Nuprl Lemma : r-list-sum_functionality
∀[L1,L2:ℝ List]. r-list-sum(L1) = r-list-sum(L2) supposing (||L1|| = ||L2|| ∈ ℤ) ∧ (∀i:ℕ||L1||. (L1[i] = L2[i]))
Proof
Definitions occuring in Statement :
r-list-sum: r-list-sum(L),
req: x = y,
real: ℝ,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
le: A ≤ B,
decidable: Dec(P),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
r-list-sum: r-list-sum(L),
cand: A c∧ B,
true: True,
subtract: n - m,
iff: P ⇐⇒ Q
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,
istype-less_than,
req_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
real_wf,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
req_weakening,
r-list-sum_wf,
nil_wf,
int_seg_wf,
int_seg_properties,
product_subtype_list,
colength-cons-not-zero,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
length_of_cons_lemma,
le_weakening2,
length_wf,
non_neg_length,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
cons_wf,
istype-nat,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
list_wf,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
add-is-int-iff,
false_wf,
reduce_cons_lemma,
radd_functionality,
req_wf,
select_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
less_than_wf,
istype-universe,
add-subtract-cancel,
true_wf,
select_cons_tl,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
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,
applyLambdaEquality,
productElimination,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
because_Cache,
baseClosed,
productIsType,
equalityIstype,
sqequalBase,
functionIsType,
promote_hyp,
hypothesis_subsumption,
instantiate,
addEquality,
applyEquality,
dependent_set_memberEquality_alt,
imageElimination,
baseApply,
closedConclusion,
intEquality,
pointwiseFunctionality,
hyp_replacement,
productEquality,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[L1,L2:\mBbbR{} List].
r-list-sum(L1) = r-list-sum(L2) supposing (||L1|| = ||L2||) \mwedge{} (\mforall{}i:\mBbbN{}||L1||. (L1[i] = L2[i]))
Date html generated:
2019_10_29-AM-10_20_23
Last ObjectModification:
2019_09_18-PM-05_14_17
Theory : reals
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