv{} M| \mleq{} n supposing |z| \mleq{} (n * M)
Date html generated:
2019_06_20-PM-01_18_54
Last ObjectModification:
2019_02_12-PM-02_04_40
Theory : int_2
Home
Index