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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