Nuprl Lemma : converges-to-rexp
∀x:ℝ. lim n→∞.approx-rexp(x;n) = e^x
Proof
Definitions occuring in Statement :
approx-rexp: approx-rexp(x;n),
rexp: e^x,
converges-to: lim n→∞.x[n] = y,
real: ℝ,
all: ∀x:A. B[x]
Definitions unfolded in proof :
approx-rexp: approx-rexp(x;n),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
real: ℝ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
has-value: (a)↓,
decidable: Dec(P),
or: P ∨ Q,
top: Top,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat: ℕ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
ge: i ≥ j ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
efficient-exp-ext,
fastexp: i^n,
uiff: uiff(P;Q),
cand: A c∧ B,
rgt: x > y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
subtype_rel: A ⊆r B,
rexp: e^x,
pi1: fst(t),
exp-exists-ext,
int-to-real: r(n),
divide: n ÷ m,
rsum: Σ{x[k] | n≤k≤m},
canonical-bound: canonical-bound(r),
absval: |i|,
rmul: a * b,
rabs: |x|,
accelerate: accelerate(k;f),
imax: imax(a;b),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
btrue: tt,
bfalse: ff,
reg-seq-mul: reg-seq-mul(x;y),
int-rdiv: (a)/k1,
fact: (n)!,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m,
rnexp: x^k1,
eq_int: (i =z j),
canon-bnd: canon-bnd(x),
genrec: genrec,
rlessw: rlessw(x;y),
quick-find: quick-find(p;n),
radd: a + b,
rdiv: (x/y),
rinv: rinv(x),
mu-ge: mu-ge(f;n),
reg-seq-inv: reg-seq-inv(x),
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
reg-seq-adjust: reg-seq-adjust(n;x),
nil: [],
it: ⋅,
exp-ratio: exp-ratio(a;b;n;p;q),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
map: map(f;as),
list_ind: list_ind,
from-upto: [n, m),
radd-list: radd-list(L),
length: ||as||,
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
assert: ↑b,
int_upper: {i...},
unit: Unit,
bool: 𝔹,
converges-to: lim n→∞.x[n] = y,
rneq: x ≠ y,
rational-lower-approx: (below x within 1/n)
Lemmas referenced :
cheap-real-upper-bound,
divide_wfa,
istype-less_than,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
value-type-has-value,
int-value-type,
rleq_wf,
int-to-real_wf,
real_wf,
decidable__lt,
istype-top,
istype-void,
less_than_wf,
rless_transitivity2,
rexp-increasing,
rless-int,
rleq_weakening_rless,
rexp_wf,
rless_functionality,
req_weakening,
rexp0,
rexp-non-decreasing,
fastexp_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
nat_properties,
ge_wf,
le_witness_for_triv,
subtract-1-ge-0,
istype-false,
rleq-int,
rleq_functionality,
rexp-radd,
subtract_wf,
radd_wf,
rmul_wf,
subtract-add-cancel,
req_functionality,
rexp_functionality,
radd-int,
itermSubtract_wf,
int_term_value_subtract_lemma,
rmul_functionality_wrt_rleq2,
rexp-positive,
itermMultiply_wf,
nat_plus_properties,
rmul_preserves_rless,
rmul-int,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
exp-positive-stronger,
exp-fastexp,
exp_step,
sq_stable__less_than,
int_term_value_mul_lemma,
exp_wf2,
le_wf,
subtype_rel_sets_simple,
exp_wf_nat_plus,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
nat_plus_wf,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
divide_wf,
add_nat_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_upper_properties,
div_rem_sum2,
decidable__equal_int,
nat_plus_subtype_nat,
rem_bounds_1,
rless_wf,
rdiv_wf,
mul_nzero,
int_entire_a,
int-rdiv_wf,
rational-approx_wf,
mul_nat_plus,
rsub_wf,
rabs_wf,
set-value-type,
int_upper_wf,
zero-add,
nequal-le-implies,
upper_subtype_nat,
neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
rneq-int,
rational-lower-approx_wf,
rational-lower-approx-property,
rational-approx-property,
rabs-difference-symmetry,
mul_bounds_1b,
rmax_wf,
rabs-rexp-difference-bound,
rleq-implies-rleq,
rmul_functionality,
rabs-of-nonneg,
mul-commutes,
rneq_functionality,
rinv_wf2,
radd-preserves-rleq,
req_transitivity,
radd_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
rinv-as-rdiv,
real_term_value_add_lemma,
rmul_comm,
rmul_preserves_rleq,
rmul-rinv,
rmax-req2,
radd_functionality_wrt_rleq,
r-triangle-inequality2,
radd-rdiv,
rdiv_functionality,
rleq-int-fractions,
mul_preserves_le,
efficient-exp-ext,
exp-exists-ext
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
applyEquality,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_set_memberEquality_alt,
closedConclusion,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
sqequalBase,
universeIsType,
inhabitedIsType,
callbyvalueReduce,
unionElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageElimination,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
intWeakElimination,
functionIsTypeImplies,
productIsType,
inlFormation_alt,
dependent_set_memberFormation_alt,
multiplyEquality,
applyLambdaEquality,
promote_hyp,
equalityElimination,
baseApply,
pointwiseFunctionality,
divideEquality,
remainderEquality,
inrFormation_alt,
functionIsType,
hypothesis_subsumption,
int_eqReduceFalseSq,
int_eqReduceTrueSq
Latex:
\mforall{}x:\mBbbR{}. lim n\mrightarrow{}\minfty{}.approx-rexp(x;n) = e\^{}x
Date html generated:
2019_10_31-AM-06_11_13
Last ObjectModification:
2019_04_03-PM-08_17_14
Theory : reals_2
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