Nuprl Lemma : divide-le
∀[a:ℕ+]. ∀[b,x:ℤ]. uiff(b ≤ (a * x);adjust_div(b;a) ≤ x)
Proof
Definitions occuring in Statement :
adjust_div: adjust_div(b;a),
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
adjust_div: adjust_div(b;a),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
le: A ≤ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
subtract: n - m,
nat: ℕ,
int_lower: {...i},
ge: i ≥ j ,
gt: i > j
Lemmas referenced :
lt_int_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
less_than'_wf,
adjust_div_wf,
subtype_rel_sets,
nequal_wf,
multiply-is-int-iff,
set_subtype_base,
int_subtype_base,
equal-wf-base,
le_wf,
nat_plus_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
add_functionality_wrt_le,
le-add-cancel2,
mul_preserves_le,
nat_plus_subtype_nat,
div_rem_sum,
subtract_wf,
mul-commutes,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
one-mul,
le_reflexive,
less-iff-le,
omega-shadow,
mul-distributes,
mul-associates,
mul-swap,
le-add-cancel-alt,
add-is-int-iff,
rem_bounds_1,
le_weakening2,
rem_bounds_2,
minus-zero,
le-add-cancel,
not-lt-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
remainderEquality,
because_Cache,
setElimination,
rename,
hypothesis,
lambdaFormation,
hypothesisEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
intEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
sqequalRule,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairEquality,
lambdaEquality,
axiomEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
applyEquality,
setEquality,
baseApply,
closedConclusion,
multiplyEquality,
addEquality,
divideEquality,
minusEquality,
sqequalIntensionalEquality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\mBbbN{}\msupplus{}]. \mforall{}[b,x:\mBbbZ{}]. uiff(b \mleq{} (a * x);adjust\_div(b;a) \mleq{} x)
Date html generated:
2017_04_14-AM-07_19_44
Last ObjectModification:
2017_02_27-PM-02_54_03
Theory : arithmetic
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