Nuprl Lemma : qsum-subsequence-qle
∀[f:ℕ ⟶ ℚ]
∀[k:ℕ]. ∀[g:ℕk + 1 ⟶ ℕ]. Σ0 ≤ i < k. f[g i] ≤ Σ0 ≤ i < g k. f[i] supposing ∀n:ℕk + 1. ∀i:ℕn. g i < g n
supposing ∀n:ℕ. (0 ≤ f[n])
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
rationals: ℚ,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
le: A ≤ B,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
qle_witness,
qsum_wf,
int_seg_wf,
le_wf,
int_seg_subtype_nat,
istype-false,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
subtract-1-ge-0,
subtract-add-cancel,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
nat_wf,
qle_wf,
int-subtype-rationals,
rationals_wf,
qsum-non-neg,
all_wf,
false_wf,
lelt_wf,
satisfiable-full-omega-tt,
squash_wf,
true_wf,
sum_unroll_base_q,
iff_weakening_equal,
equal_wf,
istype-universe,
sum_unroll_hi_q,
qadd_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__le,
subtype_rel_self,
le_weakening2,
sum_split_q,
subtype_rel_function,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
qle_functionality_wrt_implies,
qadd_functionality_wrt_qle,
qle_weakening_eq_qorder,
summand-qle-sum,
qle_reflexivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
applyEquality,
dependent_set_memberEquality_alt,
productElimination,
imageMemberEquality,
baseClosed,
productIsType,
functionIsType,
addEquality,
unionElimination,
functionEquality,
lambdaFormation,
computeAll,
voidEquality,
isect_memberEquality,
intEquality,
dependent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
lambdaEquality,
isect_memberFormation,
imageElimination,
universeEquality,
instantiate,
hyp_replacement,
applyLambdaEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbQ{}]
\mforall{}[k:\mBbbN{}]. \mforall{}[g:\mBbbN{}k + 1 {}\mrightarrow{} \mBbbN{}].
\mSigma{}0 \mleq{} i < k. f[g i] \mleq{} \mSigma{}0 \mleq{} i < g k. f[i] supposing \mforall{}n:\mBbbN{}k + 1. \mforall{}i:\mBbbN{}n. g i < g n
supposing \mforall{}n:\mBbbN{}. (0 \mleq{} f[n])
Date html generated:
2019_10_16-PM-00_33_03
Last ObjectModification:
2018_10_10-AM-11_05_30
Theory : rationals
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