Nuprl Lemma : rinv-functionality-lemma
∀x,y:ℤ. ∀a,b,n:ℕ+.
((n ≤ (a * |x|)) ⇒ (n ≤ (b * |y|)) ⇒ (|x - y| ≤ 4) ⇒ (|((4 * n * n) ÷ x) - (4 * n * n) ÷ y| ≤ (2 + (16 * a * b))))
Proof
Definitions occuring in Statement :
absval: |i|,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
absval: |i|,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
int_nzero: ℤ-o,
subtract: n - m,
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
cand: A c∧ B,
uiff: uiff(P;Q),
less_than': less_than'(a;b)
Lemmas referenced :
mul_com,
absval_sym,
rem_bounds_absval_le,
mul_preserves_le,
absval_pos,
nat_plus_subtype_nat,
mul_bounds_1a,
multiply_functionality_wrt_le,
absval-diff-symmetry,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
multiply-is-int-iff,
add_functionality_wrt_eq,
int-triangle-inequality,
add_functionality_wrt_le,
le_weakening,
le_functionality,
decidable__le,
add-commutes,
add-swap,
one-mul,
mul-associates,
mul-commutes,
mul-swap,
minus-one-mul,
add-associates,
minus-add,
mul-distributes,
div_rem_sum2,
nequal_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
decidable__equal_int,
iff_weakening_equal,
absval_mul,
true_wf,
squash_wf,
nat_plus_wf,
nat_wf,
le_wf,
int_entire_a,
absval_nat_plus,
subtract_wf,
absval_wf,
mul_cancel_in_le,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_plus_properties,
int_subtype_base,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalRule,
setElimination,
rename,
natural_numberEquality,
minusEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache,
divideEquality,
multiplyEquality,
applyEquality,
addEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
unionElimination,
dependent_set_memberEquality,
remainderEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion
Latex:
\mforall{}x,y:\mBbbZ{}. \mforall{}a,b,n:\mBbbN{}\msupplus{}.
((n \mleq{} (a * |x|))
{}\mRightarrow{} (n \mleq{} (b * |y|))
{}\mRightarrow{} (|x - y| \mleq{} 4)
{}\mRightarrow{} (|((4 * n * n) \mdiv{} x) - (4 * n * n) \mdiv{} y| \mleq{} (2 + (16 * a * b))))
Date html generated:
2016_05_18-AM-06_54_19
Last ObjectModification:
2016_01_17-AM-01_47_50
Theory : reals
Home
Index