Nuprl Lemma : div_floor_bounds
∀[a:ℤ]. ∀[n:ℤ-o].
((((a ÷↓ n) * n) ≤ a) ∧ a < ((a ÷↓ n) + 1) * n supposing 0 < n
∧ ((a ÷↓ n) + 1) * n < a ∧ (a ≤ ((a ÷↓ n) * n)) supposing n < 0)
Proof
Definitions occuring in Statement :
div_floor: a ÷↓ n,
int_nzero: ℤ-o,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
div_floor: a ÷↓ n,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
sq_type: SQType(T),
guard: {T},
and: P ∧ Q,
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,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
subtype_rel: A ⊆r B,
nat: ℕ,
has-value: (a)↓,
int_lower: {...i},
gt: i > j,
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P)
Lemmas referenced :
value-type-has-value,
int-value-type,
equal_wf,
div_rem_sum,
rem_bounds_z,
subtype_base_sq,
int_subtype_base,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
less_than'_wf,
subtract_wf,
member-less_than,
equal-wf-base-T,
absval_wf,
nat_wf,
div_floor_wf,
int_nzero_wf,
squash_wf,
true_wf,
absval_neg,
le_wf,
absval_pos,
not-gt-2,
add_functionality_wrt_lt,
le_reflexive,
minus-one-mul-top,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
zero-add,
mul-commutes,
add-commutes,
mul-distributes-right,
one-mul,
not-lt-2,
add_functionality_wrt_le,
add-is-int-iff,
set_subtype_base,
nequal_wf,
multiply-is-int-iff,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-zero,
le-add-cancel2,
add-zero,
two-mul,
decidable__lt,
le-add-cancel-alt,
mul-associates,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
remainderEquality,
hypothesisEquality,
setElimination,
rename,
because_Cache,
lambdaFormation,
independent_functionElimination,
voidElimination,
natural_numberEquality,
divideEquality,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
isect_memberFormation,
unionElimination,
equalityElimination,
productElimination,
sqequalRule,
lessCases,
axiomSqEquality,
isect_memberEquality,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
promote_hyp,
impliesFunctionality,
independent_pairEquality,
lambdaEquality,
addEquality,
multiplyEquality,
axiomEquality,
applyEquality,
callbyvalueReduce,
addLevel,
hyp_replacement,
dependent_set_memberEquality,
levelHypothesis,
minusEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}].
((((a \mdiv{}\mdownarrow{} n) * n) \mleq{} a) \mwedge{} a < ((a \mdiv{}\mdownarrow{} n) + 1) * n supposing 0 < n
\mwedge{} ((a \mdiv{}\mdownarrow{} n) + 1) * n < a \mwedge{} (a \mleq{} ((a \mdiv{}\mdownarrow{} n) * n)) supposing n < 0)
Date html generated:
2019_06_20-AM-11_25_49
Last ObjectModification:
2018_08_20-PM-09_28_54
Theory : arithmetic
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