Nuprl Lemma : mu-property
∀[f:ℕ ⟶ 𝔹]. {(↑(f mu(f))) ∧ (∀[i:ℕ]. ¬↑(f i) supposing i < mu(f))} supposing ∃n:ℕ. (↑(f n))
Proof
Definitions occuring in Statement :
mu: mu(f),
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,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
int_upper: {i...},
nat: ℕ,
all: ∀x:A. B[x],
mu: mu(f),
exists: ∃x:A. B[x],
prop: ℙ,
guard: {T},
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B
Lemmas referenced :
mu-ge-property,
subtype_rel_dep_function,
nat_wf,
bool_wf,
int_upper_wf,
subtype_rel_self,
assert_wf,
less_than_wf,
mu_wf,
assert_witness,
exists_wf,
lelt_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
natural_numberEquality,
isect_memberFormation,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
because_Cache,
lambdaFormation,
introduction,
productElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
setElimination,
rename,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
functionEquality,
dependent_set_memberEquality
Latex:
\mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. \{(\muparrow{}(f mu(f))) \mwedge{} (\mforall{}[i:\mBbbN{}]. \mneg{}\muparrow{}(f i) supposing i < mu(f))\} supposing \mexists{}n:\mBbbN{}. (\muparrow{}(f n))
Date html generated:
2016_05_14-AM-07_29_45
Last ObjectModification:
2015_12_26-PM-01_26_31
Theory : int_2
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