Nuprl Lemma : infinitesmal-zero
∀[x:ℝ]. uiff(x = r0;∀[k:ℕ+]. (|x| ≤ (r1/r(k))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
true: True,
absval: |i|,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
squash: ↓T,
rdiv: (x/y),
real: ℝ,
rsub: x - y,
reg-seq-add: reg-seq-add(x;y),
nequal: a ≠ b ∈ T ,
rminus: -(x),
reg-seq-mul: reg-seq-mul(x;y),
rabs: |x|,
int_upper: {i...},
rnonneg2: rnonneg2(x),
int-to-real: r(n),
int_nzero: ℤ-o,
sq_type: SQType(T),
subtract: n - m,
less_than': less_than'(a;b),
bdd-diff: bdd-diff(f;g),
ge: i ≥ j ,
regular-int-seq: k-regular-seq(f),
sq_stable: SqStable(P)
Lemmas referenced :
less_than'_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
rabs_wf,
nat_plus_wf,
req_wf,
req_witness,
uall_wf,
rleq_wf,
real_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
rmul_preserves_rleq,
rmul_wf,
absval_wf,
nat_wf,
rinv_wf2,
rleq-int,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
rleq_functionality,
rabs_functionality,
req_weakening,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
squash_wf,
true_wf,
rabs-int,
rmul-int,
rmul-rinv,
rmul_comm,
rmul-rdiv-cancel2,
uiff_transitivity,
rmul_preserves_rleq2,
radd-bdd-diff,
rnonneg2_functionality,
rminus_wf,
radd_wf,
rnonneg-iff,
rmul-bdd-diff-reg-seq-mul,
rminus_functionality_wrt_bdd-diff,
bdd-diff_weakening,
reg-seq-add_functionality_wrt_bdd-diff,
reg-seq-mul_wf,
int_subtype_base,
equal-wf-base,
reg-seq-add_wf,
less_than_wf,
less_than_transitivity1,
subtract_wf,
all_wf,
int_upper_wf,
le_wf,
int_term_value_minus_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
itermMinus_wf,
itermAdd_wf,
nequal_wf,
div-cancel,
decidable__equal_int,
int_upper_properties,
subtype_base_sq,
add-zero,
zero-mul,
mul-commutes,
mul-associates,
minus-one-mul,
false_wf,
req-iff-bdd-diff,
imax_ub,
imax_wf,
iff_weakening_equal,
absval_mul,
absval_nat_plus,
mul_cancel_in_le,
add-mul-special,
add-commutes,
add-swap,
int-triangle-inequality,
le_weakening,
le_functionality,
add_functionality_wrt_le,
sq_stable__le,
add_functionality_wrt_eq,
nat_plus_subtype_nat,
absval_pos,
equal_wf,
absval-non-neg,
set_subtype_base,
mul-swap,
mul-distributes,
multiply-is-int-iff,
add-is-int-iff,
minus-one-mul-top,
int_upper_subtype_nat,
mul_preserves_le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
independent_isectElimination,
inrFormation,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
baseClosed,
multiplyEquality,
imageElimination,
imageMemberEquality,
addEquality,
closedConclusion,
baseApply,
divideEquality,
dependent_set_memberEquality,
hyp_replacement,
cumulativity,
instantiate,
inlFormation,
universeEquality,
functionExtensionality,
promote_hyp,
pointwiseFunctionality
Latex:
\mforall{}[x:\mBbbR{}]. uiff(x = r0;\mforall{}[k:\mBbbN{}\msupplus{}]. (|x| \mleq{} (r1/r(k))))
Date html generated:
2017_10_03-AM-08_52_04
Last ObjectModification:
2017_07_28-AM-07_34_57
Theory : reals
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