Nuprl Lemma : qexp-exp
∀[n:ℕ]. ∀[x:ℤ]. (x ↑ n = x^n ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
rationals: ℚ,
exp: i^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
exp: i^n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
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,
equal_wf,
qexp-zero,
iff_weakening_equal,
exp0_lemma,
int-subtype-rationals,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
primrec0_lemma,
primrec-unroll,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
rationals_wf,
exp_unroll_q,
exp_wf2,
le_wf,
qmul-mul,
int-equal-in-rationals,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
qmul_wf
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,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
applyEquality,
imageElimination,
because_Cache,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
equalityElimination,
baseApply,
closedConclusion,
impliesFunctionality,
universeEquality,
dependent_set_memberEquality,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbZ{}]. (x \muparrow{} n = x\^{}n)
Date html generated:
2018_05_22-AM-00_00_12
Last ObjectModification:
2017_07_26-PM-06_49_13
Theory : rationals
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