Nuprl Lemma : mu-bound-property
∀[b:ℕ]. ∀[f:ℕb ⟶ 𝔹]. {(↑(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,
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},
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
sq_stable: SqStable(P),
not: ¬A,
false: False,
squash: ↓T,
all: ∀x:A. B[x],
mu: mu(f),
and: P ∧ Q,
cand: A c∧ B,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b
Lemmas referenced :
lelt_wf,
mu-ge-bound-property,
assert_witness,
sq_stable__not,
sq_stable__uall,
sq_stable__and,
squash_wf,
not_wf,
less_than_wf,
isect_wf,
uall_wf,
nat_wf,
bool_wf,
assert_wf,
int_seg_wf,
exists_wf,
mu-bound
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_functionElimination,
lambdaFormation,
introduction,
dependent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_pairFormation,
dependent_set_memberEquality
Latex:
\mforall{}[b:\mBbbN{}]. \mforall{}[f:\mBbbN{}b {}\mrightarrow{} \mBbbB{}].
\{(\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_29_57
Last ObjectModification:
2016_01_14-PM-09_58_51
Theory : int_2
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