Nuprl Lemma : sum-has-value
∀[n,f:Base]. {(n ∈ ℤ) ∧ (f ∈ ℕn ⟶ ℤ)} supposing (Σ(f[x] | x < n))↓
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
has-value: (a)↓,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
base: Base
Definitions unfolded in proof :
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
has-value: (a)↓,
sum_aux: sum_aux(k;v;i;x.f[x]),
sum: Σ(f[x] | x < k),
or: P ∨ Q,
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
all: ∀x:A. B[x],
lelt: i ≤ j < k,
int_seg: {i..j-},
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
sq_type: SQType(T),
le: A ≤ B
Lemmas referenced :
istype-top,
istype-void,
int_subtype_base,
value-type-has-value,
int-value-type,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
false_wf,
void_wf,
int_seg_wf,
satisfiable-full-omega-tt,
ge_wf,
less_than_wf,
less_than_transitivity1,
less_than_irreflexivity,
le_wf,
set_subtype_base,
nat_wf,
has-value_wf_base,
base_wf
Rules used in proof :
because_Cache,
isect_memberEquality,
hypothesisEquality,
baseClosed,
closedConclusion,
baseApply,
isectElimination,
lemma_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
hypothesis,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
callbyvalueLess,
unionElimination,
applyEquality,
independent_functionElimination,
computeAll,
voidEquality,
voidElimination,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
lambdaFormation,
instantiate,
functionExtensionality,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
extract_by_obid,
imageMemberEquality,
Error :lambdaFormation_alt,
imageElimination,
callbyvalueCallbyvalue,
callbyvalueReduce,
callbyvalueAdd,
Error :dependent_set_memberEquality_alt,
addEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :universeIsType,
cumulativity,
Error :productIsType,
dependent_set_memberEquality
Latex:
\mforall{}[n,f:Base]. \{(n \mmember{} \mBbbZ{}) \mwedge{} (f \mmember{} \mBbbN{}n {}\mrightarrow{} \mBbbZ{})\} supposing (\mSigma{}(f[x] | x < n))\mdownarrow{}
Date html generated:
2019_06_20-PM-01_17_53
Last ObjectModification:
2019_03_28-PM-00_07_18
Theory : int_2
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