Nuprl Lemma : div_reduce_inequality
∀[a:ℤ]
((∀n:ℕ+. ∀x:ℤ. uiff(0 ≤ (a + (n * x));0 ≤ ((a ÷↓ n) + x)))
∧ (∀n:{...-1}. ∀x:ℤ. uiff(0 ≤ (a + (n * x));0 ≤ ((a ÷↓ (-n)) + ((-1) * x)))))
Proof
Definitions occuring in Statement :
div_floor: a ÷↓ n,
int_lower: {...i},
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
le: A ≤ B,
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
int_lower: {...i},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
squash: ↓T
Lemmas referenced :
le_witness_for_triv,
istype-int,
nat_plus_wf,
int_lower_wf,
div_floor_bounds,
subtype_rel_sets,
less_than_wf,
nequal_wf,
istype-less_than,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
int_subtype_base,
istype-le,
div_floor_wf,
decidable__le,
not-le-2,
mul_preserves_le,
nat_plus_subtype_nat,
istype-void,
add_functionality_wrt_lt,
le_reflexive,
add-associates,
multiply-is-int-iff,
set_subtype_base,
add-is-int-iff,
mul-distributes,
mul-commutes,
one-mul,
zero-mul,
add-commutes,
add-swap,
mul-distributes-right,
le_functionality,
add_functionality_wrt_le,
int_lower_properties,
subtract_wf,
decidable__lt,
minus-one-mul-top,
istype-false,
not-lt-2,
minus-le,
condition-implies-le,
minus-zero,
add-zero,
zero-add,
le-add-cancel-alt,
minus-one-mul,
add-mul-special,
not-equal-2,
minus-add,
minus-minus,
le-add-cancel2,
uiff_wf,
squash_wf,
true_wf,
le_wf,
mul-associates,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
Error :isect_memberEquality_alt,
isectElimination,
extract_by_obid,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :lambdaFormation_alt,
Error :universeIsType,
independent_pairFormation,
minusEquality,
natural_numberEquality,
applyEquality,
intEquality,
because_Cache,
closedConclusion,
setElimination,
rename,
Error :setIsType,
independent_functionElimination,
voidElimination,
Error :equalityIsType4,
baseClosed,
addEquality,
multiplyEquality,
unionElimination,
baseApply,
Error :dependent_set_memberEquality_alt,
Error :inlFormation_alt,
Error :inrFormation_alt,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[a:\mBbbZ{}]
((\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbZ{}. uiff(0 \mleq{} (a + (n * x));0 \mleq{} ((a \mdiv{}\mdownarrow{} n) + x)))
\mwedge{} (\mforall{}n:\{...-1\}. \mforall{}x:\mBbbZ{}. uiff(0 \mleq{} (a + (n * x));0 \mleq{} ((a \mdiv{}\mdownarrow{} (-n)) + ((-1) * x)))))
Date html generated:
2019_06_20-AM-11_25_54
Last ObjectModification:
2018_10_27-AM-11_46_24
Theory : arithmetic
Home
Index