Nuprl Lemma : sum-nat-le-simple
∀[n:ℕ]. ∀[f:ℕn ⟶ ℕ]. ∀x:ℕn. (f[x] ≤ Σ(f[x] | x < n))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
all: ∀x:A. B[x],
guard: {T},
uimplies: b supposing a,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
le: A ≤ B
Lemmas referenced :
sum-nat-le,
sum_wf,
nat_wf,
int_seg_wf,
int_seg_properties,
nat_properties,
decidable__le,
le_wf,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than'_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
lambdaFormation,
independent_isectElimination,
because_Cache,
productElimination,
dependent_functionElimination,
dependent_set_memberEquality,
functionExtensionality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}]. \mforall{}x:\mBbbN{}n. (f[x] \mleq{} \mSigma{}(f[x] | x < n))
Date html generated:
2018_05_21-PM-00_28_24
Last ObjectModification:
2018_05_19-AM-06_59_49
Theory : int_2
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