Nuprl Lemma : qexp-sign
∀n:ℕ. ∀r:ℚ.
((0 < r ↑ n ⇐⇒ (n = 0 ∈ ℤ) ∨ 0 < r ∨ (r < 0 ∧ (↑isEven(n))))
∧ (r ↑ n < 0 ⇐⇒ r < 0 ∧ (↑isOdd(n)))
∧ (r ↑ n = 0 ∈ ℚ ⇐⇒ (r = 0 ∈ ℚ) ∧ (¬(n = 0 ∈ ℤ))))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qless: r < s,
rationals: ℚ,
isEven: isEven(n),
isOdd: isOdd(n),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
isOdd: isOdd(n),
isEven: isEven(n),
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
modulus: a mod n,
absval: |i|,
cand: A c∧ B,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
nat_plus: ℕ+
Lemmas referenced :
rationals_wf,
qless_wf,
qexp_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
int_subtype_base,
assert_wf,
isEven_wf,
isOdd_wf,
less_than_wf,
primrec-wf2,
all_wf,
iff_wf,
or_wf,
equal-wf-base,
equal-wf-T-base,
not_wf,
nat_wf,
exp_zero_q,
true_wf,
int-equal-in-rationals,
subtype_base_sq,
assert-qeq,
intformeq_wf,
int_formula_prop_eq_lemma,
int-subtype-rationals,
exp_unroll_q,
odd-implies,
subtract-add-cancel,
even-implies,
qmul-positive2,
qmul-negative,
qmul-zero,
qless_transitivity,
qless_irreflexivity,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
rename,
setElimination,
because_Cache,
sqequalRule,
functionIsType,
productIsType,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
applyEquality,
hypothesisEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
unionIsType,
equalityIsType4,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
productElimination,
equalityIsType3,
setIsType,
productEquality,
callbyvalueReduce,
sqleReflexivity,
inlFormation_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
inrFormation_alt,
promote_hyp
Latex:
\mforall{}n:\mBbbN{}. \mforall{}r:\mBbbQ{}.
((0 < r \muparrow{} n \mLeftarrow{}{}\mRightarrow{} (n = 0) \mvee{} 0 < r \mvee{} (r < 0 \mwedge{} (\muparrow{}isEven(n))))
\mwedge{} (r \muparrow{} n < 0 \mLeftarrow{}{}\mRightarrow{} r < 0 \mwedge{} (\muparrow{}isOdd(n)))
\mwedge{} (r \muparrow{} n = 0 \mLeftarrow{}{}\mRightarrow{} (r = 0) \mwedge{} (\mneg{}(n = 0))))
Date html generated:
2019_10_16-PM-00_32_02
Last ObjectModification:
2018_10_10-PM-01_06_56
Theory : rationals
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