Nuprl Lemma : iroot-lemma
∀a:ℕ. ∀n,b,k:ℕ+. ∃x:ℕ. ∃y:ℕ+. (a * y^n < (x * b)^n ∧ ((x * b)^n ≤ ((a + k) * y^n)))
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
add: n + m
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
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,
guard: {T},
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
squash: ↓T,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
sq_type: SQType(T)
Lemmas referenced :
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_wf,
istype-nat,
iroot-property,
nat_plus_subtype_nat,
iroot_wf,
add-is-int-iff,
set_subtype_base,
le_wf,
int_subtype_base,
less_than_wf,
istype-false,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
exp_wf2,
add_nat_plus,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
multiply_nat_wf,
exp_wf4,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
div_rem_sum,
nat_plus_inc_int_nzero,
rem_bounds_1,
div_bounds_1,
minus-zero,
mul-swap,
mul_nat_plus,
exp_wf_nat_plus,
divide_wf,
exp_preserves_le,
mul_bounds_1a,
squash_wf,
true_wf,
exp-of-mul,
subtype_rel_self,
iff_weakening_equal,
mul_preserves_lt,
mul_preserves_le,
add_nat_wf,
subtype_base_sq,
decidable__equal_int,
exp-difference-inequality,
subtract-add-cancel,
subtract-is-int-iff,
exp_step,
mul-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
dependent_set_memberEquality_alt,
addEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyEquality,
productElimination,
because_Cache,
closedConclusion,
baseApply,
baseClosed,
intEquality,
minusEquality,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
equalityIsType1,
multiplyEquality,
divideEquality,
imageElimination,
equalityIsType4,
productIsType,
instantiate,
universeEquality,
cumulativity,
remainderEquality,
hyp_replacement
Latex:
\mforall{}a:\mBbbN{}. \mforall{}n,b,k:\mBbbN{}\msupplus{}. \mexists{}x:\mBbbN{}. \mexists{}y:\mBbbN{}\msupplus{}. (a * y\^{}n < (x * b)\^{}n \mwedge{} ((x * b)\^{}n \mleq{} ((a + k) * y\^{}n)))
Date html generated:
2019_10_15-AM-11_24_03
Last ObjectModification:
2018_10_18-PM-11_44_15
Theory : general
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