Nuprl Lemma : reg-seq-adjust-property
∀[n:ℕ+]. ∀[x:ℝ]. ∀m:ℕ+. (m ≤ (n * |reg-seq-adjust(n;x) m|)) supposing 5 ≤ |x n|
Proof
Definitions occuring in Statement :
reg-seq-adjust: reg-seq-adjust(n;x),
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
reg-seq-adjust: reg-seq-adjust(n;x),
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
nat: ℕ,
sq_stable: SqStable(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
absval: |i|,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
sq_stable__le,
absval_wf,
top_wf,
less_than_wf,
nat_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
rnonzero-lemma1,
nat_plus_wf,
le_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
multiplyEquality,
because_Cache,
lessCases,
independent_pairFormation,
baseClosed,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
productElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_functionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
promote_hyp,
instantiate
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbR{}]. \mforall{}m:\mBbbN{}\msupplus{}. (m \mleq{} (n * |reg-seq-adjust(n;x) m|)) supposing 5 \mleq{} |x n|
Date html generated:
2019_10_16-PM-03_07_23
Last ObjectModification:
2018_08_20-PM-09_45_14
Theory : reals
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