Nuprl Lemma : real-upper-bound
∀[x:ℝ]. ∀[n:ℕ+]. (x ≤ r((((x n) + 1) ÷ 2 * n) + 1))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
divide: n ÷ m,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
rational-approx: (x within 1/n),
int_nzero: ℤ-o,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
rge: x ≥ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nat: ℕ,
int_lower: {...i},
gt: i > j,
ge: i ≥ j
Lemmas referenced :
rational-approx-property,
rabs-difference-bound-rleq,
rational-approx_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_var_lemma,
int_formula_prop_wf,
rless_wf,
le_witness_for_triv,
nat_plus_wf,
real_wf,
radd_wf,
intformeq_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_subtype_base,
int-rdiv_wf,
nequal_wf,
rmul_preserves_rleq,
rmul_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
req_weakening,
rneq-int,
set_subtype_base,
less_than_wf,
itermSubtract_wf,
itermAdd_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_functionality,
radd_functionality,
int-rdiv-req,
req_transitivity,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3,
radd-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq-int,
div_rem_sum,
decidable__le,
rem_bounds_1,
istype-less_than,
add-is-int-iff,
intformle_wf,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
false_wf,
istype-le,
multiply-is-int-iff,
rem_bounds_2,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
isectElimination,
setElimination,
rename,
hypothesis,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
addEquality,
divideEquality,
applyEquality,
multiplyEquality,
lambdaFormation_alt,
equalityIstype,
baseApply,
baseClosed,
sqequalBase,
dependent_set_memberEquality_alt,
intEquality,
imageMemberEquality,
pointwiseFunctionality,
promote_hyp,
imageElimination
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (x \mleq{} r((((x n) + 1) \mdiv{} 2 * n) + 1))
Date html generated:
2019_10_31-AM-06_10_37
Last ObjectModification:
2019_01_30-PM-02_01_18
Theory : reals_2
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