Nuprl Lemma : rem_bounds_4
∀[a:ℕ]. ∀[n:{...-1}]. ((0 ≤ (a rem n)) ∧ a rem n < -n)
Proof
Definitions occuring in Statement :
int_lower: {...i},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
remainder: n rem m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
true: True,
less_than': less_than'(a;b),
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
or: P ∨ Q,
decidable: Dec(P),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
all: ∀x:A. B[x],
nequal: a ≠ b ∈ T ,
int_lower: {...i},
nat: ℕ,
false: False,
implies: P ⇒ Q,
not: ¬A,
le: A ≤ B,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
squash: ↓T,
bfalse: ff,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_stable: SqStable(P),
cand: A c∧ B,
int_nzero: ℤ-o
Lemmas referenced :
nat_wf,
int_lower_wf,
member-less_than,
or_wf,
le-add-cancel,
add_functionality_wrt_le,
minus-zero,
minus-add,
zero-add,
add-swap,
add-commutes,
add-associates,
condition-implies-le,
not-le-2,
false_wf,
le_wf,
decidable__le,
not-equal-2,
less_than'_wf,
decidable__lt,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
istype-void,
eqff_to_assert,
set_subtype_base,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
eq_int_wf,
assert_of_eq_int,
le_antisymmetry_iff,
sq_stable_from_decidable,
add-zero,
le-add-cancel-alt,
equal-wf-base,
istype-top,
istype-int,
istype-assert,
istype-less_than,
not-lt-2,
iff_weakening_equal,
subtype_rel_self,
nequal_wf,
subtype_rel_sets,
rem-zero,
true_wf,
squash_wf,
decidable__int_equal,
add-inverse,
le_reflexive,
add_functionality_wrt_lt
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
orFunctionality,
addLevel,
independent_functionElimination,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
applyEquality,
inrFormation,
voidElimination,
lambdaFormation,
independent_pairFormation,
inlFormation,
unionElimination,
addEquality,
independent_isectElimination,
natural_numberEquality,
hypothesis,
rename,
setElimination,
remainderEquality,
isectElimination,
extract_by_obid,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lessCases,
Error :remPositive,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
equalityElimination,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :universeIsType,
imageMemberEquality,
baseClosed,
imageElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType4,
baseApply,
closedConclusion,
Error :lambdaEquality_alt,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
Error :equalityIsType1,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
remainderBounds4,
productEquality,
universeEquality,
setEquality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\{...-1\}]. ((0 \mleq{} (a rem n)) \mwedge{} a rem n < -n)
Date html generated:
2019_06_20-AM-11_24_15
Last ObjectModification:
2018_10_16-PM-03_24_54
Theory : arithmetic
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