Nuprl Lemma : rsum-as-itop
∀[n,m:ℤ]. ∀[x:{n..m-} ⟶ ℝ]. (Π(λx,y. (x + y),r0) n ≤ k < m. x[k] = Σ{x[k] | n≤k≤m - 1})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
itop: Π(op,id) lb ≤ i < ub. E[i]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
nat: ℕ,
ge: i ≥ j ,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
decidable__lt,
req_witness,
itop_wf,
real_wf,
radd_wf,
int-to-real_wf,
int_seg_wf,
rsum_wf,
subtract_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
itermAdd_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
istype-le,
istype-less_than,
nat_properties,
ge_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract-1-ge-0,
istype-nat,
int_subtype_base,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
add-commutes,
zero-add,
zero-mul,
add-zero,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add-subtract-cancel,
radd-zero-both,
req_functionality,
req_weakening,
rsum-single,
subtract-add-cancel,
rsum-split-last,
radd_functionality,
rsum-empty
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
isectElimination,
lambdaEquality_alt,
inhabitedIsType,
universeIsType,
natural_numberEquality,
sqequalRule,
applyEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
productElimination,
imageElimination,
independent_pairFormation,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
because_Cache,
addEquality,
functionIsType,
isectIsTypeImplies,
lambdaFormation_alt,
intWeakElimination,
functionIsTypeImplies,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
intEquality,
multiplyEquality,
closedConclusion,
setIsType
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (\mPi{}(\mlambda{}x,y. (x + y),r0) n \mleq{} k < m. x[k] = \mSigma{}\{x[k] | n\mleq{}k\mleq{}m - 1\})
Date html generated:
2019_10_29-AM-10_19_40
Last ObjectModification:
2019_09_19-AM-11_34_56
Theory : reals
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