Nuprl Lemma : iroot-lemma2
∀a:ℕ. ∀n,b,k:ℕ+. (∃p:ℕ × ℕ+ [let x,y = p in 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],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
spread: spread def,
product: x:A × B[x],
multiply: n * m,
add: n + m
Definitions unfolded in proof :
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
nat: ℕ,
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,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
uiff: uiff(P;Q),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
le: A ≤ B,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
iff: P ⇐⇒ Q,
subtract: n - m,
rev_implies: P ⇐ Q
Lemmas referenced :
iroot_wf,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
add_nat_plus,
istype-less_than,
decidable__lt,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
value-type-has-value,
nat_plus_wf,
set-value-type,
less_than_wf,
int-value-type,
exp-fastexp,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
divide_wf,
multiply_nat_wf,
exp_wf4,
exp_wf2,
mul_bounds_1a,
nat_wf,
le_wf,
nat_plus_subtype_nat,
istype-nat,
iroot-property,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
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,
mul_nat_plus,
exp_wf_nat_plus,
exp_preserves_le,
squash_wf,
true_wf,
exp-of-mul,
subtype_rel_self,
iff_weakening_equal,
mul_preserves_lt,
mul_preserves_le,
add_nat_wf,
istype-false,
decidable__equal_int,
exp-difference-inequality,
subtract-add-cancel,
subtract-is-int-iff,
exp_step,
add-commutes,
mul-swap,
mul-commutes
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
dependent_set_memberFormation_alt,
cut,
dependent_set_memberEquality_alt,
addEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
applyEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productElimination,
equalityIsType1,
callbyvalueReduce,
multiplyEquality,
independent_pairEquality,
productIsType,
instantiate,
cumulativity,
intEquality,
imageElimination,
divideEquality,
equalityIsType4,
universeEquality,
remainderEquality,
hyp_replacement,
minusEquality
Latex:
\mforall{}a:\mBbbN{}. \mforall{}n,b,k:\mBbbN{}\msupplus{}. (\mexists{}p:\mBbbN{} \mtimes{} \mBbbN{}\msupplus{} [let x,y = p in a * y\^{}n < (x * b)\^{}n \mwedge{} ((x * b)\^{}n \mleq{} ((a + k) * y\^{}n))])
Date html generated:
2019_10_15-AM-11_24_28
Last ObjectModification:
2018_10_18-PM-11_44_02
Theory : general
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