Nuprl Lemma : assert-is-power
∀n:ℕ+. ∀x:ℕ. (↑is-power(n;x) ⇐⇒ ∃r:ℕ. (x = r^n ∈ ℤ))
Proof
Definitions occuring in Statement :
is-power: is-power(n;x),
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
is-power: is-power(n;x),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
prop: ℙ,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True,
uiff: uiff(P;Q),
le: A ≤ B
Lemmas referenced :
iroot-property,
iroot_wf,
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
istype-int,
int-value-type,
set_subtype_base,
less_than_wf,
int_subtype_base,
istype-le,
exp_wf2,
nat_properties,
nat_plus_properties,
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-less_than,
iff_weakening_uiff,
assert_wf,
eq_int_wf,
fastexp_wf,
equal-wf-base,
assert_of_eq_int,
istype-assert,
istype-nat,
nat_plus_wf,
subtype_base_sq,
exp-fastexp,
nat_plus_subtype_nat,
squash_wf,
true_wf,
exp-le-iff,
itermAdd_wf,
int_term_value_add_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
sqequalRule,
callbyvalueReduce,
independent_isectElimination,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
productElimination,
independent_pairFormation,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
sqequalBase,
equalitySymmetry,
productIsType,
because_Cache,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
addEquality,
promote_hyp,
equalityTransitivity,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbN{}. (\muparrow{}is-power(n;x) \mLeftarrow{}{}\mRightarrow{} \mexists{}r:\mBbbN{}. (x = r\^{}n))
Date html generated:
2020_05_19-PM-10_03_42
Last ObjectModification:
2020_01_04-PM-08_29_26
Theory : num_thy_1
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