Nuprl Lemma : absval_div_decreases
∀[n:{2...}]. ∀[i:ℤ-o]. |i ÷ n| < |i|
Proof
Definitions occuring in Statement :
absval: |i|,
int_upper: {i...},
int_nzero: ℤ-o,
less_than: a < b,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
int_upper: {i...},
int_nzero: ℤ-o,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
nat: ℕ,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
absval_div_nat,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
div_rem_sum,
absval_wf,
subtype_rel_sets,
le_wf,
nequal_wf,
int_upper_properties,
int_nzero_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
rem_bounds_1,
int_nzero_wf,
member-less_than,
nat_wf,
int_upper_wf,
intformless_wf,
int_formula_prop_less_lemma,
decidable__le,
add-is-int-iff,
multiply-is-int-iff,
intformnot_wf,
int_formula_prop_not_lemma,
mul_preserves_le,
int_upper_subtype_nat,
itermMultiply_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
decidable__equal_int,
nat_properties,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
equal-wf-T-base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
setEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
baseClosed,
divideEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
multiplyEquality,
minusEquality,
equalityElimination,
lessCases,
sqequalAxiom,
imageMemberEquality,
instantiate,
cumulativity
Latex:
\mforall{}[n:\{2...\}]. \mforall{}[i:\mBbbZ{}\msupminus{}\msupzero{}]. |i \mdiv{} n| < |i|
Date html generated:
2017_04_14-AM-09_22_50
Last ObjectModification:
2017_02_27-PM-03_58_40
Theory : int_2
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