Nuprl Lemma : rsum-empty
∀[n,m:ℤ]. ∀[x:Top]. Σ{x[i] | n≤i≤m} ~ r0 supposing m < n
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
int-to-real: r(n),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rsum: Σ{x[k] | n≤k≤m},
has-value: (a)↓,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
map: map(f;as),
list_ind: list_ind,
nil: [],
it: ⋅
Lemmas referenced :
top_wf,
less_than_wf,
radd_list_nil_lemma,
map_nil_lemma,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
from-upto-nil,
int-value-type,
value-type-has-value
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
because_Cache,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
sqleReflexivity,
sqequalAxiom,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:Top]. \mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} \msim{} r0 supposing m < n
Date html generated:
2016_05_18-AM-07_46_08
Last ObjectModification:
2016_01_17-AM-02_08_42
Theory : reals
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