Nuprl Lemma : fast-rexp_wf
∀[x:ℝ]. (fast-rexp(x) ∈ {y:ℝ| y = e^x} )
Proof
Definitions occuring in Statement :
fast-rexp: fast-rexp(x),
rexp: e^x,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fast-rexp: fast-rexp(x),
has-value: (a)↓,
uimplies: b supposing a,
real: ℝ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_upper: {i...},
so_apply: x[s],
rfun: I ⟶ℝ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
le: A ≤ B,
top: Top,
rneq: x ≠ y,
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],
subinterval: I ⊆ J ,
rexp: e^x,
pi1: fst(t),
exp-exists-ext,
sq_stable: SqStable(P),
rational-upper-approx: above x within 1/n,
rge: x ≥ y,
rgt: x > y,
rational-lower-approx: (below x within 1/n),
cand: A c∧ B
Lemmas referenced :
value-type-has-value,
int-value-type,
less_than_wf,
real-has-value,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
true_wf,
nequal_wf,
int_upper_wf,
all_wf,
nat_plus_wf,
le_wf,
absval_wf,
rexp_wf,
set-value-type,
canonical-bound_wf,
real_wf,
i-member_wf,
rccint_wf,
req_functionality,
rexp_functionality,
req_weakening,
req_wf,
set_wf,
subtype_rel_set,
int_upper_subtype_nat,
false_wf,
member_rccint_lemma,
approx-arg-interval_wf,
int-to-real_wf,
rless_wf,
subtract_wf,
rdiv_wf,
rless-int,
rless-int-fractions,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermMultiply_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
rless_functionality,
int-rdiv-req,
derivative-rexp,
member_riiint_lemma,
rleq_wf,
riiint_wf,
derivative_functionality_wrt_subinterval,
sq_stable__rleq,
rabs_wf,
rational-upper-approx-property,
rational-upper-approx_wf,
rsub_wf,
equal_wf,
nat_wf,
rleq_functionality,
rabs-of-nonneg,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
rexp-positive,
rexp-non-decreasing,
canonical-bound-property,
rational-lower-approx-property,
rational-lower-approx_wf,
exp-exists-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
applyEquality,
setElimination,
rename,
hypothesisEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
addLevel,
lambdaFormation,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
because_Cache,
setEquality,
lambdaEquality,
multiplyEquality,
axiomEquality,
productElimination,
isect_memberEquality,
voidEquality,
addEquality,
inrFormation,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
productEquality,
sqleReflexivity,
imageElimination
Latex:
\mforall{}[x:\mBbbR{}]. (fast-rexp(x) \mmember{} \{y:\mBbbR{}| y = e\^{}x\} )
Date html generated:
2017_10_04-PM-10_38_26
Last ObjectModification:
2017_06_05-PM-11_59_20
Theory : reals_2
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