Nuprl Lemma : summand-le-sum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[i:ℕn]. (f[i] ≤ Σ(f[x] | x < n)) supposing ∀x:ℕn. (0 ≤ f[x])
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
all: ∀x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
less_than': less_than'(a;b),
false: False,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
isolate_summand,
le_wf,
le_witness_for_triv,
int_seg_wf,
istype-le,
istype-int,
istype-nat,
sum_wf,
ifthenelse_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
istype-false,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
le_functionality,
le_weakening,
add_functionality_wrt_le,
non_neg_sum
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
applyEquality,
because_Cache,
sqequalRule,
isect_memberEquality_alt,
productElimination,
equalityTransitivity,
independent_isectElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
universeIsType,
natural_numberEquality,
setElimination,
rename,
addEquality,
lambdaEquality_alt,
intEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
independent_pairFormation,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
imageElimination,
dependent_set_memberEquality_alt,
approximateComputation,
int_eqEquality,
Error :memTop,
productIsType,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[i:\mBbbN{}n]. (f[i] \mleq{} \mSigma{}(f[x] | x < n)) supposing \mforall{}x:\mBbbN{}n. (0 \mleq{} f[x])
Date html generated:
2020_05_19-PM-09_41_36
Last ObjectModification:
2020_01_23-PM-00_39_10
Theory : int_2
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