Nuprl Lemma : mu-bound-property+
∀[b:ℕ]. ∀[f:ℕb ⟶ 𝔹]. {(mu(f) ∈ ℕb) ∧ (↑(f mu(f))) ∧ (∀[i:ℕb]. ¬↑(f i) supposing i < mu(f))} supposing ∃n:ℕb. (↑(f n))
Proof
Definitions occuring in Statement :
mu: mu(f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
guard: {T},
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
mu-bound-property,
exists_wf,
int_seg_wf,
assert_wf,
bool_wf,
nat_wf,
mu-bound
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
productElimination,
independent_pairFormation,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality
Latex:
\mforall{}[b:\mBbbN{}]. \mforall{}[f:\mBbbN{}b {}\mrightarrow{} \mBbbB{}].
\{(mu(f) \mmember{} \mBbbN{}b) \mwedge{} (\muparrow{}(f mu(f))) \mwedge{} (\mforall{}[i:\mBbbN{}b]. \mneg{}\muparrow{}(f i) supposing i < mu(f))\} supposing \mexists{}n:\mBbbN{}b. (\muparrow{}(f n))
Date html generated:
2016_05_14-AM-07_30_00
Last ObjectModification:
2015_12_26-PM-01_26_13
Theory : int_2
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