Nuprl Lemma : rem_mag_bound
∀[a:ℤ]. ∀[n:ℤ-o].  |a rem n| < |n|
Proof
Definitions occuring in Statement : 
absval: |i|
, 
int_nzero: ℤ-o
, 
less_than: a < b
, 
uall: ∀[x:A]. B[x]
, 
remainder: n rem m
, 
int: ℤ
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
ge: i ≥ j 
, 
and: P ∧ Q
, 
top: Top
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
guard: {T}
, 
false: False
, 
not: ¬A
, 
nequal: a ≠ b ∈ T 
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
uimplies: b supposing a
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
uall: ∀[x:A]. B[x]
, 
int_nzero: ℤ-o
, 
nat_plus: ℕ+
, 
true: True
, 
squash: ↓T
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
rem_bounds_absval, 
nat_plus_inc_int_nzero, 
nat_plus_wf, 
nat_wf, 
decidable__le, 
int_subtype_base, 
equal-wf-base, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_and_lemma, 
intformless_wf, 
itermConstant_wf, 
itermVar_wf, 
intformeq_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
nat_plus_properties, 
nequal_wf, 
less_than_wf, 
subtype_rel_sets, 
absval_wf, 
rem_sym_1, 
iff_weakening_equal, 
squash_wf, 
true_wf, 
minus_minus_cancel, 
absval_sym, 
int_formula_prop_not_lemma, 
intformnot_wf, 
int_nzero_properties, 
member-less_than, 
int_nzero_wf, 
rem_sym_2, 
int_formula_prop_le_lemma, 
int_term_value_minus_lemma, 
intformle_wf, 
itermMinus_wf
Rules used in proof : 
cut, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :lambdaFormation_alt, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
applyEquality, 
hypothesis, 
sqequalRule, 
setElimination, 
rename, 
Error :universeIsType, 
intEquality, 
unionElimination, 
natural_numberEquality, 
lambdaFormation, 
independent_functionElimination, 
baseClosed, 
computeAll, 
independent_pairFormation, 
voidEquality, 
voidElimination, 
isect_memberEquality, 
int_eqEquality, 
dependent_pairFormation, 
applyLambdaEquality, 
setEquality, 
independent_isectElimination, 
lambdaEquality, 
because_Cache, 
isectElimination, 
minusEquality, 
imageElimination, 
equalitySymmetry, 
imageMemberEquality, 
equalityTransitivity, 
productElimination, 
remainderEquality, 
universeEquality, 
isect_memberFormation, 
dependent_set_memberEquality, 
closedConclusion, 
baseApply
Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}].    |a  rem  n|  <  |n|
Date html generated:
2019_06_20-PM-01_14_13
Last ObjectModification:
2018_10_03-PM-01_57_53
Theory : int_2
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