Nuprl Lemma : mu-bound-unique
∀[b:ℕ]. ∀[f:ℕb ⟶ 𝔹]. ∀[x:ℕb]. mu(f) = x ∈ ℤ supposing (↑(f x)) ∧ (∀y:ℕb. ((↑(f y)) ⇒ (y = x ∈ ℤ)))
Proof
Definitions occuring in Statement :
mu: mu(f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
nat: ℕ,
guard: {T},
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
int_seg: {i..j-},
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
assert_wf,
int_seg_wf,
mu-bound-property,
all_wf,
equal_wf,
bool_wf,
nat_wf,
mu-bound
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
productEquality,
sqequalRule,
lambdaEquality,
functionEquality,
intEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[b:\mBbbN{}]. \mforall{}[f:\mBbbN{}b {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:\mBbbN{}b]. mu(f) = x supposing (\muparrow{}(f x)) \mwedge{} (\mforall{}y:\mBbbN{}b. ((\muparrow{}(f y)) {}\mRightarrow{} (y = x)))
Date html generated:
2017_04_14-AM-09_19_01
Last ObjectModification:
2017_02_27-PM-03_55_46
Theory : int_2
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