Nuprl Lemma : ratexp_wf
∀[a:ℤ × ℕ+]. ∀[n:ℕ]. (ratexp(a;n) ∈ {r:ℤ × ℕ+| ratreal(r) = ratreal(a)^n} )
Proof
Definitions occuring in Statement :
ratexp: ratexp(x;n),
ratreal: ratreal(r),
rnexp: x^k1,
req: x = y,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
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,
ratexp: ratexp(x;n),
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
rnexp_zero_lemma,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
req_wf,
ratreal_wf,
int-to-real_wf,
subtract-1-ge-0,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
value-type-has-value,
int-value-type,
subtract_wf,
nat_plus_wf,
rnexp_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
set-value-type,
product-value-type,
istype-nat,
rdiv_wf,
rless-int,
rless_wf,
req-int-fractions2,
subtype_base_sq,
nequal_wf,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
req_functionality,
ratreal-req,
req_weakening,
ratmul_wf,
rmul_wf,
req_transitivity,
rmul_functionality,
rmul_comm,
req_inversion,
rnexp_step
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
independent_pairEquality,
unionElimination,
equalityIstype,
because_Cache,
applyEquality,
baseClosed,
sqequalBase,
int_eqReduceFalseSq,
callbyvalueReduce,
setEquality,
productEquality,
intEquality,
isectIsTypeImplies,
productIsType,
closedConclusion,
inrFormation_alt,
productElimination,
imageMemberEquality,
instantiate,
cumulativity
Latex:
\mforall{}[a:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. \mforall{}[n:\mBbbN{}]. (ratexp(a;n) \mmember{} \{r:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(r) = ratreal(a)\^{}n\} )
Date html generated:
2019_10_30-AM-09_30_09
Last ObjectModification:
2019_01_11-PM-05_13_56
Theory : reals
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