Nuprl Lemma : rem_bounds_2
∀[a:{...0}]. ∀[n:ℕ+]. ((0 ≥ (a rem n) ) ∧ ((a rem n) > (-n)))
Proof
Definitions occuring in Statement :
int_lower: {...i}
,
nat_plus: ℕ+
,
uall: ∀[x:A]. B[x]
,
gt: i > j
,
ge: i ≥ j
,
and: P ∧ Q
,
remainder: n rem m
,
minus: -n
,
natural_number: $n
Definitions unfolded in proof :
gt: i > j
,
prop: ℙ
,
all: ∀x:A. B[x]
,
uimplies: b supposing a
,
guard: {T}
,
nequal: a ≠ b ∈ T
,
nat_plus: ℕ+
,
int_lower: {...i}
,
false: False
,
implies: P
⇒ Q
,
not: ¬A
,
le: A ≤ B
,
ge: i ≥ j
,
and: P ∧ Q
,
member: t ∈ T
,
uall: ∀[x:A]. B[x]
,
or: P ∨ Q
,
decidable: Dec(P)
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
less_than: a < b
,
less_than': less_than'(a;b)
,
top: Top
,
true: True
,
squash: ↓T
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
subtype_rel: A ⊆r B
,
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
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
cand: A c∧ B
,
int_nzero: ℤ-o
,
subtract: n - m
Lemmas referenced :
decidable__lt,
equal_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
istype-void,
eqff_to_assert,
set_subtype_base,
le_wf,
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,
false_wf,
eq_int_wf,
assert_of_eq_int,
equal-wf-base,
less_than'_wf,
gt_wf,
iff_weakening_equal,
subtype_rel_self,
nequal_wf,
subtype_rel_sets,
rem-zero,
true_wf,
squash_wf,
ge_wf,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
add-zero,
zero-add,
add_functionality_wrt_le,
not-lt-2,
not-equal-2,
decidable__int_equal,
int_lower_wf,
add-inverse,
le_reflexive,
add_functionality_wrt_lt,
member-less_than,
nat_plus_wf
Rules used in proof :
isect_memberEquality,
minusEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
intEquality,
voidElimination,
independent_functionElimination,
independent_isectElimination,
lambdaFormation,
hypothesis,
rename,
setElimination,
remainderEquality,
natural_numberEquality,
isectElimination,
lemma_by_obid,
because_Cache,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
unionElimination,
extract_by_obid,
independent_pairFormation,
lessCases,
Error :remNegative,
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,
applyEquality,
Error :lambdaEquality_alt,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
Error :equalityIsType1,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
remainderBounds2,
productEquality,
universeEquality,
setEquality,
addLevel,
voidEquality,
addEquality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\{...0\}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((0 \mgeq{} (a rem n) ) \mwedge{} ((a rem n) > (-n)))
Date html generated:
2019_06_20-AM-11_24_07
Last ObjectModification:
2018_10_15-PM-03_19_08
Theory : arithmetic
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