Nuprl Lemma : realexp-nat
∀[x:{x:ℝ| r0 < x} ]. ∀[n:ℕ]. (realexp(x;r(n)) = x^n)
Proof
Definitions occuring in Statement :
realexp: realexp(x;y),
rless: x < y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
eq_int: (i =z j),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
realexp: realexp(x;y),
subtype_rel: A ⊆r B,
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
less_than_wf,
req_witness,
realexp_wf,
rless_wf,
int-to-real_wf,
rnexp_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
set_wf,
real_wf,
false_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rmul_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
btrue_wf,
req_functionality,
req_weakening,
rnexp-req,
expr_wf,
ln_wf,
req_wf,
rlog_wf,
rexp_wf,
rmul-zero-both,
rexp0,
expr-req,
rexp_functionality,
rmul_functionality,
req_inversion,
radd_wf,
rsub_wf,
itermAdd_wf,
req-iff-rsub-is-0,
req_transitivity,
radd_functionality,
rsub-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
uiff_transitivity,
realexp_functionality,
realexp-radd,
rmul-identity1,
rexp-rlog,
ln-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality,
because_Cache,
unionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
setEquality
Latex:
\mforall{}[x:\{x:\mBbbR{}| r0 < x\} ]. \mforall{}[n:\mBbbN{}]. (realexp(x;r(n)) = x\^{}n)
Date html generated:
2017_10_04-PM-10_39_53
Last ObjectModification:
2017_06_06-AM-10_59_01
Theory : reals_2
Home
Index