Nuprl Lemma : qexp-convex
∀a,b:ℚ. ((0 ≤ b) ⇒ (b ≤ a) ⇒ (∀n:ℕ+. (a - b ↑ n ≤ (a ↑ n - b ↑ n))))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qsub: r - s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
le: A ≤ B,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
qsub: r - s,
qge: a ≥ b,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
qadd_functionality_wrt_qle,
qmul_comm_qrng,
qle_witness,
qmul_preserves_qle2,
qexp_preserves_qle,
qmul_assoc_qrng,
qmul_ident,
q_distrib,
qadd_inv_assoc_q,
mon_assoc_q,
qinv_inv_q,
mon_ident_q,
qinverse_q,
qadd_ac_1_q,
qadd_comm_q,
qle_weakening_eq_qorder,
qmul_functionality_wrt_qle,
qle_functionality_wrt_implies,
qmul_one_qrng,
qmul_over_minus_qrng,
qmul_over_plus_qrng,
exp_zero_q,
iff_weakening_equal,
exp_unroll_q,
true_wf,
squash_wf,
qadd_preserves_qle,
qexp-nonneg,
add-subtract-cancel,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
false_wf,
decidable__lt,
qle_reflexivity,
qadd_wf,
qmul_wf,
less_than_wf,
rationals_wf,
int-subtype-rationals,
nat_plus_wf,
nat_plus_subtype_nat,
primrec-wf-nat-plus,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
qsub_wf,
qexp_wf,
qle_wf,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
dependent_set_memberEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
introduction,
imageMemberEquality,
baseClosed,
minusEquality,
addEquality,
productElimination,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
isect_memberFormation
Latex:
\mforall{}a,b:\mBbbQ{}. ((0 \mleq{} b) {}\mRightarrow{} (b \mleq{} a) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. (a - b \muparrow{} n \mleq{} (a \muparrow{} n - b \muparrow{} n))))
Date html generated:
2016_05_15-PM-11_10_36
Last ObjectModification:
2016_01_16-PM-09_26_37
Theory : rationals
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