Nuprl Lemma : qlog-exists
∀e:{e:ℚ| 0 < e} . ∀q:{q:ℚ| (0 ≤ q) ∧ q < 1} . {n:ℕ+| ((e ≤ 1) ⇒ (e ≤ q ↑ n - 1)) ∧ q ↑ n < e}
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
cand: A c∧ B,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
top: Top,
subtract: n - m,
true: True,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b,
uiff: uiff(P;Q),
pi1: fst(t)
Lemmas referenced :
rationals_wf,
qle_wf,
int-subtype-rationals,
qless_wf,
uniform-comp-nat-induction,
all_wf,
qexp_wf,
set_wf,
nat_plus_wf,
subtract_wf,
nat_plus_properties,
nat_properties,
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,
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,
istype-le,
nat_plus_subtype_nat,
istype-nat,
decidable__qle,
int_seg_wf,
int_seg_properties,
qlog-lemma-ext,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
qmul_wf,
qexp1,
istype-universe,
equal_wf,
qexp2,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
istype-void,
add-commutes,
istype-less_than,
le_wf,
set_subtype_base,
qmul_one_qrng,
qexp-zero,
qless_transitivity_2_qorder,
qless_irreflexivity,
sq_stable_from_decidable,
decidable__qless,
qless_functionality_wrt_implies_1,
qle_weakening_eq_qorder,
decidable__lt,
zero-le-nat,
qmul_preserves_qless,
qexp-positive,
qless_witness,
qexp-add,
subtract-add-cancel,
qmul_preserves_qle,
qexp_preserves_qle,
qle_weakening_lt_qorder,
qexp-one,
qle_reflexivity,
qle_functionality_wrt_implies,
qmul_comm_qrng,
int_formula_prop_eq_lemma,
intformeq_wf,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul,
add-associates,
qdiv_wf,
qmul_zero_qrng,
qmul-qdiv-cancel,
int_term_value_add_lemma,
itermAdd_wf,
qle_witness,
qmul_preserves_qle2,
qmul_com,
exp_zero_q,
qle_complement_qorder,
qlog-bound,
qless_transitivity_1_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
setIsType,
universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
productIsType,
sqequalHypSubstitution,
isectElimination,
thin,
closedConclusion,
natural_numberEquality,
applyEquality,
hypothesisEquality,
because_Cache,
lambdaEquality_alt,
setEquality,
productEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
isect_memberFormation_alt,
isectIsType,
functionIsType,
imageElimination,
instantiate,
cumulativity,
intEquality,
universeEquality,
inhabitedIsType,
isect_memberEquality_alt,
baseClosed,
imageMemberEquality,
equalitySymmetry,
equalityTransitivity,
baseApply,
equalityIsType4,
applyLambdaEquality,
hyp_replacement,
promote_hyp,
minusEquality,
multiplyEquality,
equalityIsType3,
addEquality
Latex:
\mforall{}e:\{e:\mBbbQ{}| 0 < e\} . \mforall{}q:\{q:\mBbbQ{}| (0 \mleq{} q) \mwedge{} q < 1\} . \{n:\mBbbN{}\msupplus{}| ((e \mleq{} 1) {}\mRightarrow{} (e \mleq{} q \muparrow{} n - 1)) \mwedge{} q \muparrow{} n < e\}
Date html generated:
2020_05_20-AM-09_27_05
Last ObjectModification:
2020_01_04-PM-10_31_51
Theory : rationals
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