Nuprl Lemma : qexp-convex3
∀a,b:ℚ. ((((0 ≤ a) ∧ (0 ≤ b)) ∨ ((a ≤ 0) ∧ (b ≤ 0))) ⇒ (∀n:ℕ+. (|a - b| ↑ n ≤ |a ↑ n - b ↑ n|)))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qabs: |r|,
qle: r ≤ s,
qsub: r - s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
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,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
sq_type: SQType(T),
false: False
Lemmas referenced :
qexp-convex2,
qmul_wf,
qmul_reverses_qle,
qle_wf,
nat_plus_wf,
or_wf,
int-subtype-rationals,
rationals_wf,
squash_wf,
true_wf,
qmul_zero_qrng,
qinv_inv_q,
iff_weakening_equal,
qexp_wf,
nat_wf,
equal_wf,
qabs-difference-symmetry,
qabs_wf,
qsub_wf,
nat_plus_subtype_nat,
qadd_wf,
qadd_comm_q,
isEven_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
qabs-qminus,
qexp-qminus,
qmul_over_plus_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
productElimination,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
isectElimination,
minusEquality,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
hyp_replacement,
equalitySymmetry,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
equalityTransitivity,
universeEquality,
setElimination,
rename,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination
Latex:
\mforall{}a,b:\mBbbQ{}. ((((0 \mleq{} a) \mwedge{} (0 \mleq{} b)) \mvee{} ((a \mleq{} 0) \mwedge{} (b \mleq{} 0))) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (|a - b| \muparrow{} n \mleq{} |a \muparrow{} n - b \muparrow{} n|)))
Date html generated:
2018_05_22-AM-00_01_45
Last ObjectModification:
2017_07_26-PM-06_50_23
Theory : rationals
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