Nuprl Lemma : sum-partial-nat
∀[n:ℕ]. ∀[f:ℕn ⟶ partial(ℕ)]. (Σ(f[x] | x < n) ∈ partial(ℕ))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
partial: partial(T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
int_seg_wf,
partial_wf,
nat_wf,
sum-unroll,
nat-partial-nat,
false_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
top_wf,
add-wf-partial-nat,
int_seg_subtype,
int_seg_properties,
decidable__lt,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
dependent_set_memberEquality,
because_Cache,
unionElimination,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
applyEquality,
applyLambdaEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} partial(\mBbbN{})]. (\mSigma{}(f[x] | x < n) \mmember{} partial(\mBbbN{}))
Date html generated:
2018_05_21-PM-00_27_55
Last ObjectModification:
2017_11_03-PM-02_26_53
Theory : int_2
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