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 :
less_than': less_than'(a;b),
le: A ≤ B,
prop: ℙ,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
nat: ℕ,
int_seg: {i..j-},
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
genrec-ap: genrec-ap,
integer-sqrt-ext,
isqrt: isqrt(x),
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
absval: |i|
Lemmas referenced :
nat_wf,
int_seg_wf,
false_wf,
isqrt_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
int_seg_properties,
int_seg_subtype_nat,
isqrt-non-decreasing,
equal_wf,
less_than_wf,
le_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
itermSubtract_wf,
itermConstant_wf,
subtract_wf,
isqrt-property,
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,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermMultiply_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
add_nat_wf,
multiply_functionality_wrt_le,
add-commutes,
add-swap,
two-mul,
less_than_functionality,
decidable__lt,
full-omega-unsat,
istype-int,
istype-void,
squash_wf,
true_wf,
istype-universe,
absval_pos,
decidable__equal_int,
iff_weakening_equal,
absval-diff-symmetry,
absval-non-neg,
subtype_base_sq,
int_subtype_base,
istype-false,
absval_wf,
add-mul-special,
zero-mul,
integer-sqrt-ext
Rules used in proof :
cut,
addEquality,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
unionElimination,
dependent_functionElimination,
productElimination,
rename,
setElimination,
sqequalRule,
hypothesis,
independent_isectElimination,
because_Cache,
natural_numberEquality,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
multiplyEquality,
productEquality,
dependent_set_memberEquality,
minusEquality,
applyLambdaEquality,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
Error :universeIsType,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :productIsType,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity
Latex:
\mforall{}a,b:\mBbbN{}. (|isqrt(a) - isqrt(b)| \mleq{} isqrt(|a - b|))
Date html generated:
2019_06_20-PM-02_37_32
Last ObjectModification:
2019_06_12-PM-00_26_11
Theory : num_thy_1
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