Nuprl Lemma : qexp-qminus
∀n:ℕ. ∀a:ℚ. (-(a) ↑ n = if isEven(n) then a ↑ n else -(a ↑ n) fi ∈ ℚ)
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qmul: r * s,
rationals: ℚ,
isEven: isEven(n),
nat: ℕ,
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x],
minus: -n,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
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,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
assert: ↑b,
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
btrue: tt,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb
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,
rationals_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
qmul_wf,
isEven_wf,
equal_wf,
ite_rw_true,
iff_weakening_equal,
bool_wf,
eqtt_to_assert,
qexp_wf,
le_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int-subtype-rationals,
even-implies,
qmul_assoc,
qmul_ac_1_qrng,
qinv_inv_q,
odd-or-even,
assert_of_bor,
isOdd_wf,
odd-implies,
exp_zero_q,
ifthenelse_wf,
squash_wf,
true_wf,
exp_unroll_q,
qmul_over_minus_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
unionElimination,
because_Cache,
minusEquality,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_set_memberEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
universeEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbQ{}. (-(a) \muparrow{} n = if isEven(n) then a \muparrow{} n else -(a \muparrow{} n) fi )
Date html generated:
2018_05_22-AM-00_01_39
Last ObjectModification:
2017_07_26-PM-06_50_18
Theory : rationals
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