Nuprl Lemma : regular-consistency
∀[x,y:ℝ]. ∀[n,m:ℕ+]. ((m * |(x n) - y n|) ≤ ((n * |(x m) - y m|) + (4 * n) + (4 * m)))
Proof
Definitions occuring in Statement :
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
real: ℝ,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
sq_stable: SqStable(P),
squash: ↓T,
true: True,
top: Top,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
subtract: n - m,
regular-int-seq: k-regular-seq(f),
sq_type: SQType(T)
Lemmas referenced :
less_than'_wf,
absval_wf,
subtract_wf,
nat_plus_wf,
real_wf,
sq_stable__le,
nat_wf,
nat_plus_properties,
decidable__le,
less_than_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
absval_sym,
iff_weakening_equal,
le_functionality,
le_weakening,
add_functionality_wrt_le,
int-triangle-inequality,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-commutes,
add-mul-special,
add-swap,
zero-mul,
zero-add,
add-zero,
subtype_base_sq,
int_subtype_base,
equal_wf,
absval_pos,
nat_plus_subtype_nat,
absval_mul,
mul-distributes,
mul-associates,
mul-swap,
mul-distributes-right
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
addEquality,
multiplyEquality,
setElimination,
rename,
hypothesis,
applyEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
functionExtensionality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
voidEquality,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
universeEquality,
instantiate,
cumulativity
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[n,m:\mBbbN{}\msupplus{}]. ((m * |(x n) - y n|) \mleq{} ((n * |(x m) - y m|) + (4 * n) + (4 * m)))
Date html generated:
2017_10_02-PM-07_13_47
Last ObjectModification:
2017_07_28-AM-07_20_03
Theory : reals
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