Nuprl Lemma : rsum-single
∀[n:ℤ]. ∀[x:{i:ℤ| i = n ∈ ℤ} ⟶ ℝ]. (Σ{x[i] | n≤i≤n} = x[n])
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rsum: Σ{x[k] | n≤k≤m},
has-value: (a)↓,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
value-type-has-value,
int-value-type,
req_witness,
rsum_wf,
equal-wf-base,
subtype_rel_sets,
lelt_wf,
int_subtype_base,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
real_wf,
from-upto-single,
map_cons_lemma,
map_nil_lemma,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
real-valueall-type,
cons_wf,
nil_wf,
evalall-reduce,
valueall-type-real-list,
radd-list_wf-bag,
list-subtype-bag,
subtype_rel_self,
radd_wf,
radd_list_nil_lemma,
req_wf,
int-to-real_wf,
req_weakening,
req_functionality,
radd-list-cons,
uiff_transitivity,
radd_comm,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
addEquality,
natural_numberEquality,
setElimination,
rename,
lambdaFormation,
productElimination,
applyLambdaEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
independent_functionElimination,
functionEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[x:\{i:\mBbbZ{}| i = n\} {}\mrightarrow{} \mBbbR{}]. (\mSigma{}\{x[i] | n\mleq{}i\mleq{}n\} = x[n])
Date html generated:
2017_10_03-AM-08_58_20
Last ObjectModification:
2017_07_28-AM-07_38_06
Theory : reals
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