Nuprl Lemma : dense-in-reals-implies-accel
∀X:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);X)
⇒ (∀x:ℝ. ∀y:{y:ℝ| y = x} . ((X x) ⇒ (X y)))
⇒ (∀x:ℝ. ∀k:ℕ+. ∃y:ℝ. ((y = accelerate(3;x) ∈ (ℕ+k ⟶ ℤ)) ∧ (X y))))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
riiint: (-∞, ∞),
req: x = y,
accelerate: accelerate(k;f),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
istype: istype(T),
rless: x < y,
sq_exists: ∃x:A [B[x]],
guard: {T},
rneq: x ≠ y,
rev_implies: P ⇐ Q,
cand: A c∧ B,
real: ℝ,
le: A ≤ B,
less_than': less_than'(a;b),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T)
Lemmas referenced :
dense-in-reals-iff,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_wf,
real_wf,
req_wf,
subtype_rel_self,
dense-in-interval_wf,
riiint_wf,
subtype_rel_dep_function,
i-member_wf,
member_riiint_lemma,
blended-real-req,
rleq_weakening_rless,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
blended-real-agrees,
blended-real_wf,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat_plus,
istype-false,
accelerate_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
div-cancel2,
subtype_base_sq,
nequal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsType,
setIsType,
because_Cache,
applyEquality,
instantiate,
universeEquality,
cumulativity,
setEquality,
closedConclusion,
inrFormation_alt,
productIsType,
equalityIstype,
intEquality,
baseApply,
baseClosed,
sqequalBase,
equalitySymmetry,
functionExtensionality,
equalityTransitivity
Latex:
\mforall{}X:\mBbbR{} {}\mrightarrow{} \mBbbP{}
(dense-in-interval((-\minfty{}, \minfty{});X)
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| y = x\} . ((X x) {}\mRightarrow{} (X y)))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. \mforall{}k:\mBbbN{}\msupplus{}. \mexists{}y:\mBbbR{}. ((y = accelerate(3;x)) \mwedge{} (X y))))
Date html generated:
2019_10_30-AM-07_20_30
Last ObjectModification:
2019_01_31-AM-11_44_33
Theory : reals
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