Nuprl Lemma : isqrt-convex
∀a,b:ℕ. (|isqrt(a) - isqrt(b)| ≤ isqrt(|a - b|))
Proof
Definitions occuring in Statement :
isqrt: isqrt(x),
absval: |i|,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
subtract: n - m
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
isqrt: isqrt(x),
integer-sqrt-ext,
genrec-ap: genrec-ap,
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
absval: |i|
Lemmas referenced :
isqrt-non-decreasing,
int_seg_subtype_nat,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
isqrt_wf,
false_wf,
int_seg_wf,
nat_wf,
isqrt-property,
subtract_wf,
itermConstant_wf,
itermSubtract_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
le_wf,
less_than_wf,
equal_wf,
mul-distributes,
mul-distributes-right,
add-associates,
minus-one-mul,
mul-associates,
mul-commutes,
mul-swap,
one-mul,
le_functionality,
add_functionality_wrt_le,
le_weakening,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
add_nat_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
multiply_functionality_wrt_le,
add-commutes,
add-swap,
two-mul,
less_than_functionality,
decidable__lt,
lelt_wf,
squash_wf,
true_wf,
absval_pos,
decidable__equal_int,
iff_weakening_equal,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma,
int_subtype_base,
absval_wf,
add-mul-special,
zero-mul,
integer-sqrt-ext
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
hypothesis,
sqequalRule,
setElimination,
rename,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addEquality,
dependent_set_memberEquality,
productEquality,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
minusEquality,
applyLambdaEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}a,b:\mBbbN{}. (|isqrt(a) - isqrt(b)| \mleq{} isqrt(|a - b|))
Date html generated:
2018_05_21-PM-07_53_12
Last ObjectModification:
2017_07_26-PM-05_30_43
Theory : general
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