Nuprl Lemma : rexp-small_wf
∀[x:{x:ℝ| |x| ≤ (r1/r(4))} ]. (rexp-small(x) ∈ {y:ℝ| y = e^x} )
Proof
Definitions occuring in Statement :
rexp-small: rexp-small(x),
rexp: e^x,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rexp-small: rexp-small(x),
rexp-approx-for-small-ext,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P)
Lemmas referenced :
rexp-approx-for-small-ext,
accelerate-rational-approx,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rexp_wf,
subtype_rel_self,
all_wf,
rleq_wf,
rdiv_wf,
rless-int,
sq_exists_wf,
rabs_wf,
rsub_wf,
nat_plus_properties,
intformand_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
rless_wf,
int-to-real_wf,
real_wf,
nat_plus_wf,
sq_stable__rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
hypothesis,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
hypothesisEquality,
setElimination,
rename,
applyEquality,
instantiate,
functionEquality,
because_Cache,
setEquality,
inrFormation_alt,
productElimination,
lambdaFormation_alt,
int_eqEquality,
independent_pairFormation,
multiplyEquality,
setIsType,
imageMemberEquality,
baseClosed,
inhabitedIsType,
functionExtensionality,
closedConclusion,
applyLambdaEquality,
imageElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
axiomEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| |x| \mleq{} (r1/r(4))\} ]. (rexp-small(x) \mmember{} \{y:\mBbbR{}| y = e\^{}x\} )
Date html generated:
2019_10_30-AM-11_41_07
Last ObjectModification:
2019_02_08-PM-02_11_19
Theory : reals_2
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