Nuprl Lemma : rroot-exists
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} . (∃y:{ℝ| (((↑isEven(i)) ⇒ (r0 ≤ y)) ∧ (y^i = x))})
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
isEven: isEven(n),
int_upper: {i...},
assert: ↑b,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
converges: x[n]↓ as n→∞,
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_upper: {i...},
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rroot-exists1-ext,
rroot-exists-part2,
converges-iff-cauchy,
nat_wf,
assert_wf,
isEven_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rnexp_wf,
int_upper_subtype_nat,
false_wf,
le_wf,
set_wf,
real_wf,
int_upper_wf,
constant-rleq-limit,
sq_stable__rleq,
unique-limit,
rnexp_zero_lemma,
constant-limit,
req_weakening,
rmul-limit,
converges-to_wf,
subtract_wf,
nat_properties,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
less_than_wf,
primrec-wf2,
equal_wf,
converges-to_functionality,
rmul_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
rmul_comm,
req_functionality,
rnexp_unroll,
rmul-one-both
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberFormation,
isectElimination,
independent_pairFormation,
productEquality,
functionEquality,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality,
independent_isectElimination,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}i:\{2...\}. \mforall{}x:\{x:\mBbbR{}| (\muparrow{}isEven(i)) {}\mRightarrow{} (r0 \mleq{} x)\} . (\mexists{}y:\{\mBbbR{}| (((\muparrow{}isEven(i)) {}\mRightarrow{} (r0 \mleq{} y)) \mwedge{} (y\^{}i = x))\})
Date html generated:
2017_10_03-AM-10_39_20
Last ObjectModification:
2017_07_28-AM-08_15_44
Theory : reals
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