Nuprl Lemma : mu-unique
∀[f:ℕ ⟶ 𝔹]. ∀[x:ℕ]. mu(f) = x ∈ ℤ supposing (↑(f x)) ∧ (∀y:ℕx. (¬↑(f y)))
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],
not: ¬A,
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,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
less_than: a < b
Lemmas referenced :
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
not_wf,
all_wf,
assert_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
nat_wf,
bool_wf,
subtype_rel_dep_function,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
mu-bound-unique
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
applyEquality,
because_Cache,
lambdaFormation,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
independent_functionElimination
Latex:
\mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:\mBbbN{}]. mu(f) = x supposing (\muparrow{}(f x)) \mwedge{} (\mforall{}y:\mBbbN{}x. (\mneg{}\muparrow{}(f y)))
Date html generated:
2016_05_14-AM-07_30_08
Last ObjectModification:
2016_01_14-PM-09_58_12
Theory : int_2
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