Nuprl Lemma : sum-nat-less
∀[n:ℕ]. ∀[f:ℕn ⟶ ℕ]. ∀[b:ℤ]. {∀x:ℕn. (f[x] ≤ (b - 1))} supposing Σ(f[x] | x < n) < b
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
int_seg_wf,
less_than'_wf,
subtract_wf,
less_than_wf,
sum_wf,
nat_wf,
isolate_summand,
sum-nat,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
int_seg_properties,
decidable__le,
lelt_wf,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformeq_wf,
itermAdd_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intEquality,
functionEquality,
voidElimination,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
voidEquality,
computeAll
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}]. \mforall{}[b:\mBbbZ{}]. \{\mforall{}x:\mBbbN{}n. (f[x] \mleq{} (b - 1))\} supposing \mSigma{}(f[x] | x < n) < b
Date html generated:
2017_04_14-AM-09_20_53
Last ObjectModification:
2017_02_27-PM-03_56_44
Theory : int_2
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